Delhi trip , along with Agra, Udaipur and Jaipur.
Excellent trip in Rajashtan thanks also to our driver Rajender! He took us to every single place to see! Very courteous, helpfull! Answering all our questions! The car always kept clean and tidy! Thank you Rajender!!
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A bill to proclaim October as Hindu Heritage Month annually in Canada's Ontario province has been introduced in the provincial assembly in Toronto.
When the ritual of kumari puja was started by Ramakrishna Paramhansa, it was meant to reinforce the essential divinity of the girl child. It is critical that this association is reimagined and refurbished in the current context to make it relevant.
Durga Saptashati is a composition that weaves in many basic tenets of life and spiritual thought in its narrative. This piece of poetry with its engaging narration and vivid imagery could suggest these tenets to those for whom the arcane and pithy language of the Upanishads was inaccessible. Some of the takeaways relevant for our times are discussed here.
'Postcards from My Periods' is their latest project through which they aim at bringing forth stories of young girls and their first period.
Navratri is a 10 day festival in which nine different forms of goddess Durga are worshipped on nine consecutive nights and the tenth day is referred as Dassera.
With the Navratri festival starting on Tuesday, the Vishwa Hindu Parishad (VHP) has started putting up banners at the entrances of select garba venues across the state, asking non-Hindus not to participate in the dance programmes as it is restricted "only for Hindus".


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Previous message: E. Shaun Russell: "Re: A plea for restraint"
>>they had murdered another person? Denial? That wasn't the "real"
>So you're basically of the "let's stake Angel" camp.


Our 1/4 Wheel weighs in at approximately 2.5lbs - 3lbs. If scraping the Happy Accident you can expect to serve ~50 people. If you are serving it sliced (with wine and crackers for example) you can expect to serve ~ 30 people.
Note: Due to variation in our hand-made process the weight of the wheel you receive will vary slightly.


In Starlink: Battle for Atlas, you control one of several pilots in a very unique starship. Your ship will have a corresponding real life representation that you can alter on the go. Swap out lasers for missiles to bring down that thick skinned boss, then switch back to lasers to go blast your way through an asteroid field. The choices are yours, and they're endless.
Exclusive to the Nintendo Switch version, you can even fly as legendary starship pilot Starfox!
Description: In Starlink: Battle for Atlas, you control one of several pilots in a very unique starship.


Be it an ethnic wear, or a western attire, these light and colorful blue shaded Punjbai Juttis would give you a lovely appearance. Wear it for its simplicity and be class apart among the gathering. Our Products are now exclusively available in Kimaya, Amaara & Kayra.


You'll often find her name mentioned in our "Catch Your Colleague Doing Something Great" section or sharing her brilliant Irish sayings. So it comes as no surprise that our very own Denise Trimble has been honored at the Irish Echo's Golden Bridges Conference. Check her out pictured above with Mayor Marty Walsh!
Golden Bridges spotlights community, business, education, arts and culture initiatives in northwest Ireland and promotes partnerships with their peers in Boston and Massachusetts. Denise is among those being honored for forging transatlantic links that build our communities and countries.


text:
Louisiana-native tops the world's best kayak anglers for a Made in USA crown
Posted by wpalert | Dec 19, 2016 | Product Review | 0 |
Lessard Wins Kayak Championship for USA
Park Falls, WI (December 19, 2016) — The most international title in fishing is headed home to America, after Steve Lessard took home his second Hobie Fishing Worlds in his home state of Louisiana. Anglers from five continents competed in a fierce field of 49 for a shot at kayak fishing's crowning achievement.
"It's intimidating," says Lessard. "You've got these guys from around the world coming in with beautiful, hand-painted lures. They're fishing with custom made rods and $500 reels, and they're all super-skilled. A lot of them are multi-species masters, with hundreds of species under their belt.
"Transitioning to redfish in Louisiana is not that hard for them."
To level the playing field, Lessard loaded his kayak with a one-two punch from St. Croix Rods.
The tournament took place over three days, with each angler measuring one red and one trout per day. And with a loaded field, Lessard relied on a variety of strategies and techniques to get the job done, from Carolina-rigging to topwater popping corks. Lessard's weapons of choice? A medium-action 7'6" St. Croix Avid Inshore to reach out and hit big reds; a medium-light St. Croix Legend Elite for cracking finicky trout on topwater.
"The Legend Elite is perfectly suited for a small topwater lure," Lessard says. "The Legend Elite really helps me keep those trout hooked-up. Redfish have a tendency to somehow miss the hook or come off once they get hooked on a treble, but the Legend Elite keeps them on and buttoned up. My first fish of the tournament, on Day One, came off that rod."
But Lessard wasn't only fighting finicky trout and a bevy of international competitors. At this world championship, he was fighting vicious weather from a ripping north wind and rapidly flowing currents that took the marshes out to the Gulf of Mexico at 5 mph.
"The wind was horrible on Day Two," he adds. "I took a chance and gambled on going south, even though not many people were doing that because the current made it nearly impossible to come back north. And, I could only get so close to the big reds when they're on a flat that's two feet deep. Otherwise, you start pushing mud. So, I was literally making 100 foot casts to reach the fish. That long rod, the Avid Inshore, really lets you cast the bait out there. And I can set the hook all the way out using a popper on braided line with no problem."
Masterminded for freshwater fishing, cherished by secret society of saltwater anglers, the Legend Elite lives as a culmination of St. Croix's finest materials and technologies.
Day Three saw Lessard in pole position, 10-inches ahead of Australia's 2013 Hobie World's Champion Richard Somerton. But with a 15 degree drop in water temperature, the stage was set for a tough day on the water to bring home the gold.
"The pressure was really on then," Lessard tells. "I had everybody in the state of Louisiana rooting for me. I had radio shows calling me with sportsmen listening in all over the state. Richard Somerton and I were going head to head. I was trying like heck to defend our home water for the USA, and we were both trying to become the first to win two world championships."
Following an early morning finesse pattern that landed his keeper trout on an extremely slow-rolling swim bait, Lessard ventured out for the kicker fish—a bull red—to put away the victory.
"I saw one on the flats, and I made a long, 80 foot cast to reach him. Then, I started feeding the line out, letting the wind carry my cork into position. I gave it subtle, slight pops and when I got close to where I saw the fish, the cork went under. I guarantee you that I had 120 feet of line out on that 7'6" rod when he hit. And after a battle that took us from a flat to a ditch to the main canal, I was able to pull him in—a 27-inch red. At that point, I almost lost it. I knew I had 46-inches for the day and that Richard would need 56-inches to beat me."
Somerton wouldn't make it. The revered Aussie tallied in 16-inches behind the American, and Lessard took home his second world title.
When asked to compare his Louisiana win with his 2014 title in Holland, Lessard brought it all back home. "In Holland, I imagined the pike as redfish," he says. "I would imagine myself in a Louisiana Marsh…and the situation worked out really well for me. Back then, I actually borrowed a three-piece St. Croix travel rod to help me win it."
Mr. Lessard, we're imaging a three-pete, St. Croix Rods front and center. We're proud that you're using the "Best Rods on Earth"…
Somewhere between friendly and fierce, Lessard exchanges a handshake with second place finisher, Australian Richard Somerton, another passionate St. Croix user.
PreviousBass Edge's The Edge – Episode 245 – Cliff Crochet
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Kayak Evolution: The 'Ultimate Fishing Machine' by Old Town

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Donated by Chef Warren's - a Southern Pines Bistro!
Finest Quality Available. Everything Fresh, Everyday.
Visit chefwarrensbistro.com for menus and reservations. Like them on Facebook and follow on Twitter to keep up with events!


BERLIN — German Chancellor Angela Merkel told Russian President Vladimir Putin on the phone on Saturday that a four-way summit to discuss the situation in eastern Ukraine would not take place until there was real progress on the Minsk peace plan.
However the chancellor told Putin that a summit by leaders from France, Germany, Russia and Ukraine in Kazakhstan's capital Astana or another city could not be confirmed at this stage, he added.
Merkel's spokesman said the chancellor told Putin that all sides needed to make their contribution to implement the peace plan. "That includes that Russia uses its influence on the separatists in order to reach consensual solutions," he said.
More than 4,700 people have been killed in fighting between Kiev's forces and pro-Russian rebels in eastern Ukraine since last April. The conflict has provoked the worst crisis in relations between Russia and the West since the Cold War.


text:
Q: Fine points about the nature of bias current in oamps and experiments In order to build a sensitive application, I am struggling to understand the behavior of the bias (and offset) current in oamps.
I always thought unconsciously that the input terminals of the oamp act more or less like a current source, and that the bias+offset current is the current sunk.
In fact, this may well be what LTspice thinks because the results of simulating the schematic below is a constant current through the cap (until Vcc+ is reached and then the current falls to 0).
The schematic below is a method proposed in the Art of Electronics to measure the bias current of very low bias current oamps. They says that this "worked well" for them. According to the book, the bias current is simply proportional to the voltage increasing rate by unit of time at the output terminal, which is easy to understand and seems also to enforce the idea that the oamp sinks a constant current.

Now, is this true? (question 1)
I've built the previous schematic and tested many different oamps. It appears that the truth is seemingly far from being that simple:
First, the voltage increasing rate at the output terminal is never linear, but seems to follow an exponential decreasing rate (like charging a cap at a constant voltage).
In one oamp (seemingly an imitation of the TLC662), the output voltage stabilizes about 1 V below Vcc+.
In the other oamps (OPA627,  TLC2272 and other), the output voltage stabilizes somewhere between 0 and Vcc+, and this threshold depends upon Vcc+; for example, for the OPA627, at 8V, the threshold is about 1.5V. At 20V, it is about 3V.
So, I looked at the schematic of the OPA6627 in the datasheet, I found that the input terminals are the gates of Jfets. This appeals the following 2d question: How is the gate of a jfet supposed to sink any current, except, perhaps, a few femtoamps due to leakage through the dielectric insulators, and disregarding the terminal capacitance loading ?
I would be happy to get thorough explanations of the oamp behavior described here.

A: The LM358 inputs are the bases of PNP transistors, so they source current related to the bias current of the front end. This will be a more-or-less constant current within the common mode range of the amplifier. If you go outside the CM range, the transistor may be biased off and won't source as much base current. This is generally true of bipolar-input op-amps, though some may sink current rather than source it (eg. everyone's favorite, the LM741). Here is the expected behavior of the input:

as you can see it's close to -10nA (sourcing) near 0V common-mode voltage, with only a +/-25% variation for voltages between -14V and +12V (with +/-18V supplies).
MOSFET-input amplifiers leak very little current across the insulator- what leakage there is will mostly be related to the protection network on the chip (usually not well documented). The TLC2272 schematic shows a differential pair of MOSFETs but there is a protection network on each input pin, and the single MOSFETs are probably each an array of individual transistors.
JFETs have an isolation junction so that will leak, but often the current is rather small.

A:
the oamp sinks a constant current. Now, is this true? (question 1)

Depends on the op amp.

First, the voltage increasing rate at the output terminal is never
  linear, but seems to follow an exponential decreasing rate (like
  charging a cap at a constant voltage).

The LM358 sources almost constant current up to ~1.3 V below Vcc (the upper common mode input voltage limit), producing a linear voltage ramp.

In one oamp (seemingly an imitation of the TLC662), the output voltage
  stabilizes about 1 V below Vcc+. In the other oamps (OPA627, TLC2272
  and other), the output voltage stabilizes somewhere between 0 and
  Vcc+, and this threshold depends upon Vcc+; for example, for the
  OPA627, at 8V, the threshold is about 1.5V. At 20V, it is about 3V.

Depending on the configuration of the input stage, bias current variation with common mode input can be quite non-linear. 'Rail to Rail' bipolar input op amps have complementary transistors which cause the bias current to jump from negative to positive as input voltage crosses the mid-point.

How is the gate of a jfet supposed to sink any current, except,
  perhaps, a few femtoamps due to leakage through the dielectric
  insulators, and disregarding the terminal capacitance loading ?

There is only leakage current in a FET Gate, but in the OP627 it can be up to 10 pA at 25 °C and 50 nA over the operating temperature range. This is not reliable. Some bipolar input op amps have lower and more predictable bias current at high temperature!

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Landscape Photography E Books from Alex Hare.
Here you'll find a collection of highly readable, helpful and informative e-books covering specific aspects of photography, from camera techniques to location guides.
At the moment I have my Guide to Night & Low Light Photography available. This is my first e-book and others will follow covering many aspects of creative landscape photography.
This is an expanding section with more publications to follow.


text:
In regards to global warming, Sandra Ibrahim wrote in the Sept. 11 Sentinel ("Letter: Don't listen to climate delusion") that "people need to think for themselves and calm down."
It might be helpful if the letter writer would read the current September 2019 National Geographic magazine, which has over 70 pages of articles, graphs and pictures titled "The Arctic is Heating Up." The magazine has been a highly respected publication for decades and has many examples of what is happening in the Arctic climate, which has seemingly gone off the rails this summer. Most of Alaska had the hottest temperatures ever recorded in July. There is no longer sea ice in Alaskan waters and ships for the first time in history can go through the Arctic Ocean all the way to the Atlantic.
Time magazine's current issue dated Sept. 25 has dedicated 100 pages called "Special Climate Issue" that deals with global issues of our warming planet and what will need to be done and what future generations are going to face.
Hopefully, Ibrahim will go to the library or purchase these publications so she can become better informed about global warming.
President Trump, two years ago decided to ignore climate scientists and go with his "gut" decision and pull the U.S. out of the Paris Climate Accord, which was affirmed by 195 other nations. He, along with his talk show cronies, Rush Limbaugh and Sean Hannity continue to mock and insult anyone who believes in global warming. Over 75 percent of Americans are concerned by global warming according to most polls, especially college students.
Planet Earth has become extremely fragile in recent decades with these global warming issues and booming world population growth. We have been instructed to be good stewards of the Earth by our creator. It's time to act so our grandchildren and their children will look back and thank us for trying to save the Earth.
Wally Rooks
Park Township


Super easy to throw together on a weeknight with just a few common ingredients, this is comfort food fusion at its finest. This dish keeps that essential ricotta cheesiness in tact, (because, let's be honest, that's what we all want in a lasagna), but swaps out the marinara for your favorite chunky salsa. And the no-bake noodles aren't simply for shortening your prep time – they also soak up extra moisture from the salsa that might otherwise water it down. We're sure you'll find, just like we did, that this is the perfect thing to bring along to that next potluck, but be forewarned – it disappears fast!
Preheat oven to 350º F and lightly grease a 9x13 baking dish.
Brown beef in a large skillet and drain off excess oil, then stir in taco seasoning and set aside.
In a medium bowl, mix egg with ricotta cheese until well combined. Set aside.
Place 4 lasagna noodles in bottom of baking dish. Spread 1/3 of ricotta over noodles, then evenly spoon 1/3 of ground beef mixture over ricotta. Evenly spread 1/3 of salsa over beef, then layer 1/3 of cheddar cheese over beef. Repeat steps, finishing with cheddar cheese.
Cover with foil and bake for 30-40 minutes, or until lasagna is heated through. Let cool 5-10 minutes, top with sour cream and fresh cilantro, (optional), serve and enjoy!


What a lovely, lovely book!
I was looking forward to reading "Chuck the Rooster Loses His Voice", as a markedly light-hearted change to the usual fare I'm offered as a reviewer, and I have to admit I loved every minute of it. This is definitely a book I would buy for my own toddler, without hesitation - a great little 10-minute children's story, told in verse, about a group of farmyard friends who come together to devise an exciting way to help out the rooster when he falls ill. Perfect for bedtime reading by Mum or Dad; the illustration is beautiful - fun, colourful and vivid - and there are some great characters. And importantly, like all good children's stories, it features, at its core, a warm message of friendship, teamwork, responsibility and difference, all told with a touch of pop-culture thrown in. The creators of this book have set out to achieve all of these things and pulled it off perfectly.
If I am to mention one flaw (and I do so hate to pick any fault with "Chuck the Rooster..."), it is that, while the language is certainly age-appropriate, perhaps the odd word is not one I would have chosen, and could be improved here and there. At times, also, the verse doesn't flow, and its rhyming is slightly off. This may be primarily due to the first language of the author; given the nature of this book and its intended reading, I'm not sure this would matter too much. Still, as a perfectionist, I would have loved to see flawless verse, which is no less than this book deserves. With the exception of these small details, "Chuck the Rooster..." is near-perfect, and I loved it. If you have toddler-age kids, I highly recommend reading this book to them - they'll love it.


text:
Q: Simultaneous pressing of keys in safari using webdriver? I'm trying to test a keyboard shortcut on mac safari using webdriver (c# V2.39.0), the shortcut involves pressing a key whilst another key is pressed.
This is easy to do using the actions api but this is not supported for safari. Is there any other way I can test this? I've had a look at the Javascrip executor but I didn't get very far.
Thanks


Our commitment towards providing utmost satisfaction to the clients enables us to manufacture, export and supply a superior quality range of Cleaning Kitchen Sponges. These sponges are used for washing oily and greasy utensils in both commercial as well as non-commercial kitchens. Our offered sponges are manufactured at our advanced production unit using the best quality materials that are sourced from the authentic vendors of the market. In addition to this, our offered Cleaning Kitchen Sponges are offered at an affordable prices to our prestigious clients.


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Exciting news – Congress passed a new Lands Act last week, authorizing three new national park units and moving our…
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text:
\section{Introduction}
\label{sec:introduction}
In this paper, we study disjoint paths routing problems.
In this setting, we are given an undirected graph $G$ and a collection of source-destination pairs $\mathcal M = \{(s_1, t_1), \hdots, (s_k, t_k)\}$.
The goal is to select a maximum-sized subset $\mathcal M' \subseteq \mathcal M$ of the pairs that can be \emph{routed}, where a routing of $\mathcal M'$ is a collection $\mathcal P$ of paths such that, for each pair $(s_i, t_i) \in \mathcal M'$, there is a path in $\mathcal P$ connecting $s_i$ to~$t_i$.
In the {\sc Maximum Edge Disjoint Paths (MaxEDP)} problem, a routing~$\mathcal P$ is \emph{feasible} if its paths are pairwise edge-disjoint, and in the {\sc Maximum Node Disjoint Paths (MaxNDP)} problem the paths in $\mathcal P$ must be pairwise vertex-disjoint.

Disjoint paths problems are fundamental problems with a long history and significant connections to optimization and structural graph theory.
The decision version of {\sc MaxEDP}/{\sc MaxNPD} asks whether all of the pairs can be routed.
Karp~\cite{Karp1975} showed that, when the number of pairs is part of the input, the decision problem is $\mathsf{NP}$-complete.
In undirected graphs, {\sc MaxEDP} and {\sc MaxNDP} are solvable in polynomial time when the number of pairs is a fixed constant; this is a very deep result of Robertson and Seymour~\cite{RobertsonSeymour1995} that builds on several fundamental results in structural graph theory from their graph minors project.

In this paper, we consider the optimization problems {\sc MaxEDP} and {\sc MaxNDP} when the number of pairs are part of the input.
In this setting, the best approximation ratio for {\sc MaxEDP} is achieved by an~$\mathcal O(\sqrt{n})$-approximation~algorithm~\cite{ChekuriEtAl2006,KolliopoulosS04},
where $n$ is the number of nodes, whereas the best hardness for undirected graphs is only $\Omega(\log^{1/2 - \varepsilon}{n})$ \cite{andrews2010inapproximability}.
Bridging this gap is a fundamental open problem that seems quite challenging at the moment.

Most of the results for routing on disjoint paths use a natural multi-commodity flow relaxation as a starting point.
A well-known integrality gap instance due to Garg et al.~\cite{GargEtAl1997} shows that this relaxation has an integrality gap of $\Omega(\sqrt{n})$, and this is the main obstacle for improving the $\mathcal O(\sqrt{n})$-approximation~ratio in general graphs.
The integrality instance on an $n \times n$ grid (of treewidth $\Theta(\sqrt{n})$) exploits a topological obstruction in the plane that prevents a large integral routing; see Fig.~\ref{fig:lp}.
This led Chekuri et al. \cite{cns-tw} to studying the approximability of {\sc MaxEDP} with respect to the \emph{tree-width} of the underlying graph.
In particular, they pose the following conjecture:
\begin{conjecture}[\cite{chekuri2009note}]
\label{conj:tw}
  The integrality gap of the standard multi-commodity flow relaxation for {\sc MaxEDP} is~$\Theta(w)$, where $w$ is the treewidth of the graph.
\end{conjecture}
Recently, Ene et al.~\cite{EneEtAl2016} showed that {\sc MaxEDP} admits an $\mathcal O(w^3)$-approximation algorithm on graphs of treewidth at most $w$.
Theirs is the best known approximation ratio in terms of $w$, improving on an earlier~$\mathcal O(w\cdot 3^w)$-approximation algorithm due to Chekuri et al.
This shows that the problem seems more amenable on ``tree-like'' graphs.

However, for $w=\omega(n^{1/6})$, the bound is weaker than the bound of $\mathcal O(\sqrt{n})$.
In fact, {\sc EDP} remains $\mathsf{NP}$-hard even for graphs of \emph{constant} treewidth, namely treewidth $w = 2$~\cite{NishizekiEtAl2001}.
This further rules out the existence of a fixed-parameter algorithm for {\sc MaxEDP} parameterized by~$w$, assuming $\mathsf{P}\not=\mathsf{NP}$.
Therefore, to obtain fixed-parameter tractability results as well as better approximation guarantees, one needs to resort to parameters stronger than treewidth.

Another route to bridge the large gap between approximation lower and upper bounds for {\sc MaxEDP} is to allow the paths to have \emph{low congestion} $c$: that is, instead of requiring the routed paths to be pairwise disjoint, at most $c$ paths can use an edge.
In their groundbreaking work, Raghavan and Thompson~\cite{RaghavanThompson1987} introduced the technique of randomized rounding of LPs to obtain polynomial-time approximation algorithms for combinatorial problems. Their approach allows to route $\Omega(\opt)$ pairs of paths with congestion $\mathcal O\left(\frac{\log{n}}{\log\log{n}}\right)$.
This extensive line of research \cite{andrews:low-congestion,chuzhoy:edp-const-congestion,KawarabayashiKobayashi2011}  has culminated in a $\log^{\mathcal O(1)} k$-approximation algorithm with congestion~$2$ for {\sc MaxEDP}~\cite{chuzhoy2012polylogarithmic}.  A slightly weaker result also holds for \textsc{MaxNDP}~\cite{chekuri-ene:ndp-congestion}.

\subsection{Motivation and Contribution}
The goal of this work is to study disjoint paths problems under another natural measure for how ``far'' a graph is from being a tree. In particular, we propose to examine \textsc{MaxEDP} and \textsc{MaxNDP} under the \emph{feedback vertex set number}, which for a graph~$G$ denotes the smallest size $r$ of a set $R$ of $G$ for which $G - R$ is a forest.  Note that the treewidth of $G$ is at most $r+1$.  Therefore, given the $\mathsf{NP}$-hardness of {\sc EDP} for $w = 2$ and the current gap between the best known upper bound $\mathcal O(w^3)$ and the linear upper bound suggested by Conjecture~\ref{conj:tw}, it is interesting to study the stronger restriction of bounding the feedback vertex set number $r$ of the input graph.
Our approach is further motivated by the fact that \textsc{MaxEDP} is efficiently solvable on trees by means of the algorithm of Garg, Vazirani and Yannakakis~\cite{GargEtAl1997}. Similarly, \textsc{MaxNDP} is easy on trees (see Theorem~\ref{thm:maxndp-fpt-kr}).

Our main insight is that one can in fact obtain bounds in terms of $r$ that either strengthen the best known bounds or are almost tight (see Table~\ref{tab:results}). It therefore seems that parameter~$r$ correlates quite well with the ``difficulty'' of disjoint paths problems.

Our first result allows the paths to have small congestion: in this setting, we strengthen the result, obtained by the classic randomized LP-rounding approach of Raghavan and Thompson~\cite{RaghavanThompson1987}, that one can always route~$\Omega(\opt)$ pairs with congestion $\mathcal O\left(\frac{\log{n}}{\log\log{n}}\right)$ with constant probability.
\begin{theorem}
\label{thm:highprob}
For any instance $(G,\mathcal M)$ of {\sc MaxEDP}, one can efficiently find a routing of $\Omega(\opt)$ pairs with congestion $\mathcal O\left(\frac{\log{kr}}{\log\log{kr}}\right)$ with constant probability; in other words, there is an efficient $\mathcal O(1)$-approximation algorithm for {\sc MaxEDP} with congestion $\mathcal O\left(\frac{\log{kr}}{\log\log{kr}}\right)$.
\end{theorem}

Our second main result builds upon Theorem~\ref{thm:highprob} and uses it as a subroutine.
We show how to use a routing for {\sc MaxEDP} with low congestion to obtain a polynomial-time approximation algorithm for {\sc MaxEDP} \emph{without congestion} that performs well in terms of $r$.
\begin{theorem}
\label{thm:fvs}
  The integrality gap of the multi-commodity flow relaxation for {\sc MaxEDP} with~$k$ terminal pairs is  $\mathcal{O}(\sqrt{r}\cdot\log^{1.5} rk)$ for graphs with feedback vertex set number~$r$.
  Moreover, there is a polynomial time algorithm that, given a fractional solution to the relaxation of value $\mathsf{opt}$, it constructs an integral routing of size  $\mathsf{opt} / \mathcal{O}(\sqrt{r}\cdot\log^{1.5} rk)$.
\end{theorem}
In particular, our algorithm strengthens the best known approximation algorithm for {\sc MaxEDP} on general graphs~\cite{ChekuriEtAl2006} as always $r\leq n$, and indeed it matches that algorithm's performance up to polylogarithmic factors.  Substantially improving upon our bounds would also improve the current state of the art of \textsc{MaxEDP}.  Conversely, the result implies that it suffices to study graphs with close to linear feedback vertex set number in order to improve the currently best upper bound of~$\mathcal{O}(\sqrt{n})$ on the approximation ratio \cite{ChekuriEtAl2006}.

Our algorithmic approaches harness the forest structure of $G - R$ for any feedback vertex set $R$.
However, the technical challenge comes from the fact that the edge set running between $G - R$ and $R$ is unrestricted.
Therefore, the ``interaction'' between $R$ and $G - R$ is non-trivial, and flow paths may run between the two parts in an arbitrary manner and multiple times.  In fact, we show that \textsc{MaxEDP} is already $\mathsf{NP}$-hard if $R$ consists of a \emph{single node} (Theorem~\ref{thm:edp-nphard-r2}); this contrasts the efficient solvability on forests~\cite{GargEtAl1997}.

In order to overcome the technical hurdles we propose several new concepts, which we believe could be of interest in future studies of disjoint paths or routing problems.

In the randomized rounding approach of Raghavan and Thompson \cite{RaghavanThompson1987}, it is shown that the probability that the congestion on any fixed edge is larger than $c\frac{\log n}{\log\log n}$ for some constant~$c$ is at most $1/n^{O(1)}$. Combining this with the fact that there are at most $n^2$ edges, yields that every edge has bounded congestion w.h.p. The number of edges in the graph may, however, be unbounded in terms of $r$ and $k$. Hence, in order to to prove Theorem~\ref{thm:highprob}, we propose a non-trivial \emph{pre-processing step} of the optimum LP solution that is applied prior to the randomized rounding.  In this step, we aggregate the flow paths by a careful rerouting so that the flow ``concentrates'' in $O(kr^2)$ nodes (so-called \emph{hot spots}) in the sense that if all edges incident on hot spots have low congestion then so have all edges in the graph.  Unfortunately, for any such hot spot the number of incident edges carrying flow may still be unbounded in terms of $k$ and $r$.  We are, however, able to give a refined probabilistic analysis that suitably relates the probability that the congestion bound is exceeded to the amount of flow on that edge.  Since the total amount of flow on each hot spot is bounded in terms of $k$, the probability that \emph{all} edges incident on the same hot spot have bounded congestion is inverse polynomial in~$r$ and~$k$.

The known $\mathcal{O}(\sqrt{n})$-approximation algorithm for \textsc{MaxEDP} by Chekuri et al. \cite{ChekuriEtAl2006} employs a clever LP-rounding approach. If there are many long paths then there must be a single node carrying a significant fraction of the total flow and a good fraction of this flow can be realized by integral paths by solving a single-source flow problem. If the LP solution contains many short flow paths then greedily routing these short paths yields the bound since each such path blocks a bounded amount of flow.  In order to prove Theorem~\ref{thm:fvs}, it is natural to consider the case where there are many paths visiting a large number of nodes in $R$.  In this case, we reduce to a single-source flow problem, similarly to the approach of Chekuri et al.
The case where a majority of the flow paths visit only a few nodes in $R$ turns out more challenging, since any such path may still visit an unbounded number of edges in terms of $k$ and $r$.  We use two main ingredients to overcome these difficulties.  First, we apply our Theorem~\ref{thm:highprob} as a building block to obtain a solution with logarithmic congestion while losing only a constant factor in the approximation ratio.  Second, we introduce the concept of \emph{irreducible routings with low congestion} which allows us exploit the structural properties of the graph and the congestion property to identify a sufficiently large number of flow paths blocking only a small amount of flow.

Note that the natural greedy approach of always routing the shortest conflict-free path gives only $\mathcal{O}(\sqrt{m})$ for \textsc{MaxEDP}.  We believe that it is non-trivial to obtain our bounds via a more direct or purely combinatorial approach.

Our third result is a fixed-parameter algorithm for {\sc MaxNDP} in~$k + r$.
\begin{theorem}
\label{thm:maxndp-fpt-kr}
  {\sc MaxNDP} can be solved in time $(8k+8r)^{2r+2}\cdot \mathcal O(n)$ on graphs with feedback vertex set number~$r$ and $k$ terminal pairs.
\end{theorem}
This run time is polynomial for constant $r$.  We also note that for small~$r$, our algorithm is asymptotically significantly faster than the fastest known algorithm for NDP, by Kawarabayashi and Wollan~\cite{KawarabayashiWollan2010}, which requires time at least \emph{quadruple-exponential} in $k$ \cite{AdlerEtAl2011}.
Namely, if $r$ is at most triple-exponential in $k$, our algorithm is asymptotically faster than theirs. We achieve this result by the idea of so-called \emph{essential pairs} and \emph{realizations}, which characterizes the ``interaction'' between the feedback vertex set $R$ and the paths in an optimum solution.
Note that in our algorithm of Theorem~\ref{thm:maxndp-fpt-kr}, parameter $k$ does not appear in the exponent of the run time at all.
Hence, for small values of $r$ our algorithm is also faster than reducing {\sc MaxNDP} to {\sc NDP} by guessing the subset of pairs to be routed (at an expense of $2^k$ in the run time) and using Scheffler's~\cite{Scheffler1994} algorithm for {\sc NDP} with run time $2^{O(r\log r)}\cdot \mathcal O(n)$.

Once a fixed-parameter algorithm for a problem has been obtained, the existence of a polynomial-size kernel comes up.
Here we note that {\sc MaxNDP} does not admit a polynomial kernel for parameter $k+r$, unless~$\mathsf{NP}\subseteq \mathsf{coNP}/poly$~\cite{BodlaenderEtAl2011a}.

Another natural question is whether the run time $f(k,r)\cdot n$ in Theorem~\ref{thm:maxndp-fpt-kr} can be improved to~$f(r)\cdot n^{\mathcal O(1)}$.
We answer this question in the negative, ruling out the existence of a fixed-parameter algorithm for {\sc MaxNDP} parameterized by~$r$ (assuming $\mathsf{FPT}\not=\mathsf{W}[1]$):
\begin{theorem}
\label{thm:ndp-w1hard-r}
  {\sc MaxNDP} in unit-capacity graphs is $\mathsf{W}[1]$-hard parameterized by $r$.
\end{theorem}
This contrasts the known result that NDP is fixed-parameter tractable in~$r$~\cite{Scheffler1994}---which further stresses the relevance of understanding this parameter.

For {\sc MaxEDP}, we prove that the situation is, in a sense, even worse:
\begin{theorem}
\label{thm:edp-nphard-r2}
  {\sc MaxEDP} is $\mathsf{NP}$-hard for unit-capacity graphs with $r = 1$ and \textsc{EDP} is $\mathsf{NP}$-hard for unit-capacity graphs with $r=2$.
\end{theorem}
This theorem also shows that our algorithms are relevant for small values of $r$, and they nicely complement the $\mathsf{NP}$-hardness for {\sc MaxEDP} in capacitated trees~\cite{GargEtAl1997}.

Our results are summarized in Table~\ref{tab:results}.
\begin{table}[htpb]
  \centering
  \begin{tabular}{clcccc}
    \toprule
      const.     & param.    & EDP                      & {\sc MaxEDP}            & NDP & {\sc MaxNDP}\\
    \midrule
      $r = 0$    &           & poly~\cite{GargEtAl1997} & poly~\cite{GargEtAl1997} & poly~\cite{Scheffler1994} & poly~(Thm.~\ref{thm:maxndp-fpt-kr})\\
      $r = 1$    &    & \emph{open}                              & {$\mathsf{NP}$-hard~(Thm.~\ref{thm:edp-nphard-r2})}          & poly~\cite{Scheffler1994} & poly~(Thm.~\ref{thm:maxndp-fpt-kr})\\
      $r \geq 2$ &           & $\mathsf{NP}$-hard~(Thm.~\ref{thm:edp-nphard-r2}) & $\mathsf{NP}$-hard~(Thm.~\ref{thm:edp-nphard-r2}) & poly~\cite{Scheffler1994} & poly~(Thm.~\ref{thm:maxndp-fpt-kr}) \\
      \midrule
                 & $r$       & \multicolumn{2}{c}{para-$\mathsf{NP}$-hard~(Thm.~\ref{thm:edp-nphard-r2})} & $\mathsf{FPT}$~\cite{Scheffler1994} & $\mathsf{W}[1]$-hard~(Thm.~\ref{thm:ndp-w1hard-r})\\
                 &           & \multicolumn{2}{c}{$\mathcal{O}(\sqrt{r}\cdot \log^{1.5}{kr})$-approx~(Thm.~\ref{thm:fvs})} & \multicolumn{2}{c}{exact $(k+r)^{\mathcal O(r)}$~(Thm.~\ref{thm:maxndp-fpt-kr})} \\
                 &           &\multicolumn{2}{c}{$\mathcal O(1)$-approx. w.cg. $\mathcal O\left(\frac{\log{kr}}{\log\log{kr}}\right)$} (Thm.~\ref{thm:highprob})\\
    \bottomrule
  \end{tabular}
  \caption{Summary of results obtained in this paper.}
  \label{tab:results}
\end{table}

\medskip
\noindent
\textbf{Related Work.}
Our study of the feedback vertex set number is in line with the general attempt to obtain bounds for MaxEDP (or related problems) that are independent of the input size.  Besides the above-mentioned works that provide bounds in terms of the \emph{tree-width} of the input graph, G{\"u}nl{\"u}k \cite{Gunluk2007} and Chekuri et al.~\cite{flow-cut-gaps-chekuri13} give bounds on the \emph{flow-cut gap} for the closely related integer multicommodity flow problem that are logarithmic with respect to the \emph{vertex cover number} of a graph. This improved upon earlier bounds of~$\mathcal O(\log n)$~\cite{LeightonRao1999} and $\mathcal O(\log k)$ \cite{flow-cut-theorem-aumann98,geometry-graphs-linial95}.
As every feedback vertex set is in particular a vertex cover of a graph, our results generalize earlier work for disjoint path problems on graphs with bounded vertex cover number.
Bodlaender et al.~\cite{BodlaenderEtAl2011a} showed that {\sc NDP} does not admit a polynomial kernel parameterized by vertex cover number \emph{and} the number~$k$ of terminal pairs, unless $\mathsf{NP}\subseteq \mathsf{coNP}/poly$ ; therefore, {\sc NDP} is unlikely to admit a polynomial kernel in $r + k$ either.
Ene et al.~\cite{EneEtAl2016} showed that {\sc MaxNDP} is $\mathsf{W}[1]$-hard parameterized by treedepth, which is another restriction of treewidth that is incomparable to the feedback vertex set number.

The basic gap in understanding the approximability of MaxEDP has led to several improved results for special graph classes, and also our results can be seen in this light. For example, polylogarithmic approximation algorithms are known for graphs whose global minimum cut value is $\Omega(\log^5 n)$ \cite{RaoZhou2010}, for bounded-degree
expanders~\cite{BroderEtAl1994,BroderEtAl1999,KleinbergRubinfeld1996,LeightonRao1999,Frieze2000}, and for Eulerian planar or 4-connected planar graphs~\cite{KawarabayashiKobayashi2011}.
Constant factor approximation algorithms are known for capacitated trees~\cite{GargEtAl1997,ChekuriEtAl2007}, grids and grid-like graphs~\cite{AumannRabani1995,AwerbuchEtAl1994,KleinbergTardos1995,KleinbergTardos1998}.
For planar graphs, there is a constant-factor approximation algorithm with congestion 2~\cite{SeguinCharbonneauShepherd2011}.
Very recently, Chuzhoy et al.~\cite{Chuzhoy-etal:NDP-planar} gave a $\tilde{\mathcal{O}}(n^{9/19})$-approximation algorithm for MaxNDP on \emph{planar} graphs.
However, improving the $\mathcal O(\sqrt{n})$-approximation algorithm for \textsc{MaxEDP} remains elusive even for \emph{planar} graphs.

\section{Preliminaries}
\label{sec:preliminaries}
We use standard graph theoretic notation.
For a graph $G$, let $V(G)$ denote its vertex set and~$E(G)$ its edge set.
Let $G$ be a graph.
A \emph{feedback vertex set} of $G$ is a set $R\subseteq V(G)$ such that $G - R$ is a forest.
A \emph{minor} of~$G$ is a graph $H$ that is obtained by successively contracting edges from a subgraph of $G$ (and deleting any occurring loops).
A class $\mathcal G$ of graphs is \emph{minor-closed} if for any graph in $\mathcal G$ also all its minors belong to $\mathcal G$.

For an instance $(G,\mathcal M)$ of {\sc MaxEDP/MaxNPD}, we refer to the vertices participating in the pairs $\mathcal M$ as \emph{terminals}.
It is convenient to assume that~$\mathcal M$ forms a matching on the terminals; this can be ensured by making several copies of a terminal and attaching them as leaves.

\medskip
\noindent
\textbf{Multi-commodity flow relaxation.}
We use the following standard multi-commodity flow relaxation for {\sc MaxEDP} (there is an analogous relaxation for {\sc MaxNDP}).
We use $\mathcal P(u, v)$ to denote the set of all paths in $G$ from $u$ to $v$, for each pair $(u, v)$ of nodes. Since the pairs~$\mathcal M$ form a matching, the sets~$\mathcal P(s_i, t_i)$ are pairwise disjoint.
Let $\mathcal P = \bigcup_{i = 1}^k \mathcal P(s_i, t_i)$.
The LP has a variable $f(P)$ for each path $P \in \mathcal P$ representing the amount of flow on $P$.
For each pair $(s_i, t_i) \in \mathcal M$, the LP has a variable $x_i$ denoting the total amount of flow routed for the pair (in the corresponding IP, $x_i$ denotes whether the pair is routed or not).
The LP imposes the constraint that there is a flow from $s_i$ to $t_i$ of value $x_i$.
Additionally, the LP has constraints that ensure that the total amount of flow on paths using a given edge (resp. node for {\sc MaxNDP}) is at most 1.

\begin{figure}[htb]
  \begin{center}
    \begin{boxedminipage}{0.38\textwidth}
      \vspace{-0.1in}
      \begin{align*}
	                     & (\textnormal{{\sc MaxEDP} LP})                           &\\
	   ~ \max \quad & \sum_{i = 1}^k x_i                                  &\\
	   \text{s.t.} \quad & \sum_{P \in \mathcal P(s_i, t_i)} f(P) = x_i \leq 1 & i=1,\hdots,k,~\\
                         & \sum_{P:\; e \in P} f(P) \leq 1       & e \in E(G)~\\\
                         & f(P) \geq 0                                         & P \in \mathcal P~
     \end{align*}
    \end{boxedminipage}
    \hspace{0.1in}
    \begin{boxedminipage}{0.4\textwidth}
      \centering
      \includegraphics[scale=0.63]{gap-example}
    \end{boxedminipage}
  \end{center}
  \caption{Multi-commodity flow relaxation for {\sc MaxEDP}.
  Right: $\Omega(\sqrt{n})$ integrality gap for {\sc MaxEDP}~\cite{GargEtAl1997}: any integral routing routes at most one pair, whereas a multi-commodity flow can send $1/2$ unit of flow for each pair $(s_i, t_i)$ along the canonical path from $s_i$ to~$t_i$ in the grid.}
\label{fig:lp}
\end{figure}

It is well-known that the relaxation {\sc MaxEDP LP} can be solved in polynomial time, since there is an efficient separation oracle for the dual LP (alternatively, one can write a compact relaxation).
We use $(f, \mathbf{x})$ to denote a feasible solution to {\sc MaxEDP LP} for an instance $(G, \mathcal M)$ of {\sc MaxEDP}.
For each terminal $v$, let $x(v)$ denote the total amount of flow routed for $v$ and we refer to~$x(v)$ as the \emph{marginal value} of $v$ in the multi-commodity flow $f$.

We will use the following result by Chekuri et al.~\cite[Sect. 3.1]{ChekuriEtAl2006}; see also Proposition 3.3 of Chekuri et al.~\cite{cns-tw-corr}.
\begin{proposition}
\label{lem:single-node-routing}
  Let $(f,\bf{x})$ be a fractional solution to the LP relaxation of a \textsc{MaxEDP} instance $(G,\mathcal M)$.
  If some node $v$ is contained in all flow paths of~$f$, then we can find an integral routing of size at least $\frac{1}{12}\sum_{i}x_i$ in polynomial time.
\end{proposition}

\section{Bi-Criteria Approximation for MaxEDP with Low Congestion}
\label{sec:bi-crit-appr}
We present a randomized rounding algorithm that will lead to the proof of Theorem~\ref{thm:highprob}.

\subsection{Algorithm}
\label{sec:rand-rounding}
Consider an instance $(G,\mathcal M)$ of {\sc MaxEDP}.
Let $R$ be a 2-approximate minimum feedback vertex set of $G$ and let $r = |R|$; note that such a set $R$ can be obtained in polynomial time~\cite{BafnaEtAl1999}.

For the sake of easier presentation, we will assume in this section that the feedback vertex set $R$ contains all terminal nodes from $\mathcal M$.  This can be achieved by temporarily adding the set of terminals to the feedback vertex set $R$.  Also note that this assumption increases the bound of Theorem~\ref{thm:highprob} by at most a constant factor.

First, solve the corresponding {\sc MaxEDP LP}.
We obtain an optimal solution~$(f,\mathbf{x})$.
For each~$(s_i, t_i)\in\mathcal M$ we further obtain a set $\mathcal{P}'(s_i, t_i) = \{P \in \mathcal{P}(s_i, t_i) \mid~f(P)>0\}$ of positive weighted paths that satisfy the LP constraints.
Note that the total set ${\mathcal P' = \bigcup_{i = 1}^k \mathcal P'(s_i, t_i)}$ is of size polynomially bounded in the input size.
In what follows, we will modify~$\mathcal P'$ and then select an (unweighted) subset~$\mathcal S$ of~$\mathcal P'$ that will form our integral solution.

Each $P\in\mathcal{P'}$ has the form $(r_1,\dots,r_2,\dots,r_\ell)$ where $r_1,\dots,r_\ell$ are the nodes in~$R$ that are traversed by $P$ in this order.  The paths~$(r_j,\dots,r_{j+1})$ with $j=1,\dots,\ell-1$ are called \emph{subpaths of}~$P$.
For every subpath $P'$ of~$P$, we set~$f(P')=f(P)$.
Let~$\mathcal J$ be the multi-set of all subpaths of all paths in $\mathcal P'$.
Let $F = G-R$ be the forest obtained by removing~$R$.

We now modify some paths in~$\mathcal P'$, one by one, and at the same time construct a subset~$H$ of nodes that we will call ``hot spots''.
At the end, every subpath in $\mathcal J$ will contain at least one hot spot.

Initially, let $H=\emptyset$.
Consider any tree~$T$ in $F$ and fix any of its nodes as a root. Then let~$\mathcal{J}_T$ be the multi-set of all subpaths in $\mathcal J$ that, excluding the endpoints, are contained in~$T$. For each subpath $P\in \mathcal{J}_T$, define its \emph{highest node}~$h(P)$ as the node on~$P$ closest to the root. Note that $P\cap T=P\cap F$ is a path.
Now, pick a subpath~$P\in \mathcal{J}_T$ that does not contain any node in $H$ and whose highest node $h(P)$ is \emph{farthest away} from the root. Consider the multi-set~$\mathcal{J}[P]$ of all subpaths in $\mathcal{J}_T$ that are identical to $P$ (but may be subpaths of different flow paths in $\mathcal{P}'$). Note that the weight~$f(\mathcal{J}[P]) := \sum_{P\in\mathcal{J}[P]} f(P)$ of $\mathcal{J}[P]$ is at most $1$ by the constraints of the LP.
Let $u,v\in R$ be the endpoints of $P$. We define $\mathcal{J}_{uv}$ as the set of all subpaths in $\mathcal{J}\setminus \mathcal{J}[P]$ that have $u$ and $v$ as their endpoints and that do not contain any node in~$H$.

Intuitively speaking, we now aggregate flow on $P$ by rerouting as much flow as possible from~$\mathcal{J}_{uv}$ to~$P$.  To this end, we repeatedly perform the following operation as long as $f(\mathcal{J}[P])<1$ and $\mathcal{J}_{uv}\not=\emptyset$.  We pick a path~$P'$ in $\mathcal J$ that contains a subpath in $\mathcal{J}_{uv}$.  We reroute flow from $P'$ by creating a new path~$P''$ that arises from $P'$ by replacing its subpath between $u$ and $v$ with $P$, and assign it the weight $f(P'')=\min\{f(P'),1-f(\mathcal{J}[P])\}$.  Then we set the weight of (the original path) $P'$ to $\max\{0, f(P')+f(\mathcal{J}[P])-1\}$.  We update the sets~$\mathcal{P'}$,~$\mathcal{P}'(s_i,t_i)$,~$\mathcal{J}$,~$\mathcal{J}_T$, $\mathcal{J}[P]$ and $\mathcal{J}_{uv}$ accordingly.

As soon as $f(\mathcal{J}[P])=1$ or $\mathcal{J}_{uv}=\emptyset$, we add $h(P)$ to $H$. Then, we proceed with the next $P\in\mathcal{J}_T$ not containing a hot spot and whose highest node $h(P)$ is farthest away from the root. If no such $P$ is left we consider the next tree~$T$ in $F$.

At the end, we create our solution~$\mathcal S$ by randomized rounding:
We route every terminal pair~$(s_i, t_i)$ with probability $x_i$.  In case $(s_i,t_i)$ is routed, we randomly select a path from~$\mathcal P'(s_i, t_i)$ and add it to~$\mathcal S$ where the probability that path $P$ is taken is $f(P)/x_i$.

\subsection{Analysis}
First, observe that~$\mathbf{x}$ did not change during our modifications of the paths, as the total flow between any terminal pair did not change.
Thus, the expected number of pairs routed in our solution is~$\sum_{i=1}^k x_i\ge \opt$. Using the Chernoff bound, the probability that we route less than $\opt/2$ pairs is at most $e^{-1/8\opt}<1/2$, assuming that $\opt>8$.
Secondly, we bound the congestion of our solution---our second criterion.
\begin{lemma}
\label{thm:congestiononfisatmost2}
  The congestion of flow $f$ is at most~2.
\end{lemma}
\begin{proof}
In our algorithm,
we increase the flow only along flow subpaths that are pairwise edge-disjoint.
To see this, consider two distinct flow subpaths $P$ and $P'$ on which we increase the flow.
Assume, without loss of generality, that $P$ was considered before $P'$ by the algorithm.
If there was an edge $e$ lying on~$P$ and $P'$, then both subpaths traverse the same tree in forest~$F$.
Hence, the path from $e$ to $h(P')$ would visit $h(P)$, and~$h(P)$ would be an internal node of $P'$.
This yields a contradiction, as~$h(P)$ was already marked as a hot spot when $P'$ was considered.
This shows that we increased the flow along any edge by at most one unit, and, hence, $f$ has congestion at most~2.
\end{proof}

We now bound the congestion of the integral solution obtained by randomized rounding.  In the algorithm, we constructed a set $H$ of hot spots.  As a part of the analysis, we will now extend this set as follows.  We build a sub-forest $F'$ of $F$ consisting of all edges of $F$ that lie on a path connecting two hot spots.  Then we add to $H$ all nodes that have degree at least~3 in $F'$.  Since the number of nodes of degree~3 in any forest is at most its number of leaves and since every leaf of $F'$ is a hot spot, it follows that this can at most double the size of $H$.  Finally, we add the set $R$ of all feedback vertex nodes to $H$.
\begin{lemma}
\label{thm:onlyfewhotspots}
  The number $|H|$ of hot spots is $\mathcal O(kr^2)$.
\end{lemma}
\begin{proof}
  It suffices to show that the number of hot spots added to $H$ by the algorithm is $\mathcal O(kr^2)$.  To this end, fix two nodes $u,v\in R$ and consider the set of flow subpaths $P$ with end nodes $u$ and $v$ for which we added~$h(P)$ to $H$.  Due to the aggregation of flows in our algorithm, all except possibly one of the subpaths are saturated, that is, they carry precisely one unit of flow. Since no two of these subpaths are contained in a same flow path of $f$ and since the flow value of $f$ is bounded from above by $k$, we added only~$\mathcal O(k)$ hot spots for the pair $u,v$.  Since there are at most $r^2$ pairs in $R$, the claim follows.
\end{proof}

\begin{definition}
  A hot spot $u\in H$ is \emph{good} if the congestion on any edge incident on $u$ is bounded by $c\cdot \frac{\log{kr}}{\log\log{kr}}$, where $c$ is a sufficiently large constant; otherwise, $u$ is \emph{bad}.
\end{definition}

\begin{lemma}\label{lem:hotspot-prob}
  Let $u\in H$ be a hot spot. Then the probability that $u$ is bad is at most $1/(k^2r^3)$.
\end{lemma}
\begin{proof}
  Let $e_1=uv_1,\dots,e_{\ell}=uv_{\ell}$ be the edges incident on $u$ and let
  $f_i$ be the total flow on edge $uv_i$ for~$i=1,\dots,\ell$.  By Lemma~\ref{thm:congestiononfisatmost2}, we have that $f_i\leq 2$.
  Since any flow path visits at most two of the edges incident on~$u$, the total flow $\sum_{i=1}^{\ell}f_i$ on the edges incident on
  $u$ is at most $2k$.

  For any $i=1,\dots,\ell$, we have that
  $f_i=\sum_{P\colon P\ni{e_i}}f(P)$, where $P$ runs over the set of
  all paths connecting some terminal pair and containing $e_i$.  Let
  $f_{ij}=\sum_{P\in\mathcal{P}(s_j,t_j)\colon P\ni e_i}f(P)$ be the
  total amount of flow sent across~$e_i$ by terminal pair $(s_j,t_j)$.
  Recall that $x_j$ is the total flow sent for terminal pair $(s_j,t_j)$.  The
  probability that the randomized rounding procedure picks path $P$
  with $P\in\mathcal{P}(s_j,t_j)$ is precisely
  $x_j\cdot\frac{f(p)}{x_j}=f(p)$.  Given the disjointness of the
  respective events, the probability that pair $(s_j,t_j)$ routes a
  path across~$e_i$ is precisely~$f_{ij}$.  Let~$X_{ij}$ be the binary
  random variable indicating whether pair $(s_j,t_j)$ routes a path
  across~$e_i$. Then $\prob{X_{ij}=1}=f_{ij}$.  Let
  $X_i=\sum_{j}X_{ij}$ be the number of paths routed across $e_i$ by
  the algorithm.  By linearity of expectation, we have that
  $\expct{X_i}=\sum_{j}\expct{X_{ij}}=\sum_{j}f_{ij}=f_i$.

  Fix any edge~$e_i$. Set $\delta=c\cdot\frac{\log{kr}}{\log\log{kr}}$ and $\delta'=2\frac{\delta}{f_i}-1$.  Note that for fixed~$i$, the
  variables~$X_{ij}$ are independent.  Hence, by the Chernoff bound,
  we have that
    \begin{align*}
      \prob{X_i\geq c\cdot\frac{\log{kr}}{\log\log{kr}}} & \leq\prob{X_i\geq(1+\delta')f_i} < \left(\frac{e^{\delta'}}{(1+\delta')^{1+\delta'}}\right)^{f_i} \\
                                                         & \leq \left(\frac{f_i}{2}\right)^{2\delta}\cdot \left(\frac{\delta}{e}\right)^{-2\delta}
                                                         \leq f_i e^{-c'\log\log{kr}\cdot\frac{\log{kr}}{\log\log{kr}}}
                                                           \leq \frac{f_i}{2k^3r^3}\, .
    \end{align*}
Here, we use that $f_i\leq 2$ for the second last inequality and for the last inequality we pick $c'$ sufficiently large by making $c$ and $k$ sufficiently large. (Note that {\sc MaxEDP} can be solved efficiently for constant $k$.)

Now, using the union bound, we can infer that the probability that any of the edges incident on $u$ carries more than $\delta$ paths is at most $\sum_{i}f_i/(2k^3r^3)\leq (2k)/(2k^3r^3)=1/(k^2r^3)$.
\end{proof}

\begin{lemma}\label{lem:hot-spot-congestion}
  Assume that every hot spot is good. Then the congestion on any edge is bounded by $2c\frac{\log{kr}}{\log\log{kr}}$.
\end{lemma}
\begin{proof}
  Consider an arbitrary edge $e=uv$ that is not incident on any hot
  spot.  In particular, this means that~$e$ lies in the forest
  $F=G-R$.  A hot spot $z$ in $F$ is called \emph{direct} to $u$ (or
  $v$) if the path in $F$ from $z$ to $u$ (or $v$) neither contains
  $e$ nor any hot spot other than $z$.

  Now observe that there can be only one hot spot $z$ direct to $u$
  and only one hot spot $z'$ direct to $v$.  If there was a second hot
  spot $z''\neq z$ direct to $u$ then there would have to be yet
  another hot spot at the node where the path $P_z$ from $z$ to $u$
  joins the path from $z''$ to $u$ contradicting the choice of $z$.
  Let $P_{z'}$ be the path from $z'$ to $v$ in $F$.  Moreover, let
  $e_z$ be the edge incident on $z$ on path $P_z$ and let $e_{z'}$ be
  the edge incident on~$z'$ on path $P_{z'}$.

  Now let $P$ be an arbitrary path that is routed by our algorithm and
  that traverses $e$.  It must visit a hot spot.  If $P$ visited neither
  $z$ nor $z'$, then $P$ would contain a hot spot direct to $u$
  or to $v$ that is distinct from~$z$ and $z'$---a contradiction.  Therefore, $P$
  contains $e_z$ or $e_z'$.  The claim now follows from the
  fact that this holds for any path traversing $e$, that $z$ and $z'$
  are good, and that therefore at most
  $2c\frac{\log{kr}}{\log\log{kr}}$ paths visit $e_z$ or~$e_z'$.
\end{proof}

\begin{theorem}
  The algorithm from Sect.~\ref{sec:rand-rounding} produces---with constant probability---a routing with $\Omega(\opt)$ paths, such that the congestion is $\mathcal O\left(\frac{\log{kr}}{\log\log{kr}}\right)$.
\end{theorem}
\begin{proof}
  As argued above, we route less than $\opt/2$ paths with probability at most $1/2$.  By Lemma~\ref{thm:onlyfewhotspots}, there are~$\mathcal O(kr^2)$ hotspots.  The probability that at least one of these hot spots is bad is ${\mathcal O(kr^2/(k^2r^3))=\mathcal O(1/(kr))}$, by Lemma~\ref{lem:hotspot-prob}.  Hence, with constant probability, we route at least $\opt/2$ pairs with congestion at most~$2c\frac{\log{kr}}{\log\log{kr}}$, by Lemma~\ref{lem:hot-spot-congestion}.
\end{proof}

\section{Refined Approximation Bound for MaxEDP}\label{sec:approx-algo-maxedp}
In this section, we provide an improved approximation guarantee for {\sc MaxEDP} \emph{without} congestion, thereby proving Theorem~\ref{thm:fvs}. (In contrast to the previous section, we do not assume here that all terminals are contained in the feedback vertex set.)

\subsection{Irreducible Routings with Low Congestion}\label{sec:double-path-cover}

We first develop the concept of \emph{irreducible routings with low congestion}, which is (besides Theorem~\ref{thm:highprob}) a key ingredient of our strengthened bound on the approximability of \textsc{MaxEDP} based on the feedback vertex number.

Consider any multigraph~$G$ and any set~$\mathcal P$ of (not necessarily simple) paths in~$G$ with congestion~$c$.
We say that an edge~$e$ is \emph{redundant in $\mathcal{P}$} if there is an edge~$e'\neq e$ such that the set of paths in $\mathcal P$ \emph{covering} (containing) $e$ is a subset of the set of paths in~$\mathcal P$ covering~$e'$.

\begin{definition}
  Set $\mathcal P$ is called an \emph{irreducible routing
    with congestion $c$} if each edge belongs to at most $c$
  paths of $\mathcal P$ and there is no edge redundant in~$\mathcal P$.
\end{definition}
In contrast to a feasible routing of an {\sc MaxEDP} instance, we do not require an irreducible routing to connect a set of terminal pairs.
If there is an edge $e$ redundant in $\mathcal{P}$, we can apply the following \emph{reduction rule}:
We contract~$e$ in $G$ and we contract~$e$ in every path of $\mathcal P$ that covers~$e$.
By this, we obtain a minor~$G'$ of~$G$ and a set~$\mathcal P'$ of paths that consists of all the contracted paths and of all paths in $\mathcal P$ that were not contracted.
Thus, there is a one-to-one correspondence between the paths in~$\mathcal P$ and~$\mathcal P'$ .

We make the following observation about $\mathcal{P}$ and $\mathcal{P}'$.
\begin{observation}\label{obs:redToDPC}
Any subset of paths in $\mathcal P'$ is edge-disjoint in $G'$ if and only if the corresponding subset of paths in $\mathcal P$ is edge-disjoint in~$G$.
\end{observation}

Since the application of the reduction rule strictly decreases the number of redundant edges, an iterative application of this rule yields an irreducible routing on a minor of the original graph.
\begin{theorem}\label{thm:double-path-covers}
  Let $\mathcal G$ be a minor-closed class of multigraphs and let $p_{\mathcal G} > 0$.
  If for each graph $G\in\mathcal G$ and every non-empty irreducible routing $\mathcal S$ of $G$ with congestion~$c$ there exists a path in~$\mathcal S$ of length at most~$p_{\mathcal G}$, then the average length of the paths in $\mathcal S$ is at most $c\cdot p_{\mathcal G}$.
\end{theorem}
\begin{proof}
Take a path~$P_0$ of length at most $p_{\mathcal G}$. Contract all edges of $P_0$ in $G$ and obtain a minor $G'\in\mathcal G$ of $G$.  For each path in $\mathcal S$ contract all edges shared with $P_0$ to obtain a set~$\mathcal S'$ of paths.
Remove $P_0$ along with all degenerated paths from~$\mathcal S'$, thus $|\mathcal S'| < |\mathcal S|$.
Note that~$\mathcal S'$ is an irreducible routing of $G'$ with congestion~$c$.
We repeat this reduction procedure recursively on~$G'$ and $S'$ until $S'$ is empty which happens after at most $|\mathcal{S}|$ steps.
At each step we decrease the total path length by at most $c\cdot p_{\mathcal G}$. Hence, the total length of paths in $\mathcal{S}$ is at most~$|\mathcal{S}| \cdot c \cdot p_{\mathcal G}$.
\end{proof}

As a consequence of Theorem~\ref{thm:double-path-covers}, we get the following result for forests.
\begin{lemma}\label{lem:path-length-forest}
  Let $F$ be a forest and let $\mathcal{S}$ be a non-empty irreducible routing of $F$ with congestion~$c$.  Then the average path length in~$\mathcal{S}$ is at most $2c$.
\end{lemma}
\begin{proof}
 We show that $\mathcal{S}$ contains a path of length as most~$2$. The lemma follows immediately by applying Theorem~\ref{thm:double-path-covers}.

  Take any tree in $F$, root it with any node and consider a leaf~$v$ of maximum depth.
  Let~$e_1$ and $e_2$ be the first two edges on the path from $v$ to the root.
  By definition of irreducible routing,
  the set of all paths covering $e_1$ is not a subset of the paths covering~$e_2$, hence,
  $e_1$ is covered by a path which does not cover~$e_2$.
  Since all other edges incident to~$e_1$ end in a leaf, this path has length at most~$2$.
\end{proof}

Note that the bound provided in Lemma~\ref{lem:path-length-forest} is actually tight up to a constant.
Let $c\geq 1$ be an arbitary integer. Consider a graph that is a path of length~$c-1$ with a star of $c-1$ leafs attached to one of its end points.
The $c-1$ many paths of length~$c$ together with the $2c-2$ many paths of length~$1$ form an irreducible routing with congestion $c$. The average path length is $((c-1)c+(2c-2))/(3c-3)=(c+2)/3$.

\subsection{Approximation Algorithm}\label{sec:bound-size-feedb}
Consider an instance $(G,\mathcal M)$ of \textsc{MaxEDP}, and let $r$ be the size of a feedback vertex set $R$ in $G$.  Using our result of Sect.~\ref{sec:bi-crit-appr}, we can efficiently compute a routing $\mathcal{P}$ with congestion~$c\coloneqq \mathcal O\left(\frac{\log{kr}}{\log\log{kr}}\right)$ containing~$\Omega(\opt)$ paths.

Below we argue how to use the routing $\mathcal P$ to obtain a feasible routing of cardinality $\Omega\left(|\mathcal{P}|/(c^{1.5} \sqrt{r})\right)$, which yields an overall approximation ratio of~$\mathcal{O}\left(\sqrt{r}\cdot\log^{1.5} rk\right)$; that will prove Theorem~\ref{thm:fvs}.

Let $r'=\sqrt{r/c}$. We distinguish the following cases.

\medskip
\noindent
\textbf{Case 1:} At least half of the paths in $\mathcal P$ visit at most $r'$ nodes of the feedback vertex set $R$.
Let $\overline{\mathcal P}$ be the subset of these paths.
As long as there is an edge~$e$ not adjacent to~$R$ that is redundant in~$\mathcal P'$,
we iteratively apply the reduction rule from Sect.~\ref{sec:double-path-cover} on~$e$.
Let $G'$ be the obtained minor of $G$ with forest $F'=G'-R$, and let $\mathcal P'$ be the obtained set of (not necessarily simple) paths corresponding to~$\overline{ \mathcal P}$.
By Observation~\ref{obs:redToDPC}, it suffices to show that there is a subset ${\mathcal{P}}_0'\subseteq{\mathcal{P}'}$ of pairwise edge-disjoint paths of size~$|\mathcal P_0| = \Omega\left(|\mathcal{P}|/( c r') \right)$
in order to obtain a feasible routing for~$(G,\mathcal M)$ of size~$\Omega\left(|\mathcal{P}|/( c r') \right)$.

To obtain~$\mathcal{P}_0'$,
we first bound the total path length in~$\mathcal P'$.
Removing~${R}$ from ${G'}$ ``decomposes'' the set ${P}'$ into a set~${\mathcal{S}:=\{S\textrm{ is a connected component of }P\cap F\mid P\in\mathcal{P}'\,\}}$ of
subpaths lying in $F'$.
Observe that $\mathcal{S}$ is an irreducible set of~$F'$ with congestion~$c$, as the reduction rule is not applicable anymore.
(Note that a single path in~$\mathcal{P'}$ may lead to many paths in the cover~$\mathcal{S}$ which are considered distinct.)
Thus, by Lemma~\ref{lem:path-length-forest}, the average path length in $\mathcal{S}$ is at most~$2c$.

Let $P$ be an arbitrary path in $\mathcal{P}'$.  Each edge on $P$ that is \emph{not} in a subpath in~$\mathcal{S}$ is incident on a node in~${R}$, and each node in ${R}$ is incident on at most two edges in~$P$.  Together with the fact that $P$ visits at most~$r'$ nodes in $R$ and
that the average length of the subpaths in $\mathcal{S}$ is at most~$2c$,
we can upper bound the total path length $\sum_{P\in\mathcal{P'}}|P|$ by~$|\mathcal{P'}| r' (2c+2)$.
Let $\mathcal P''$ be the set of the $|\mathcal{P}'|/2$ shortest paths in~$\mathcal{P}'$.
Hence, each path in~$\mathcal P''$ has length at most~$4 r' (c+1)$.

We greedily construct a feasible solution $\mathcal{P}_0'$ by iteratively picking an arbitrary path~$P$ from~$\mathcal{P}''$ adding it to $\mathcal{P}_0'$ and removing all paths from $\mathcal{P}''$ that share some edge with~$P$ (including $P$ itself).  We stop when~$\mathcal{P}''$ is empty.  As $\mathcal{P}''$ has congestion~$c$,
we remove at most~$4r'c(c+1)$ paths from~$\mathcal{P}''$ per iteration. Thus,~$|\mathcal{P}_0'|\geq|\mathcal{P''}|/(4r'c(c+1))=\Omega\left(|\mathcal{P}|/( c^{1.5} \sqrt{r}\right)$.

\medskip
\noindent
\textbf{Case 2:} At least half of the paths in $\mathcal P$ visit at least $r'$ nodes of the feedback vertex set $R$.
Let $\mathcal P'$ be the subset of these paths.
Consider each path in~$\mathcal{P}'$ as a flow of value $1/c$ and let~$f$ be the sum of all these flows.
Note that $f$ provides a feasible solution to the \textsc{MaxEDP} LP relaxation for $({G}, M)$ of value at least~$|{\mathcal{P}}|/(2c)$.
Note that each such flow path contributes~$1/c$ unit of flow to each of the $r'$ nodes in ${R}$ it visits.
Since every flow path in~$f$ has length at least $r'$, the total inflow of the nodes in ${R}$ is at least $|f|r'$.
By averaging, there must be a node $v\in {R}$ of inflow at least $r'|f|/r = |f|/r'$.
Let $f'$ be the subflow of $f$ consisting of all flow paths visiting $v$.
This subflow corresponds to a feasible solution $(f',\bf{x'})$ of the LP relaxation of value at least
$|f|/r'\geq |\mathcal{P}|/(2cr')$.
Using Proposition~\ref{lem:single-node-routing}, we can recover an integral feasible routing of size at least~$\frac{1}{12}\sum_{i}x_i'\geq |{\mathcal{P}}|/(24cr')
=\Omega\left(|\mathcal{P}|/( c^{1.5} \sqrt{r}\right)$.

This completes the proof of~Theorem~\ref{thm:fvs}.
 \qed

\section{Fixed-Parameter Algorithm for MaxNDP}
\label{sec:exact-ndp}

We give a fixed-parameter algorithm for {\sc MaxNDP} with run time $(k+r)^{\mathcal O(r)}\cdot n$, where $r$ is the size of a minimum feedback vertex set in the given instance~$(G,\mathcal M)$.
A feedback vertex set $R$ of size $r$ can be computed in time $2^{O(r)}\cdot \mathcal O(n)$~\cite{LokshtanovEtAl2015}.
By the matching assumption, each terminal in~$\mathcal M$ is a leaf.  We can thus assume that none of the terminals is contained in~$R$.

Consider an optimal routing $\mathcal{P}$ of the given {\sc MaxNDP} instance. Let $\mathcal{M}_R\subseteq\mathcal{M}$ be the set of terminal pairs that are connected via $\mathcal{P}$ by a path that visits at least one node in~$R$.  Let $P\in\mathcal{P}$ be a path connecting a terminal pair $(s_i,t_i)\in\mathcal{M}_R$.  This path has the form $(s_i,\dots,r_1,\dots,r_2,\dots,r_\ell,\dots,t_i)$, where $r_1,\dots,r_\ell$ are the nodes in $R$ that are traversed by $P$ in this order.  The pairs $(s_i,r_1),(r_\ell,t_i)$ and~$(r_j,r_{j+1})$ with $j=1,\dots,\ell-1$ are called \emph{essential} pairs for $P$.  A node pair is called \emph{essential} if it is essential for some path in $\mathcal{P}$.  Let $\mathcal{M}_e$ be the set of essential pairs.

Let $F$ be the forest that arises when deleting $R$ from the input graph $G$.  Let~$(u,v)$ be an essential pair.  A $u$-$v$ path $P$ in $G$ is said to \emph{realize} $(u,v)$ if all internal nodes of $P$ lie in $F$.  A set $\mathcal{P}'$ of paths is said to \emph{realize}~$\mathcal{M}_e$ if every pair in $\mathcal{M}_e$ is realized by some path in $\mathcal{P}'$ and if two paths in $\mathcal{P}'$ can only intersect at their end nodes. Note that the optimal routing $\mathcal{P}$ induces a natural realization of~$\mathcal{M}_e$, by considering all maximal subpaths of paths in $\mathcal{P}$ whose internal nodes all lie in~$F$.
Conversely, for any realization~$\mathcal{P}'$ of~$\mathcal{M}_e$, we can concatenate paths in~$\mathcal{P}'$ to obtain a feasible routing that connects all terminal pairs in~$\mathcal{M}_R$. Therefore, we consider $\mathcal{P}'$ (slightly abusing notation) also as a feasible routing for $\mathcal{M}_R$.

In our algorithm, we first guess the set~$\mathcal{M}_e$ (and thus $\mathcal{M}_R$).
Then, by a dynamic program, we construct two sets of paths, $\mathcal{P}_e$ and $\mathcal{P}_F$
where $\mathcal{P}_e$ realizes~$\mathcal{M}_e$ and~$\mathcal{P}_F$ connects in~$F$ a subset of $\overline{\mathcal{M}}_R:=\mathcal{M}\setminus\mathcal{M}_R$. In our algorithm, the set $\mathcal{P}_e\cup\mathcal{P}_F$ forms a feasible routing that maximizes $|\mathcal{P}_F|$ and routes all pairs in~$\mathcal{M}_R$.  (Recall that we consider the realization $\mathcal{P}_e$ of~$\mathcal{M}_e$ as a feasible routing for~$\mathcal{M}_R$.)

Now assume that we know set $\mathcal{M}_e$. We will describe below a dynamic program that computes an optimum routing in time $2^{\mathcal O(r)}(k+r)^{\mathcal O(1)}n$.  For the sake of easier presentation, we only describe how to compute the cardinality of such a routing.

We make several technical assumptions that help to simplify the presentation.  First, we modify the input instance as follows. We subdivide every edge incident on a node in $R$ by introducing a single new node on this edge.  Note that this yields an instance equivalent to the input instance.  As a result, every neighbor of a node in $R$ that lies in $F$, that is, every node in $N_G(R)$, is a leaf in~$F$.
Moreover, the set~$R$ is an independent set in $G$.  Also recall that we assumed that every terminal is a leaf. Therefore, we may assume that $R$ does not contain any terminal.  We also assume that forest $F$ is a rooted tree, by introducing a dummy node (which plays the role of the root) and arbitrarily connecting this node to every connected component of $F$ by an edge.  In our dynamic program, we will take care that no path visits this root node. We also assume that $F$ is an ordered tree by introducing an arbitrary order among the children of each node.

For any node $v$, let $F_v$ be the subtree of $F$ rooted at~$v$.
Let $\ch{v} := \deg_F(v)-1$ be the number of children of~$v$
and let $v_1,\dots v_{\ch{v}}$ be the (ordered) children of~$v$.
Then, for $i=1,\dots,\ch{v}$, let~$F_v^i$ denote the subtree of~$F_v$ induced by the union of~$v$ with the subtrees $F_{v_1},\dots,F_{v_i}$.
For leaves~$v$, we define $F_v^0$ as $F_v=v$.

We introduce a dynamic programming table $T$.
It contains an entry for every~$F_v^i$ and every subset~$\mathcal{M}_e'$ of~$\mathcal{M}_e$.
Roughly speaking, the value of such an entry is the solution to the subproblem, where we restrict the forest to~$F_v^i$, and the set of essential pairs to~$\mathcal{M}_e'$.
More precisely, table~$T$ contains five parameters. Parameters~$v$ and $i$ describing $F_v^i$, parameter~$\mathcal{M}_e'$, and two more parameters~$u$ and~$b$.
Parameter
$u$ is either a terminal,
or a node in $R$,
and~$b$ is in one of the three states: $\textrm{\textit{free}}$, $\textrm{\textit{to-be-used}}$, or $\textrm{\textit{blocked}}$.
The value~$T[v,i,\mathcal{M}_e',u,b]$ is the maximum cardinality of a set $\mathcal{P}_F$ of paths with the following properties:
\begin{enumerate}
\item\vspace{-0.5em} $\mathcal{P}_F$ is a feasible routing of some subset of $\overline{\mathcal{M}}_R$.
\item $\mathcal{P}_F$ is completely contained in $F_v^i$.
\item There is an additional set $\mathcal{P}_e$ of paths with the following properties:
\begin{enumerate}
\item $\mathcal{P}_e$ is completely contained in $F_v^i\cup R$ and node-disjoint from the paths in $\mathcal{P}_F$.
\item $\mathcal{P}_e$ is a realization of $\mathcal{M}_e'\cup\{(u,v)\}$ if $b=\textrm{\textit{to-be-used}}$.
Else, it is a realization of~$\mathcal{M}_e'$.
\item There is no path in $\mathcal{P}_e\cup\mathcal{P}_F$ visiting $v$ if $b=\textrm{\textit{free}}$.
\end{enumerate}
\end{enumerate}
If no such set $\mathcal{P}_F$ exists then $T[v,i,\mathcal{M}_e',u,b]$ is $-\infty$.

Note that the parameter~$u$ is only relevant when $b=\textrm{\textit{to-be-used}}$ (otherwise, it can just be ignored).
Observe that~$T[v,i,\mathcal{M}_e',u,\textrm{\textit{blocked}}]\ge T[v,i,\mathcal{M}_e',u,\textrm{\textit{free}}]\ge T[v,i,\mathcal{M}_e',u,\textrm{\textit{to-be-used}}]$.
Below, we describe how to compute the entries of $T$ in a bottom-up manner.

In the base case $v$ is a leaf. We set $T[v,0,\emptyset,u,\textrm{\textit{free}}]=0$.
Then we set~$T[v,0,\mathcal{M}_e',u,\textrm{\textit{blocked}}]=0$ if $\mathcal{M}_e'$ is either empty, consists of a single pair of nodes in~$R\cap N_G(v)$,
or consists of a single pair where one node is~$v$ and the other one is in~$R\cap N_G(v)$.
Finally, we set $T[v,0,\emptyset,u,\textrm{\textit{to-be-used}}]=0$ if $u=v$ or $u$ is in~$R\cap N_G(v)$.
For all other cases where $v$ is a leaf, we set $T[v,i,\mathcal{M}_e',u,b]=-\infty$.

For the inductive step, we consider the two cases $i=1$ and $i>1$.
Let~$i=1$. It holds that~$T[v,1,\mathcal{M}_e',u,\textrm{\textit{to-be-used}}]=T[v_1,\ch{v},\mathcal{M}_e',u,\textrm{\textit{to-be-used}}]$ since the path in $\mathcal{P}_e$ realizing $(u,v)$ has to start at a leaf node of $F_{v_1}$.
It also holds that $T[v,1,\mathcal{M}_e',u,\textrm{\textit{blocked}}]$ and $T[v,1,\mathcal{M}_e',u,\textrm{\textit{free}}]$ are
equal to~$T[v_1,\ch{v},\mathcal{M}_e',u,\textrm{\textit{blocked}}]$.

Now, let $i>1$. In a high level view, we guess which part of $\mathcal{M}_e'$ is realized in $F_{v}^{i-1}\cup R$ and which part is realized in $F_{v_i}\cup R$.
For this,
we consider every tuple~$(\mathcal{M}_{e1}',\mathcal{M}_{e2}')$ such that $\mathcal{M}_{e1}' \uplus \mathcal{M}_{e2}'$ is a partition of~$\mathcal{M}_e'$. By our dynamic programming table, we find a tuple that maximizes our objective.
In the following, we assume that we guessed~$(\mathcal{M}_{e1}',\mathcal{M}_{e2}')$ correctly. Let us consider the different cases of $b$ in more detail.

For $b=\textrm{\textit{free}}$, node $v$ is not allowed to be visited by any path, especially by any path in~$F_{v}^{i-1}\cup R$. Hence,~$T[v,i,\mathcal{M}_e',u,\textrm{\textit{free}}]$ is equal to \[T[v,i-1,\mathcal{M}_{e1}',u,\textrm{\textit{free}}] + T[v_i,\ch{v_i},\mathcal{M}_{e2}',u,\textrm{\textit{blocked}}]~.\]

In the case of $b=\textrm{\textit{to-be-used}}$, we have to realize $(u,v)$ in~$F_{v}^{i}\cup R$. For this, there are two possibilities: Either $(u,v)$ is realized by a path
in~$F_{v}^{i-1}\cup R$, or there is a realizing path that first goes through~$F_{v_i}\cup R$ and then reaches $v$ via the edge $(v_i, v)$.
Hence, for the first case, we consider \[T[v,i-1,\mathcal{M}_{e1}',u,\textrm{\textit{to-be-used}}] + T[v_i,\ch{v_i},\mathcal{M}_{e2}',u,\textrm{\textit{blocked}}],\]
for the second case, we consider \[T[v,i-1,\mathcal{M}_{e1}',u,\textrm{\textit{free}}] + T[v_i,\ch{v_i},\mathcal{M}_{e2}',u,\textrm{\textit{to-be-used}}]~.\]
Maximizing over both,
we obtain~$T[v,i,\mathcal{M}_e',u,\textrm{\textit{to-be-used}}]$.

For the case of $b=\textrm{\textit{blocked}}$, we will consider two subcases.
In the first subcase, there is no path in~$\mathcal{P}_e\cup\mathcal{P}_F$ going through edge $(v_i,v)$, hence, we get
\[T[v,i-1,\mathcal{M}_{e1}',u,\textrm{\textit{blocked}}] + T[v_i,\ch{v_i},\mathcal{M}_{e2}',u,\textrm{\textit{blocked}}]~.\]
In the second subcase, there is a path~$P$ in $\mathcal{P}_e\cup\mathcal{P}_F$ going through edge $(v_i,v)$.
Since $P$ is connecting two leafs in~$F_{v}^{i}$, a part of $P$ is in~$F_{v}^{i-1}\cup R$ and the other part is in~$F_{v_i}\cup R$.
If $P\in\mathcal{P}_e$, then it is realizing a pair of~$\mathcal{M}_{e}'$. Hence, for every pair~$(u_1,u_2)\in\mathcal{M}_{e}'$, we have to consider the term
\[T[v,i-1,\mathcal{M}_{e1}'-(u_1,u_2),u_1,\textrm{\textit{to-be-used}}] + T[v_i,\ch{v_i},\mathcal{M}_{e2}'-(u_1,u_2),u_2,\textrm{\textit{to-be-used}}]\]
and the symmetric term where we swap $u_1$ and $u_2$.
If $P\in\mathcal{P}_F$, then it is realizing a terminal pair of~$\overline{\mathcal{M}}_R$. Hence, for every pair~$(u_1,u_2)\in\overline{\mathcal{M}}_R$ we get the term
\[1+ T[v,i-1,\mathcal{M}_{e1}',u_1,\textrm{\textit{to-be-used}}] + T[v_i,\ch{v_i},\mathcal{M}_{e2}',u_2,\textrm{\textit{to-be-used}}]\]
and the symmetric term where we swap $u_1$ and $u_2$. Note that we count the path realizing~$(u_1,u_2)$ in our objective.
Maximizing over all the terms of the two subcases, we obtain~$T[v,i,\mathcal{M}_e',u,\textrm{\textit{to-be-used}}]$.

Let us analyze the run time of algorithm described in Sect.~\ref{sec:exact-ndp}.
In order to guess $\mathcal{M}_e$, we enumerate all potential sets of essential pairs.
There are at most $(2k+r+1)^{2r}$ candidate sets to consider, since each pair contains a node in~$R$, and each node in $R$ is paired with at most two other nodes each of which is either a terminal or another node in~$R$.
For each particular guess $\mathcal{M}_e$, we run the above dynamic program.
The number of entries in $T$---as specified by the five parameters $v$, $i$, $\mathcal{M}_e', u$ and $b$---for each fixed $\mathcal{M}_e$ is at most~$(\sum_{v\in V(F)}\deg_F(v))\times 2^{2r}\times(2k+r)\times 3$.
For the computation of each such entry, we consider all combinations of at most~$2^{2r}$ partitions of $\mathcal{M}_e'$ with either at most $r$ essential pairs in~$\mathcal{M}_e'$, or with at most~$k$ terminal pairs in~$\overline{\mathcal{M}}_R$.  Altogether, this gives a run time of~$(8k+8r)^{2r+2}\cdot \mathcal O(n)$.
This finishes the proof of Theorem~\ref{thm:maxndp-fpt-kr}.

\section{Parameterized Intractability of {\sc MaxNDP} for Parameter $r$}
\label{sec:ndp-w1hard-r}
In this section we show that {\sc MaxNDP} is $\mathsf{W}[1]$-hard parameterized by the size~$r$ of a feedback vertex set.
This reduction was originally devised for parameter treedepth, by Ene et al.~\cite{EneEtAl2016}; here we notice that the same reduction also works for parameter $r$.
(Both treedepth and feedback vertex set number are restrictions of treewidth, but they are incomparable to each other.)

For sake of completeness, we include the reduction here, and argue about the feedback vertex set number of the reduced graph.
The reduction is from the~$W[1]$-hard \textsc{Multicolored Clique} problem~\cite{FellowsEtAl2009}, where given a graph
$G$, an integer~$k$, and a partition $V = V^1 \uplus V^2 \uplus \ldots \uplus V^k$,
we are to check if there exists $k$-clique in $G$ with exactly one
vertex in every set~$V^i$. By adding dummy vertices, we can assume that $|V^i| = n$ for every $i=1,\hdots,k$, and that $n, k \geq 2$.

\medskip
\noindent
\textbf{Construction.}
Given an instance $(G,k,(V^i)_{i=1}^k)$ of \textsc{Multicolored Clique}, we aim at constructing an equivalent instance $(H,\mathcal M,\ell)$ of {\sc MaxNDP}.

We start with a construction, for every set $V^i$, a gadget $W^i$ as follows.
First, for every $v \in V^i$ we construct a $(k-1)$-vertex path $X_v^i$ on vertices~$x_{v,1}^i, x_{v,2}^i, \ldots,  x_{v,i-1}^i, x_{v,i+1}^i, \ldots, x_{v,k}^i$.
Second, we select an arbitrary vertex~$u^i \in V_i$.
Third, for every $v \in V^i \setminus \{u^i\}$, we add a vertex~$s^i_v$ adjacent to the first vertex of~$X_v^i$ (i.e., $x_{v,1}^i$ and $x_{u^i,1}^i$ if $i > 1$ or $x_{v,2}^i$ and $x_{u^1,2}^i$ if $i=1$),
  a vertex~$t^i_v$ adjacent to the last vertex of~$X_v^i$ (i.e., $x_{v,k}^i$ and $x_{u^i,k}^i$ if $i < k$ or~$x_{v,k-1}^i$ and $x_{u^i,k-1}^i$ if $i=k$), and make $(s^i_v,t^i_v)$ a terminal pair.
This concludes the description of the gadget $W^i$. By $\mathcal M_{st}$ we denote the set of terminal pairs constructed in this step.

To encode adjacencies in $G$, we proceed as follows. For every pair $1 \leq i < j \leq k$, we add a vertex $p_{i,j}$, adjacent
to all vertices $x_{v,j}^i$ for $v \in V_i$ and all vertices $x_{u,i}^j$ for $u \in V_j$. For every edge $vu \in E(G)$ with $v \in V_i$ and $u \in V_j$,
we add a terminal pair $(x_{v,j}^i, x_{u,i}^j)$. Let $\mathcal M_x$ be the set of terminal pairs constructed in this step; we have $\mathcal M = \mathcal M_{st} \cup \mathcal M_x$.

Finally, we set the required number of paths $\ell := k(n-1) + \binom{k}{2}$. This concludes the description of the instance $(H,\mathcal M,\ell)$.

\medskip
\noindent
\textbf{From a clique to disjoint paths.}
Assume that the input \textsc{Multicolored Clique} instance is a ``yes''-instance, and let $\{v^i~|~i = 1,\hdots,k\}$ be a clique in $G$ with $v^i \in V^i$ for $i = 1,\hdots,k$.
We construct a family of~$\ell$ vertex-disjoint paths as follows. First, for $i = 1,\hdots,k$ and every $v \in V^i \setminus \{u^i\}$, we route a path from~$s^i_v$ to $t^i_v$
through the path $X_v^i$ if $v \neq v^i$, and through the path $X_{u^i}^i$ if $v = v^i$. Note that in this step we have created~$k(n-1)$ vertex-disjoint paths
connecting terminal pairs, and in every gadget $W^i$ the only unused vertices are vertices on the path~$X_{v^i}^i$.
To construct the remaining $\binom{k}{2}$ paths, for every pair $1 \leq i < j \leq k$ we take the $3$-vertex path from $x_{v^i,j}^i$ to $x_{v^j,i}^j$ through $p_{i,j}$;
note that the assumption that $v^iv^j \in E(G)$ ensures that $(x_{v^i,j}^i, x_{v^j,i}^j)$ is indeed a terminal pair in $\mathcal M$.

\medskip
\noindent
\textbf{From disjoint paths to a clique.}
In the other direction, let $\mathcal P$ be a family of $\ell$ vertex-disjoint paths connecting terminal pairs in $H$.
Let $\mathcal P_{st} \subseteq \mathcal P$ be the set of paths connecting terminal pairs from $\mathcal M_{st}$, and similarly define $\mathcal P_x$.
First, observe that the set $P = \{p_{i,j}~|~1 \leq i < j \leq k\}$ separates every terminal pair from~$\mathcal M_x$. Hence, every path from $\mathcal P_x$ contains at least one vertex from $P$.
Since $|P| = \binom{k}{2}$, we have~$|\mathcal M_x| \leq \binom{k}{2}$, and, consequently,~$|\mathcal P_{st}| \geq \ell - \binom{k}{2} = k(n-1) = |\mathcal M_{st}|$.
We infer that $\mathcal P_{st}$ routes all terminal pairs in $\mathcal M_{st}$ without using any vertex of $P$, while $\mathcal P_x$ routes $\binom{k}{2}$ pairs from $\mathcal P_x$, and every path from $\mathcal P_x$ contains exactly one vertex from $P$.

Since the paths in $\mathcal P_{st}$ cannot use any vertex in $P$, every such path needs to be contained inside one gadget~$W^i$. Furthermore, observe that a shortest path between terminals $s_{v,a}^i$ and $t_{v,a}^i$ inside $W^i$ is either~$X_{u^i}^i$ or~$X_v^i$,
      prolonged with the terminals at endpoints, and thus contains $k+1$ vertices.
Furthermore, a shortest path between two terminals in $\mathcal M_x$ contains three vertices. We infer that the total number of vertices on paths in $\mathcal P$ is at least
\begin{align*}
|\mathcal P_{st}| \cdot (k+1) + |\mathcal P_x| \cdot 3 &= k(n-1)(k+1) + 3\binom{k}{2}\\
&= k\left(n(k-1) + 2(n-1)\right) + \binom{k}{2} = |V(H)| \enspace .
\end{align*}
We infer that every path in $\mathcal P_{st}$ consists of $k+1$ vertices, and every path in $\mathcal P_x$ consists of three vertices.
In particular, for $i = 1,\hdots,k$ and $v \in V^i \setminus \{u^i\}$, the path in $\mathcal P_{st}$ that connects $s_v^i$ and $t_v^i$ goes either through~$X_v^i$ or~$X_{u^i}^i$.
Consequently, for $i = 1,\hdots, k$ there exists a vertex $v^i \in V^i$ such that the vertices of~$W^i$ that do not lie on any path from $\mathcal P_{st}$ are exactly the vertices on the path $X_{v^i}^i$.

We claim that $\{v^i~|~i =1,\hdots,k\}$ is a clique in $G$. To this end, consider a pair $1 \leq i < j \leq k$.
Since~$|\mathcal P_x| = \binom{k}{2}$, there exists a path in $\mathcal P_x$ that goes through~$p_{i,j}$. Moreover, this path has exactly three vertices. Since the only neighbours of $p_{i,j}$ that are not used
by paths from $\mathcal P_{st}$ are $x_{v^i,j}^i$ and $x_{v^j,i}^j$, we infer that $(x_{v^i,j}^i, x_{v^j,i}^j) \in \mathcal M$ and, consequently, $v^iv^j \in E(G)$. This concludes the proof of the correctness of the construction.

\medskip
\noindent
\textbf{Bounding the feedback vertex set number.}
We are left with a proof that $H$ has bounded feedback vertex set number.

To this end, first observe that $H-P$ contains $k$ connected components, being the gadgets~$W^i$. Second, observe that the deletion of the endpoints of the path~$X_{u^i}^i$ from the gadget $W^i$ breaks $W^i$ into connected components being paths on at most $k+1$ vertices.
Consequently,~$H$ has a feedback vertex set~$R$ consisting of~$P$ and $\{x_{u_i,1}^i,x_{u_i,k}^i\in V(W^i)~|~i=1,\hdots,k\}$, of size $|R| = \mathcal O(k^2)$.
This finishes the proof of Theorem~\ref{thm:ndp-w1hard-r}. \qed

\section{Hardness of Edge-Disjoint Paths in Almost-Forests}
\label{sec:hardnessofedpinalmostforests}
In this section we show that {\sc EDP} (and hence {\sc MaxEDP}) is $\mathsf{NP}$-hard already in graphs that are almost forests, namely, in graphs that are forests after deleting two nodes.
That is, we prove Theorem~\ref{thm:edp-nphard-r2}.

\begin{proof}[Proof of Theorem~\ref{thm:edp-nphard-r2}]
  We first show $\mathsf {NP}$-hardness of EDP for $r=2$.
  We reduce from the problem {\sc Edge 3-Coloring} in cubic graphs, which is $\mathsf{NP}$-hard~\cite{Holyer1981}.
  Given a cubic graph~$H$, we construct a complete bipartite graph $G$, where one of the two partite classes of $V(G)$ consists of three nodes $\{v_1,v_2,v_3\}$, and the other partite class consists of $V(H)$.
  As terminal pairs, we create the set~${\mathcal M = \{(s,t)~|~\{s,t\}\in E(H)\}}$; in words, we want to connect a pair of nodes by a path in $G$ if and only if they are connected by an edge in~$H$.
  This completes the construction of the instance $(G,\mathcal M)$ of {\sc MaxEDP}.
  Notice that $G$ has a feedback vertex set of size $r = 2$, since removing any size-2 subset of $\{v_1,v_2,v_3\}$ from $G$ yields a forest.

  Regarding correctness of the reduction, we show that $H$ is 3-edge-colorable if and only if \emph{all} pairs in~$\mathcal M$ can be routed in $G$.

  In the forward direction, suppose that $H$ is 3-edge-colorable.
  Let~${\varphi:E(H)\rightarrow\{1,2,3\}}$ be a proper 3-edge-coloring of $H$.
  For $c = 1,2,3$, let~${E_c\subseteq E(H)}$ be the set of edges that receive color~$c$ under $\varphi$.
  Then there is a routing in $G$ that routes all terminal pairs $\{(s,t)\in\mathcal M~|~\{s,t\}\in E_c\}$ exclusively via the node~$v_c$ (and thus via paths of length~2).
  Notice that this routing indeed yields edge-disjoint paths, for if there are distinct vertices~$s,t_1,t_2\in V(H)$ and edges $e_1 = \{s,t_1\},e_2 = \{s,t_2\}\in E(H)$, then $e_1,e_2$ receive distinct colors under~$\varphi$ (as $\varphi$ is proper), and so the two terminal pairs $\{s,t_1\},\{s,t_2\}$ are routed via distinct nodes~$c_1,c_2\in\{v_1,v_2,v_3\}$, and thus also via edge-disjoint paths.

  In the backward direction, suppose that all terminal pairs in $\mathcal M$ can be routed in~$G$.  Since~$H$ is cubic, any node $s\in V(H)$ is contained in three terminal pairs.  Therefore, no path of the routing can have a node in~$V(H)$ as an internal node and thus all paths in the routing have length~2.
  Then this routing naturally corresponds to a proper 3-edge-coloring $\varphi$ of~$H$, where any terminal pair $\{s,t\}$ routed via $c$ means that we color the edge $\{s,t\}\in E(H)$ with color ~$c$ under $\varphi$.

  \medskip
In order two show $\mathsf{NP}$-hardness of \textsc{MaxEDP} for $r=1$, we also reduce from \textsc{Edge 3-Coloring} in cubic graphs and perform a similar construction as described above: This time, we construct a bipartite graph $G$ with one subset of the partition being~$\{v_1,v_2\}$, the other being $V(H)$, and the set $\mathcal{M}$ of terminal pairs being again specified by the edges of $H$.  This completes the reduction. The resulting graph $G$ has a feedback vertex set of size $r=1$.

We claim that $H$ is 3-colorable if and only if we can route $n=|V(H)|$ pairs in $G$.

In the forward direction, suppose that $H$ is 3-edge-colorable.
Let~${\varphi:E(H)\rightarrow\{1,2,3\}}$ be a proper 3-edge-coloring of $H$.
For $c = 1,2,3$, let~${E_c\subseteq E(H)}$ be the set of edges that
receive color $c$ under $\varphi$.  Then there is a routing in $G$
that routes all f $\{(s,t)\in\mathcal M~|~\{s,t\}\in E_c\}$
exclusively via the node $v_c$ (and thus via paths of length~2) for
the colors $c=1,2$. (The terminals corresponding to edges receiving
color 3 remain unrouted.)

The reasoning that the resulting routing is feasible is analogous to the case of~$r=2$.  Since for each of the~$n$ terminals exactly two of the three terminal pairs are routed, this means that precisely $n$ terminal pairs are routed overall.

In the backward direction, suppose that $n$ terminal pairs in $\mathcal M$ can be routed in~$G$.
Since any terminal~$v$ in $G$ is a node in $V(H)$ has therefore has degree two in $G$, this means that at most two paths can be routed for $v$.  As~$n$ terminal pairs are realized, this also means that \emph{exactly} two paths are routed for each terminal.  Hence, none of the paths in the routing has length more than two. Otherwise, it would contain an internal node in~$V(H)$, which then could not be part of two other paths in the routing.
Then this routing naturally corresponds to a partial edge-coloring of~$H$, where any terminal pair $\{s,t\}$ routed via $c$ means that we color the edge $\{s,t\}\in E(H)$ with color $c$.  Since each terminal $v$ in $V(H)$ is involved in exactly two paths in the routing, exactly one terminal pair for $v$ remains unrouted. Hence, exactly one edge incident on~$v$ in $H$ remains uncolored in the partial coloring.  We color all uncolored edges in~$H$ by color 3 to obtain a proper~3-coloring.
\end{proof}

Thus, we almost close the complexity gap for {\sc EDP} with respect to the size of a minimum feedback vertex set, only leaving the complexity of the case $r = 1$ open.
We conjecture that this case can be solved in polynomial time.

redpajama_set_name: RedPajamaArXiv


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