Chondromalacia mostly occurs in the sesamoid joints such as the knee joint (chondromalacia patellae) or the first metatarsal phalangeal joint (Chondromalacia sesamoid).
Chondromalacia patellae is the most common form of chondromalacia.
Mal-alignment of the tendons and bones in the affected joint due to quadricep muscle weakness may contribute to altered biomechanical function; ultimately cartilage and joint damage is the result.
Cartilage or joint damage such as osteo-arthritis, may contribute to the joint pain; however, chondromalacia may occur in younger athletes.
Being overweight or obese can contribute to increased pressure on the joints during activity.
Poor foot function, such as collapsing arches, may result in altered biomechanics. This may cause cartilage degeneration in the affected joints.
A generalised deep pain in the knees; either, below, behind or on the sides of the knee cap.
Swelling of the knee joint.
Clinical assessment involving physical palpation of the area, muscle and joint assessment.
Full biomechanical assessment involving gait analysis.
Full medical history and pain history may be required to rule out other diagnoses.
Orthotics: If the chondromalacia is biomechanically related, Interpod orthotics may be used to re-align the foot and improve biomechanics to relieve symptoms during weight bearing and activity.
Footwear: Supportive and cushioned footwear can maintain the functioning of your Interpod orthotics.
Taping: Taping or strapping can be used to maintain joint alignment during activity.
Rest: Resting from aggravating activities may be advised by your practitioner.
Pain medication: Using NSAIDs such as ibuprofen as advised by your practitioner for temporary pain relief.
Muscle exercises: Muscle exercises may be prescribed by your practitioner to improve the strength of muscles surrounding the affected joint.
Weight loss: maintaining a healthy body weight may relieve excess pressure on the joints.
Surgery: Once all other treatment options have been exhausted, then surgery may be an option.
Orthotics: Continue to wear your Interpod orthotics for work and exercise to maintain function of your feet and to prevent biomechanical abnormalities contributing to chondromalacia.
Footwear: Continue to wear appropriate supportive shoes during activity to maximise the function of your Interpod orthotics and prevent excessive movement of the joints in your feet.
BMI: Maintain a healthy body weight to reduce excess pressure on your feet and knees.
Frowen, O'Donnell, Lorimer & Burrow. (2010). Neale's Disorders of the foot (8th Ed.), Elsevier Limited, pgs. 319 & 356-357.


text:
Eglantyne Mary Jebb (1889–1978), MA, the Third Principal of Froebel College (1932–1955)
Henry Marvell Carr	(1894–1970)
Froebel College
© the copyright holder. Photo credit: University of Roehampton
H 80.5 x W 67.5 cm
FR ART 03
Chair (4,611)
Dress (2,427)
Hair (5,934)
Necklace (875)
Portrait (4,031)
Principal (25)
Red Dress (49)
Royal Academician (25,921)
Silvery (7)
Woman (8,848)
Eglantyne Jebb (1)
Eglantyne Mary Jebb (1)
Froebel College (3)
Roehampton (1)
Roehampton Lane, London, Greater London SW15 5PJ England
This venue is closed to the public.
Henry Marvell Carr
A 3.7 Anti-Aircraft Gun of 393/72 Heavy Anti-Aircraft Regiment, RA, CMF IWM (Imperial War Museums)
Eglantyne Mary Jebb (1889–1978), MA, the Third Principal of Froebel College (1932–1955) Froebel College
Paul Crombie, Clerk of the Court of Alderney (1967–1975) Alderney Court House
Council of the College, 1946–1947 Hunterian Museum
General Sir Harold Franklyn, KCB, DSO, MC, Colonel of the Regiment (1939–1949) The Green Howards Regimental Museum
View in Rome, Italy Russell-Cotes Art Gallery & Museum
Brigadier James Noel Tetley (1898–1971), DSO, TD, DL, LLD, Pro-Chancellor of the University of Leeds (1956–1965) The Stanley & Audrey Burton Gallery, University of Leeds
Hairstyles, cosmetics and body art

redpajama_set_name: RedPajamaCommonCrawl


text:
London production of 'The Phantom of the Opera' announces new cast
The show will welcome two West End legends to play the lead role on separate stages of the run, as well as a new official and alternate Christine. With music by Andrew Lloyd Webber, the piece tells the unfortunate tale of a disfigured genius who insanely falls in love with a young beautiful ingenue.
adam linstead, all i ask of you, andrew lloyd webber, andrew york, angel of music, Anouk Van Laake, Connor Carson, Corina Clark, earl carpenter, Edward Court, Eilish Harmon-Beglan, Emma Harris, Eve Shanu-Wilson, Grace Hume, greg castiglioni, her majesty's theatre, Hollie Aires, holly anne hull, Hywel Dowsell, Jacob Hughes, James Gant, James Hume, Jasmine Wallis, jean valjean, jon robyns, kelly glyptis, Lily De-La-Haye, Lizzie Wofford, masquerade, matt blaker, matt harrop, Michael Robert-Lowe, Michelle Cornelius, music of the night, musical theatre, musicals, Olivia Holland-Rose, paige blankson, Ralph Watts, Serina Faull, Simon Whitaker, Skye Weiss, the phantom of the opera, think of me, Thomas Holdsworth, Tim Morgan, Tim Southgate, west end, Yukina Hasebe
'El Fantasma de la Ópera' publica nuevas imágenes de la producción londinense y extiende periodo de reserva
El espectáculo ha lanzado una nueva serie de imágenes en las que aparecen tanto los nuevos miembros del reparto como los actuales, así como su impresionante escenografía. Con música de Andrew Lloyd Webber y coreografías de Gillian Lynne, el musical cuenta la trágica historia de un genio deforme locamente enamorado de una joven soprano cuyo […]
adam linstead, all i ask of you, andrew lloyd webber, angel of music, cameron mackintosh, chandelier, charles hart, christine daae, covent garden, drama, ellie young, francesca ellis, gillian lynne, greg castiglioni, haymarket, her majesty's theatre, holly anne hull, kelly glyptis, killian donnelly, london, london production, love, love story, lucy st louis, masquerade, matt blaker, matt harrop, music of the night, opera, phantom, phantom of the opera, picadilly circus, point of no return, raoul, richard stilgoe, soho, think of me, unrequitted love, west end
'The Phantom of the Opera' releases new London production images and extends booking
The show has launched a new set of pictures featuring both both new cast and ongoing cast members as well as its impressive set design. With music by Andrew Lloyd Webber and coreographies by Gillian Lynne, the musical tells the tragic story of a deformed genius madly in love with a young soprano whose heart […]
adam linstead, all i ask of you, andrew lloyd webber, angel of music, cameron mackintosh, chandelier, charles hart, christine daae, covent garden, drama, ellie young, francesca ellis, gillian lynne, greg castiglioni, haymarket, her majesty's theatre, holly anne hull, kelly glyptis, killian donnelly, london, london production, love, love story, lucy st louis, masquerade, matt blaker, matt harrop, music of the night, opera, phantom, phantom of the opera, picadilly circus, point of no return, raoul, richard stilgoe, soho, think of me, unrequitted love, west end


text:
Fraser Los
Writing, Editing and Communications
The Lingering Legacy of the Nathan E. Stewart
fraser / April 18, 2017
April 2017: Hakai Magazine
In the waning light of November 14, 2016, the tugboat Nathan E. Stewart was finally lifted from the chilly water off British Columbia's Athlone Island—32 days after it ran aground on its way south from Alaska.
When the tug struck the rocks on Edge Reef shortly after 1:00 a.m. on October 13, it was pushing a 10,000-tonne-capacity fuel barge. The barge was empty, but the tug still leaked about 110,000 liters of diesel and more than 2,200 liters of engine lubricants into the waters near Bella Bella, a town of 1,450 people in the traditional territory of the Heiltsuk First Nation. Strong winds and high tides pushed the fuel across Seaforth Channel and into Gale Creek, the site of an ancient Heiltsuk village, polluting critical marine habitats and harvesting sites.
"It was horrific—seeing the ocean full of diesel, and that sickening odor," recalls Jordan Wilson, a Heiltsuk coastal guardian watchman, and one of the first on the scene. "All the hydraulic oils and engine oils—everything was just oozing out of that boat for weeks."
With each tide, the toxins washed higher up on shorelines of surrounding islands. "That's where I'd go to harvest food to share with my family and friends," adds Wilson, "but not anymore."
For many people outside the community, it was just another media story—a snapshot of a tragic event from a remote place. For locals, the spill lives on in myriad ways, with authorities still feeling the effects of a months-long drain on already strapped financial and human resources, and fishers unable to harvest manila clams and other species that may have been contaminated.
April 18, 2017 in Environment, Transportation, Water. Tags: First Nations, Heiltsuk Nation, marine management, oil spill
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As retailers pulled their merchandise from shelves, co-founders Domenico Dolce and Stefano Gabbana went on camera to apologize to the Chinese people.
Nov. 26, 2018 7:30 a.m.
The company blamed hackers for the anti-Chinese insults, but the explanation felt flat to many and the damage was done. The Milan designers cancelled the Shanghai runway show, meant as a tribute to China, as their guest list of Asian celebrities quickly joined the protests.
London-based Brand Finance consultancy estimates that the scandal could wipe up to 20 per cent off the Dolce&Gabbana brand's value of $937 million, which already places it out of the top 50 global apparel brands. They are 36th among all Italian brands, extending to sectors like automotive.
That could shake Dolce&Gabbana's financial health. The privately held company does not release its individual sales figures. But Chinese consumers are responsible for a third of all luxury spending around the globe, according to a recent study by Bain consultancy. That will grow to 46 per cent of forecast sales of an estimated 365 billion euros ($412 billion) by 2025, fueled by millennials and the younger Generation Z set, who will make a growing percentage of their purchases online.


China rice bran oil brand Arawana drew hundreds of shoppers at its launch in Japan.
China rice bran oil brand Arawana has made its first move into the ultra-competitive Japan market with its first supermarket launch.
Its debut was accompanied by huge advertisements on skyscrapers in Tokyo's Shibuya Crossing, and a grand product launch event attracted hundreds of shoppers at the "il gusto Dolce Vita" store.
This is quite an unusual phenomenon as, while Japan is a major consumer of rice bran oil, its market is dominated by domestic brands.
Japan is known to be the originator of rice bran oil, with a century of history.
At the International Conference on Rice Bran Oil (ICRBO) 2016 in Tokyo, Arawana competed with several rice bran oil brands from Japan, India, Vietnam and Thailand, and won the "Quality Award". With this exposure and recognition, word about Arawana rice bran oil soon spread, and even Japanese students were reportedly buying Arawana rice bran oil from China.
Arawana is China's leading brand of edible oil, whose parent company Wilmar International Limited ranks number 239 in the Fortune 500 List 2016. As one of the world's largest grain and oil groups, Wilmar has an annual revenue of almost US$41.4b.
Arawana started to focus on rice bran oil in 2006. The R&D team created "enzymatic degumming", a patented technology of low-temperature extraction, to naturally reserve oryzanol and phytosterol — two core nutrients of rice bran oil.
The average content of both oryzanol and phytosterol have been measured be 10,000 mg/kg, which is identified to be a good balance between nutrient reservation and quality. Arawana rice bran oil is, therefore, known for its quality, colour and flavour.
In the not too distant future, Arawana will further join hands with Japanese retailers to expand in the market. It plans to sell its products in 10,000 supermarkets in Tokyo and Osaka within two years.
Japan consumes 80,000 tons of rice bran oil every year, displaying huge consumption potential. This market potential is one of the main reasons for Arawana to compete in the market with dominant domestic brands.


text:
Blog / News / What Is Seltzer Water in The UK and Where Can You Buy it?...
What Is Seltzer Water in The UK and Where Can You Buy it?
To put it simply, seltzer is just carbonated water. So in other words, water with a bit of a kick. Just like sparkling water, seltzer water is natural spring water that's infused with CO2 under pressure to make it fizzy. Sparkling water, carbonated water and seltzer are terms that can be used interchangeably, with seltzer water being more commonly used in the US, and sparkling water favoured in the UK.
Seltzers and sparkling water are often flavoured with fruits to give them added flavour, making them a healthy and refreshing alternative to soda. DASH infuses all of its sparkling water with real, wonky fruit, making it the perfect refreshing pick me up throughout the day.
Benefits of Seltzer Water
Because it's made with natural ingredients, there are many benefits to seltzer water:
It's hydrating - Seltzer water is made with natural spring water, so it hydrates you the way a regular glass of still water would
No calories, sugar or sweetener - Many seltzer waters, like DASH, don't contain calories, sugar or sweetener, making it a healthier option
It's a great substitute for fizzy drinks - Seltzer waters still allow you to get your fizz fix, while avoiding any nasty additives
It's a perfect soft drink to serve at a party - Seltzer water and sparkling water can feel like an elegant option, which is why they're a popular soft drink to serve at weddings
Diabetes friendly - Seltzer waters that are free from sugars and calories can be drank by people with diabetes
What Are The Differences Between Normal Seltzer Water and Hard Seltzer Water ?
The difference between normal seltzer water and hard seltzer water is pretty straightforward. Hard seltzer water is just normal seltzer water that's been mixed with alcohol for an extra buzz. Hard seltzers are an excellent alternative to other alcoholic drinks as they are usually lower in calories, lower in sugar and more hydrating than the likes of beer or wine. It's also often lower in alcohol than other drinks, and is usually safe to drink for people with specific diets such as vegan.
Hard Seltzer Recipes
If you've got an occasion coming up and you're looking for a lighter way to drink alcohol, hard seltzer is definitely something to try out. There are plenty of pre-made hard seltzers you can buy in the supermarkets, but it can be way more fun and personal to make your own at home. And because seltzer is just like sparkling water, you can use DASH drinks to make them, too. We've included a few recipes to give you some inspiration down below.
Cucumber-lime hard seltzer
Light and refreshing, a cucumber-lime hard seltzer is the perfect tipple for a summer's day.
- 3 slices of cucumber
- 1 handful of ice cubes
- Juice of a whole lime
- 25 ml of vodka or another spirit of your choice
- 1 can of DASH cucumber sparkling water
Grab a glass and fill it with a handful of ice cubes and your garnish of 3 slices of cucumber. Pour in the 25 ml of vodka, then squeeze in the juice of a full lime. To finish, pour in a cold can of DASH cucumber sparkling water, stir and enjoy.
Raspberry gin hard seltzer
Fruity and delicious, A homemade raspberry gin seltzer is the perfect alternative to a cocktail!
- 1 handful of raspberries
- 1 slice of lime
- 25 ml of gin
- 1 can of DASH raspberry sparkling water
Grab a glass and pop your handful of raspberries in the bottom. Muddle them up until they form a puree, this helps intensify the flavour of the drink. Once this is done, pour in a handful of ice cubes and add in the 25 ml of gin. Top off with a can of DASH raspberry sparkling water and a squeeze of fresh lime.
Lemon & Basil Hard Seltzer
Citrusy with a twist, this homemade lemon and basil hard seltzer will definitely pack a punch!
- Ice
- 2 fresh basil leaves
- 25 ml of vodka
- 1 Lemon flavoured can of DASH sparkling water
- 1 lemon
Take your glass and pop in your two fresh basil leaves, then crush them up slightly in the bottom. Add your ice, 25 ml of vodka and can of lemon flavoured DASH sparkling water and stir. Finish it off with a slice of fresh lemon and get sipping!
Where To Buy Seltzer Water In The UK?
There are plenty of places where you can buy seltzer water or sparkling water in the UK. For example, DASH sparkling water is readily available to buy in bundles online, or you can even start a subscription and receive 20% off with each purchase. Bundles are also available in a variety of different flavours, whether you're after a range to try from or you're just looking for a restock on your favourite. You can purchase DASH sparkling water in popular UK supermarkets, too, including Tesco and Waitrose.


text:
.class public Lcom/samsung/android/settings/devicevisibility/DeviceVisibilityReceiver;
.super Landroid/content/BroadcastReceiver;
.source "DeviceVisibilityReceiver.java"

# direct methods
.method public constructor <init>()V
    .locals 0

    invoke-direct {p0}, Landroid/content/BroadcastReceiver;-><init>()V

    return-void
.end method

# virtual methods
.method public onReceive(Landroid/content/Context;Landroid/content/Intent;)V
    .locals 8

    const/4 v7, 0x2

    const/4 v5, 0x1

    const/4 v6, 0x0

    if-eqz p2, :cond_0

    if-nez p1, :cond_2

    :cond_0
    const-string/jumbo v4, "DeviceVisibilityReceiver"

    if-nez p2, :cond_1

    const-string/jumbo v3, "intent is null"

    :goto_0
    invoke-static {v4, v3}, Landroid/util/secutil/Log;->secD(Ljava/lang/String;Ljava/lang/String;)I

    return-void

    :cond_1
    const-string/jumbo v3, "context is null"

    goto :goto_0

    :cond_2
    invoke-virtual {p2}, Landroid/content/Intent;->getAction()Ljava/lang/String;

    move-result-object v0

    invoke-virtual {p1}, Landroid/content/Context;->getContentResolver()Landroid/content/ContentResolver;

    move-result-object v3

    invoke-static {v3}, Lcom/samsung/android/settings/devicevisibility/DeviceVisibilityUtils;->getDBInt(Landroid/content/ContentResolver;)I

    move-result v1

    const-string/jumbo v3, "android.intent.action.AIRPLANE_MODE"

    invoke-virtual {v0, v3}, Ljava/lang/String;->equals(Ljava/lang/Object;)Z

    move-result v3

    if-eqz v3, :cond_5

    const-string/jumbo v3, "DeviceVisibilityReceiver"

    const-string/jumbo v4, "AIRPLANE_MODE_CHANGED"

    invoke-static {v3, v4}, Landroid/util/secutil/Log;->secD(Ljava/lang/String;Ljava/lang/String;)I

    invoke-static {p1}, Lcom/samsung/android/settings/devicevisibility/DeviceVisibilityUtils;->getAirPlaneModeStatus(Landroid/content/Context;)I

    move-result v3

    if-ne v3, v5, :cond_4

    if-ne v1, v5, :cond_3

    invoke-static {p1, v7, v6}, Lcom/samsung/android/settings/devicevisibility/DeviceVisibilityUtils;->setDBInt(Landroid/content/Context;ILjava/lang/String;)V

    :cond_3
    :goto_1
    return-void

    :cond_4
    if-ne v1, v7, :cond_3

    invoke-static {p1, v5, v6}, Lcom/samsung/android/settings/devicevisibility/DeviceVisibilityUtils;->setDBInt(Landroid/content/Context;ILjava/lang/String;)V

    goto :goto_1

    :cond_5
    const-string/jumbo v3, "com.samsung.android.oneconnect.DEVICE_VISIBILITY_STATE_CHAGED"

    invoke-virtual {v0, v3}, Ljava/lang/String;->equals(Ljava/lang/Object;)Z

    move-result v3

    if-eqz v3, :cond_3

    const-string/jumbo v3, "DEVICE_VISIBILITY_STATE_VALUE"

    invoke-virtual {p2, v3, v1}, Landroid/content/Intent;->getIntExtra(Ljava/lang/String;I)I

    move-result v2

    const-string/jumbo v3, "DeviceVisibilityReceiver"

    new-instance v4, Ljava/lang/StringBuilder;

    invoke-direct {v4}, Ljava/lang/StringBuilder;-><init>()V

    const-string/jumbo v5, "INTENT_DEVICE_VISIBILITY_STATE_CHANGED state + "

    invoke-virtual {v4, v5}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;

    move-result-object v4

    invoke-virtual {v4, v2}, Ljava/lang/StringBuilder;->append(I)Ljava/lang/StringBuilder;

    move-result-object v4

    invoke-virtual {v4}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;

    move-result-object v4

    invoke-static {v3, v4}, Landroid/util/secutil/Log;->v(Ljava/lang/String;Ljava/lang/String;)I

    invoke-static {p1, v2, v6}, Lcom/samsung/android/settings/devicevisibility/DeviceVisibilityUtils;->setDBInt(Landroid/content/Context;ILjava/lang/String;)V

    goto :goto_1
.end method

redpajama_set_name: RedPajamaGithub


sounds fun. any thoughts on bringing back the splitboard category for this year?
Dan – This year we are going to be focussing our attentions on building out the 3 day event and we will be keeping the divisions the same as last year. these divisions closely match the divisions of other skimo races our the west.


If anyone besides me remembers the #midterms, we did not vote to trash and burn our nation. We voted to fix it. But as usual. Power Corrupts and Absolute Power Corrupts Absolutely Without compromise we might as well just blow up the whole world right now.
The #midterms is my personal #fave when POTUS dismantles the talk show host.
Actually - YOU lost control of Congress when you got #Crushed in the #midterms after 2 years of inept and cruel leadership. Bye Donnie. Off to prison after your sycophants cant protect you anymore in 2020. Get Him @SpeakerPelosi !


On Thursday the 11th of January, newly-formed alternative band, KOKO hit the stage of The Exchange in their local city of Bristol.
Last year we caught up with Harry Lee Dobson to talk about his solo music career and his new band KOKO. We've been tracking the band's movements ever since. When we were invited to spend the day with them ahead of their debut gig in Bristol, we jumped at the offer.
Lead guitarist, Danny Clark backstage.
Pre-stage grooming with lead vocalist and rhythm guitarist, Ollie Garland.
Smoke break at The Exchange.
A quick photograph before taking stage: Harry Lee Dobson, Ollie Garland and Danny Clark.
With no music yet released, they have a diverse range of songs already under their belt, allowing the boys to well and truly own the night – performing all of their original music and getting the packed-out venue going!
Supporting acts included Bristol band, Summoned by Ashes and London-based band, The Gold.
"It was an amazing night and the crowd was insane! The energy of our performance really hit the crowd and got everyone going," added Danny Clark, KOKO's lead guitarist.
KOKO's bassist and backing vocalist, Harry Lee Dobson.
KOKO, lighting up The Exchange.
With their first single set to be released within the coming months and already creating a stir in the industry, these lads are going be the ones to watch out for this year.
All photographs by Chloe Dunscombe.


The fifth edition of the Dawson Daily News Print & Publishing Symposium again brought together an eclectic mixture of talented artists employing a wide range of styles and techniques for printmaking and content creation. With a hands-on, artisanal, and instructional approach, the workshops and open studios ranged from the traditional to the avant garde, printing on a variety of media in many unique ways, and covering the print and publishing craft from inception of ideas to final packaging of completed works.
This year the Symposium kicked off on Thursday evening with the first-ever Dawson Daily News Book Fair, a marketplace of books and publications, featuring original works from local artists and authors, who were also available to sign their work. Justin Apperly launched his art book From Dust to Oaxaca, featuring a photo essay of his six-month long motorcycle journey from Dawson to Mexico and back. There were also custom zines from Dawson artists, and books (printed and digital) from resident authors.
Mathias Macphee with a final silkscreened print.
On Saturday the Symposium proper kicked off. This year, in addition to the open studio projects and workshops inside the heritage Dawson Daily News building, there were numerous activities taking place outside in the adjoining lot under canopies and tents. Fortunately the weather (which had decidedly been on the soggy side in recent weeks) cooperated splendidly and delivered sunshine for the weekend.
In this outside area, Mary Tremonte gave hands-on demonstrations of silk screening throughout the weekend, squeeging her custom hand-drawn designs onto bandanas, as well as other occasional articles of clothing. Hung from the fringe of the canopy, the bandanas could be seen flapping like flags when the breeze stirred. Perhaps this is what drew the attention of the many kids who showed up to try their hand at the process. There was another trio of other notable silk screening art projects taking place as well. Mathias Macphee and the Silkscreen club demonstrated the production of exquisite multi-coloured prints that required multiple ink passes and meticulous calibration, while artists Michael MacLean and Elizabeth Xu worked together to execute Mountain Landscape Pattern, a large, collaborative, fabric-print installation.
The sun was more than just a meteorological bystander, it was also an active component in the work of Bill Horne. The B.C. artist demonstrated photo-stencil exposing in order to produce silk screens for printing onto various media, utilizing an enclosed "dim room" tent and an array of cool-looking custom hardware in the process.
As part of the start-to-finish scope of the symposium's topics, Nathaniel Marchand demonstrated the craft of paper making. Adding a definitive Yukon flavour, Marchand used the fibre from fireweed plants for the pulp in the beautifully-textured paper sheets he produced over the weekend.
Bii Horne tests the exposure for a photo stencil.
Inside the building, Sierra Megas gave a workshop in introductory bookbinding, introducing participants to the hand-binding of books using the Coptic stitch. Using samples of different finished binds as reference, workshop attendees were themselves able to try stitching with hands-on guidance. It was double duty for Megas, whose art exhibition Lost & Found also graced the walls of the Dawson Daily News building's front room.
The Symposium has been fortunate in the past to draw on the skills of the Writers-in-Residence from Berton House, and this year was no exception. On Saturday afternoon current resident Claire Caldwell offered a Poetry Lab to help poets, aspiring and seasoned alike, flex their lyrical muscles. The workshop employed a series of poetic experiments "to transform the raw ingredients of language into bright and bubbling concoctions." Judging from the attendance and enthusiasm, Dawson is home to a legion of aspiring poets.
Sierra Megas (standing) oversees the book binding workshop.
In the back of the building, among the vintage presses, another visiting artist, Claire Kujundzic, had an open studio throughout the weekend demonstrating collography. This process employs a collage of materials glued on to a "printing plate," usually made from a square of cardboard. When the glue is dry, it is shellacked or varnished so that it can be painted with acrylics. Paper is then pressed on to the surface to produce a print.
Two artists, one local and one visiting, who have been continuously involved with the Symposium since its inception, were back for the fifth edition. Master Printer Peter Braune from New Leaf Editions, has been the driving force behind the restoration of the vintage Chandler & Press letter press that forms the centrepiece of the event. A versatile and seasoned artisan, Braune offered letterpress and etching demonstrations as well as hands-on guidance to artists and drop-in visitors alike throughout the weekend.
Claire Kujundzic prepares materials for her collography workshop.
Dawson stalwart John Steins held an open studio in wood-block printing throughout the weekend. When not painstakingly carving away at his latest creation, Steins showed visitors other beautiful wood blocks he has produced in the past, as well as the prints made from some of them.
While the digital age was certainly not ignored (for example, Dawson Writer-at-Large Dan Dowhal gave a workshop in indie publishing employing free and easy-to-use online services like Amazon's Create Space and Lulu.com) it is the unique handcrafted works that continue to shine and capture the imagination of visitors to the Dawson Daily News Print & Publishing Symposium. Mass production will always have its place, but the Symposium demonstrates the irrepressible and irreplaceable role of artists, writers, and skilled craftspeople.
Taking a break from his papermaking workshop, Nathaniel Marchand helps the Silkscreen Club.
Master printer Peter Braune prepares a copper plate for engraving.
Basking in the glorious sunshine the weekend offered, John Steins works on carving a wood block used for printing.


text:
Q: How to call function after setState updated(callback) I want to setVoting before setVote then run sendVote. However anyway it run sendVote first. How can i solve it? I use React Hook... so it is difficult. Please help me!!! Thank you so much.
    <TouchableOpacity onPress={() => {setVoting(!voting); setVote(voting?vote-1:vote+1); sendVote();}}>
       <Ionicons
        name= {voting?'md-star':'md-star-outline'}
        size={24}
        color={'#aa4fff'}
       />
    </TouchableOpacity>

A: try the following:
useEffect(()=>{
   setVote(voting?vote-1:vote+1);
},[voting]);

useEffect(()=>{
   sendVote();
},[vote]);

...then remove the setVote and sendVote from onPress:
<TouchableOpacity onPress={() => setVoting(!voting)}>
       <Ionicons
        name= {voting?'md-star':'md-star-outline'}
        size={24}
        color={'#aa4fff'}
       />
</TouchableOpacity>

redpajama_set_name: RedPajamaStackExchange


Political season is well upon us in the United States (for my International readers, both of you, you might find this interesting as well) and one of the tried-and-true keys to developing a message in a political/issues campaign quickly is to use a message matrix.
1. What can your opponents/activists/competition say about your organization/product/etc?
2. What can your opponents/activists/competition say about themselves?
3. What can your organization say about itself/product/etc.?
4. What can your organization say about your opponents/activists/competition?
When you take the time to sit down and look at your matrix, this is the time you need to take a critical look at your organization and your competition. Be honest with your organization's strengths and weakness, and develop your potential messages accordingly. As a part of the message box, you might want to conduct a SWOT and/or PESTLE analysis, depending on your time.
If you look at your competition or activist groups in advance, a message matrix can help you organize your thoughts as you prepare questions, responses and talking points before you need them. It's always good to show the executive leadership you're prepared in advance for any media problems they might run into. It can also help you determine those messages, talking points, and possible responses on the fly if you need to.
This is just an overview of the message box, we'll get into closer detail soon enough. But keep in mind, these messages will probably need to be vetted by your legal department, executive leadership, etc.
Where Does PR Go From Here?
Public Relations has a PR problem.
And what do we in PR do about it?
I remember when I finally received my degree in PR and advertising. Not more than thirty minutes after commencement I was talking with my dad, who congratulated me on finally getting the degree and then dropped a verbal pipe bomb of his own on me.
Ladies and gentlemen, we have to step up and take back what it is public relations professionals do for a living. We have allowed this to happen. Years of not pointing out the weak actions of public relations professionals have allowed the media, and those activists who love to attack us as corporate liars, to frame us as unethical, lying bastards.
But these same people never want to be held up to the same standard of professionalism. The irony in this is that the same members of the media and activists (right or left, anti-corporate or not) who decry public relations as craven "spin masters" and liars who are busy "telling people what to think" use many of the same tactics created by public relations practitioners throughout the years on their own supporters. If we're going to hold PR people up to a predisposed view, then these detractors need to be held to the same standard.
It is because of public relations' poor image that those professionals whose work is exemplary, people who should be held up as examples of ethical practitioners with a strong work ethic are immediately under suspicion of being "just another flack." It's true that there are poor PR professionals, those who haven't been trained in PR (yes, a successful PR pro is trained beyond just the regular view of "I'm a writer" or "I'm a people person"), just like there are unethical journalists, or bad teachers, or misleading activist groups.
We are rarely brought into the upper echelon decision making, because we're not only seen as, but want to promote ourselves as, just "writers." According to a 1999 white paper by crisis management expert Jim Lukaszewski, a management consultant explained to public relations professionals why they would rarely be able to address the concerns of the C-Suite. And it was a shock to many of the PR professionals in the audience. For all of our talk about the importance of communicating and writing in public relations, few professionals are able to show the C-Suite the impact that we have on those numbers that are important to them. Finances, sales, increase in profits.
I give PRSA some credit for trying to improve the professionalism of practitioners around the country, with the APR program and now the organization's new MBA initiative, tying public relations closer to business schools (where it needs to be). This is only the start of the necessary push to improve public relations' image. It took us a long time to dig into this hole, it's going to take us a long time to get out of it.


text:
Q: GAE datastore queries sometimes not working after changing entity I'm trying to figure out an issue that I'm seeing. I'm noticing that when I change an entity object definition by adding a new field for instance, sometimes I no longer can get results from previous records that already exist in the datastore. For instance, in this example below, I can see the entity in the datastore, but I'm getting this exception:
SEVERE: No results for query: SELECT FROM com.alero.services.model.MyEntity myEntity WHERE myEntity.key = :key
I'm using JPA with the GAE datastore.
The only way I seem to be able to correct the problem and start getting results again is to delete the appengine-generated directory and then recreate the entities. Is it being cached in some manner and is there something I should be doing to allow the datastore to know that it needs to remove the cache if this is the case?
        String queryString = "select from " + MyEntity.class.getName()
                + " myentity where myentity.key = :key";

        Key keySub = KeyFactory.createKey("MyEntity", myEntityId);

        Query query = em.createQuery(queryString);
        query.setParameter("key", KeyFactory.keyToString(keySub));

        try {
            MyEntity myEntity = (MyEntity) query.getSingleResult();
            em.close();
            return myEntity;
        } catch (Exception e) {
            log.severe(e.getMessage());
            em.close();
        }

A: Actually I discovered the issue. It wasn't really related to me changing my entity. It was how I was persisting and merging my entities. I have an unowned relationship between two entities in an unowned bidirectional relationship and I was adding the references to each other before persisting my entities. This was creating an owned relationship. Therefore when I was querying with my key, I wasn't finding it because the actual key on the entity was a child key when I was trying to query it as a parent key.
So I need to persist my two entities before adding the relationships to each other in the bidirectional relationship. Then I can merge them. This creates a parent key for each entity and then I can create my relationship between the bidirectional relationship.


"I've never had a campaign where the entire nation has tried to destroy my name," he told the Houston Chronicle. "They are going after me in the most personal and vindictive way. It's gutter politics."
Aaaawwww, poor muffin. Tom De Lay complaining about gutter politics. The poor dear.


If you look carefully at the photos one picture shows the zipper 'open' and the guidelines are long enabling the tarp to be up off the ground for more ventilation and for more interior space. Another photo shows it zipped 'down' and the guidelines are supershort…. This is what I call "Storm Mode"…less room, more protection.
This tarp cost $330… They make the same tarp out of silica nylon which weighs 13 ounces and costs $165…. Speaking in round numbers, half the price, twice the weight and bulk.
Thanks for the review – this was definitely helpful for me!
Thanks for the review, just come back off the Tour Aotearoa and realized that my 1.6 kg tent was to big and way too heavy. So just order one so with luck will be here in a couple of weeks, just wondering how low the front comes down from the ground? Just in the picture it looks pretty high but the SMD pictures look like its level all round. Have you used it since and are your thoughts still positive.
The Top & Bottom Photos show it with the zipper "up" …also it is pitched a few inches up off the ground… the middle photo show it with the zipper "down" and tight to the ground…a.k.a storm mode…. if it were really bad I would advise clipping on a extra "door" …ummm…an extra big Tyvek ground cloth with one side closing the 'door' yet big enough for your pad plus bag too…….NOTE: this is the Original version…I'm pretty sure they made a Newer Version with either (1) a longer zipper (2) bigger vestibule beak…..Extra Extra NOTE: I just purchased a Mountain Laurel Designs "Trail Star" in Cuben…It is 4 oz. heavier (12 oz. total)….It is a big Pentagon with which you do "Origami" and can pitch it "small & tight for a storm….or…HUGE and open for nice weather….Super Versatile ….More protection for Epic Expeditions….Google it!….Great simple design.
The photo's are of an older model (the same as mine). The newer version that you have seen on the website comes much lower to the ground and the zip goes nearly all the way to the apex. Hope this helps.
I purchased the standard (silica nylon) Deschutes tarp. After over a year and a number of trips, no regrets. The Deschutes tarp is great with a bivy and sleeping bag. Good protection, easy to erect and packs up small. I strap their carbon fibre pole to my top tube.


— Translation: But it is rather true that the people, in so far as this term signifies a special part of the citizens, does not know what it wants.
Es macht das Volk sich auch mit Worten Lust.
— The rabble also vent their rage in words.


Telomeres form the distal parts of eukaryotic chromosomes and protect chromosomal ends from degradation and end-to-end fusion. Human telomerase reverse transcriptase (TERT) gene encodes a ribonucleoprotein enzyme that is able to extend chromosome ends that have been shortened during successive cycles of cell division. Genome-wide association studies (GWAS) associated 5p15.33, the genomic region harboring TERT, with susceptibility to several cancers, supporting the important role of telomerase in carcinogenesis.
Because genetic variation in TERT and its regulatory elements may influence cancer susceptibility, we have investigated single nucleotide polymorphisms (SNPs) in TERT as well as a polymorphic tandem repeats minisatellite of TERT termed MNS16A. We found significant associations of MNS16A variable number of tandem repeats (VNTRs) in association with colorectal cancer as well as prostate cancer. We have identified two novel variants of MNS16A, namely VNTR-364 and VNTR-212 in addition to the four previously reported VNTRs and are under way to investigate the functionality of these variants. Furthermore, we intend to determine telomere length in our three cohorts using a quantitative PCR assay for measurement of telomere length and their association with genetic variants.We aim to broadly characterize the TERT-CLPTM1L region in regard of common SNPs and minisatellites in order to assess their relevance for cancer etiology and to elucidate the functional mechanisms underlying the associations of TERT polymorphisms with cancer.


Tank Water Specialists, Tank Cleaning Services, listed under "Tank Cleaning Services" category, is located at 3 Fairview St Moonee Beach NSW, 2450, Australia and can be reached by 0266564411 phone number. Tank Water Specialists has currently 0 reviews.
Browse all Tank Cleaning Services in Moonee Beach NSW.


A few weeks ago, I was tasked with finding some stock photos of cars for a senior designer. I found myself on a website that was particularly tricky to navigate, so I did what I always do: ask said senior designer (read: someone with a lot of experience) for some help. Much to my relief, he also found the website supremely confusing. We spent around 20 minutes trying to find a very specific photo (spoiler alert: we never did find the photo we needed).
The footage was that of your typical car ad, except the "LGBTQ" version didn't show any people. At all. It was 30 seconds of a car speeding down a road at multiple different angles. Overhead. Wide shot. Closeup. Not a single person. Not one. You'd be forgiven for thinking it was an ad for one of those Tesla self-driving cars.
When the ad ended we both looked at each other with something combination of exasperation, disbelief, and amusement.
Answer: Nothing. Absolutely nothing at all.
And, honestly, that's usually been my experience with trying to find myself in advertising.
My whole life has been spent searching hours on end to find something, anything I can relate to, only to find that people would much rather look at empty cars, or empty houses, or empty… whatevers, than see someone like me.
So, how does all that exposition explain what I think about LGBTQ+ representation in advertising?
Well, I find that it's often spineless, if it exists at all.
I mean, yes. I live in California, and I definitely do see the occasional ad that is out loud and proud (so to speak). But that's only if I travel several hours from the conservative center of the state where I live to somewhere like San Francisco or LA. And those ads aren't doing anything for anyone anywhere else.
Sure, the rainbows come out every June, and there's some cute hashtag like #loveislove or #loveisequal or something with the word "love" and/or "pride" in it, but I'm not only bisexual in June.
I'm bisexual the rest of the year, too.
And why? Because we're afraid of losing customers? Because we're afraid of alienating people? Because we're afraid of the backlash?
Remember that Cheerios ad with the cute interracial family? That caused backlash. As recently as five years ago people were upset over a cereal commercial because it featured an interracial family.
Advertising is, yes, meant to sell things. But, it's also a reflection of —and an influencer of—culture. Just as much as any other form of entertainment.
Advertising gave us the modern Santa Claus, convinced women to start shaving, and told America that a diamond was the only way to propose. If we can change public opinion on things as banal as razors and diamonds, why not do it for people?
Why won't we use our powers for good? Just this once?
Representation normalizes people, humanizes them. And it just feels good to see yourself on TV.
Look, I'm used to not being represented at all, no matter what medium. I'm mixed race, I'm bisexual, and I like weird things like Ukrainian pop singers and 90s cult movies. My name is Kiku, for God's sake. My entire life has been some variation of that Key and Peele substitute teacher skit (you know the one).
So, yeah, I'm used to being alone in this world. Maybe that's part of the reason I decided to become part of the great mass media machine.
But until that day comes (if it comes at all), the day when I become a real adult and my heart dies, I'm going to wear it on my sleeve and put it in my work. And I hope that someone, somewhere, will follow me.
There's still a little part of me that wants to make the world a better place. And since I'll never be able to cure Cancer, the least I can hope to do is make someone's life a little bit easier than mine was and let them see themselves on screen.
Billy Bean—the only openly gay Major League Baseball player alive today—has a message for us all about the costs of hiding who you are & who you love.
If you don't have the right people on the team, you're in danger of producing work that is both ego and ethnocentric.


Robust and strong machines supplied by ourselves for use by Bodybuilding Champions and fitness professional alike. The plate loading series adopts perfect ergonomic design and utilizes the best acr and angles engineered for human exercise physiology. set-up-dimension:1610x1490x2100 mm. net weight:131 kg gross weight: 166 kg.


text:
\section{Introduction}\label{sec:intro}
The \textit{Minimum-Cost Edge-Disjoint Path} (EDP) problem is a classical network design problem, defined as follows.
Given an edge-weighted graph $G=(V,E)$, two terminals $s,t\in V$ and an integer parameter $k\in \mathbb{Z}_+$, find
$k$ edge-disjoint paths connecting $s$ and $t$ with minimum total cost. EDP is motivated by the following survivable network
design problem: what is the connection cost of two nodes in a network, given that any $k-1$ edges can be a-posteriori removed
from the graph. The implicit assumption in EDP is that every edge in the graph in equally vulnerable. Unfortunately, this
assumption is unrealistic in many applications, hence resulting in overly-conservative solutions.
This paper studies a natural refinement of the EDP problem called the \textit{Fault-Tolerant Path Problem} (FTP), formally
defined as follows.

\begin{framed}
\textbf{Instance: } An edge-weighted graph $G=(V,E)$, two nodes $s,t\in V$, a subset of the edges $M\subset E$ and an integer
$k\in \mathbb{Z}_+$.

\textbf{Problem: } Find minimum cost $S \subset E$ such that $S \setminus F$ contains an $s$-$t$ path for every $F\subset M$ with
$|F| \leq k$.
\end{framed}

Note that FTP becomes EDP when $M = E$. We let $\Omega(M,k)$ denote the set of possible faults, namely $\Omega(M,k) = \{A\subset M: |A| \leq k\}$.

A well-known polynomial algorithm for EDP works as follows. Assign unit capacities to all edges in $G$ and find a minimum-cost
$k$-flow from $s$ to $t$. The integrality property of the \textit{Minimum-Cost Flow} (MCF) problem guarantees that an
extreme-point optimal solution is integral, hence it corresponds to a subset of edges. It is then straightforward to verify that
this set is an optimal solution of the EDP problem (for a thorough treatment of this method we refer to the book of Schrijver~\cite{Schrijver}).

The latter algorithm raises two immediate questions concerning FTP. The first question is whether FTP admits a polynomial
time algorithm. In this paper we give a negative answer to this question conditioned on P$\neq$NP, showing that FTP is NP-hard.
In fact, the existence of constant-factor approximation algorithms is unlikely already for the restricted case of directed
acyclic graphs. Consequently, it is natural to ask whether polynomial algorithms can be obtained for restricted variants of FTP.
We devote the first part of this paper to this question.

The second question
concerns the natural fractional relaxation of FTP. As we previously observed, one natural relaxation of EDP is the
MCF problem. This relaxation admits an integrality gap of one, namely the optimal integral solution value is always equal to
the corresponding optimal fractional value. In the context of FTP it is natural to study the following fractional relaxation,
which we denote by \textit{Fractional FTP} (FRAC-FTP).

\begin{framed}
\textbf{Instance: } An edge weighted graph $G=(V,E)$, two nodes $s,t\in V$, a subset of the edges $M\subset E$ and an integer
$k\in \mathbb{Z}_+$.

\textbf{Problem: } Find a minimum cost capacity vector $x: E \rightarrow [0,1]$ such that for every $F\in \Omega(M,k)$, the
maximum $s$-$t$ flow in $G_F = (V, E\setminus F)$, capacitated by $x$, is at least one.
\end{framed}

It is obvious that by further requiring $x \in \{0,1\}^E$ we obtain the FTP problem, thus FRAC-FTP is indeed a fractional
relaxation of FTP. We later show that the integrality gap with respect to this relaxation is bounded by $k+1$. Furthermore, we show that
this bound is tight, namely that there exists an infinite family of instances with integrality gap arbitrarily close to $k+1$.
This result also leads to a simple $(k+1)$-approximation algorithm for FTP, which we later combine with an
algorithm for the case $k=1$ to obtain a $k$-approximation algorithm.

The remainder of the paper is organized as follows. In the following section we review some related work.
In Section~\ref{sec:complexity} we study the complexity of FTP.
In Section~\ref{sec:exact} we provide some exact polynomial algorithms for special cases of FTP. In
Section~\ref{sec:intgap} we relate FTP and FRAC-FTP by proving a tight bound on the interagrality gap and
show how this result leads to a $k$-approximation algorithm for FTP. In Section~\ref{sec:conclusions} we
conclude the paper and mention some interesting directions for future work.

\section{Related Work}\label{sec:related}

The shortest path problem was studied in numerous robust settings
(\cite{RSP,CompRSP,DemandR,DemandR2,MinMaxRegretSP,RRSP,ExactSubgraphSP}). Yu and Yang~\cite{RSP}
studied the shortest path problem with a discrete scenario set in the minimax and minimax-regret models.
They show that both problems are NP-Hard even for $2$ scenarios and layered graphs.
Aissi et al.~\cite{MinMaxRegretSP} extended these results by showing that both version of the
problem admit an FPTAS if the number of scenarios is bounded. They also show that both problems are
not approximable within $2-\epsilon$ for any $\epsilon > 0$ with an unbounded number of scenarios unless P$=$NP.
Adjiashvili et al.~\cite{Adjiashvili} studied a two-stage feasibility problem
in which for a given pair of terminals $s$ and $t$ in a graph $G=(V,E)$, and two parameters $f,r\in \mathbb{Z}_+$, the goal
is to choose a minimum-cardinality subset of the edges $S\subset E$ with the property that for every failure set
$F\subset E$ with $|F| \leq f$ there exists a recovery set $R\subset E \setminus F$ with $|R|\leq r$,
such that $s$ and $t$ are connected in $(S \setminus F) \cup R$. The authors give an exact algorithm for
$f=r=1$ and a $2$-approximation algorithm for $f=1$ and arbitrary $r$. Also some complexity results
are provided.
Puhl~\cite{RRSP} studied the recoverable-robust shortest path problem, a robust path problem in a
two-stage optimization setting. Hardness results were provided for discrete scenario sets, interval
scenarios and $\Gamma$-scenarios for two variants of cost function. For one variant an approximation
algorithm is given.
B\"using~\cite{ExactSubgraphSP}
developed approximation algorithms and proved hardness results for the problem of finding the smallest
number of edges in the graph which contains a shortest path according to every scenario.

The robustness model proposed in this paper is natural for various classical combinatorial optimization
problems. Of particular interest is the counterpart of the Minimum Spanning Tree problem. This problem
is closely related to the Minimum $k$-Edge Connected Spanning Subgraph (ECSS) problem, a well-understood
robust connection problem.
Gabow, Goemans, Tardos and Williamson~\cite{kEdgeConnected1}
developed a polynomial time $(1 + \frac{c}{k})$-approximation algorithm for ECSS, for some fixed
constant $c$. The authors also show that for some constant $c' < c$, the existence of a polynomial time
$(1 + \frac{c'}{k})$-approximation algorithm implies P$=$NP. An intriguing property of ECSS is
that the problem becomes easier to approximate when $k$ grows. Concretely, while for every fixed $k$,
ECSS is NP-hard to approximate within some factor $\alpha_k>1$, the latter result asserts that there
is function $\beta(k)$ tending
to one as $k$ tends to infinity such that ECSS is approximable within a factor $\beta(k)$.
This phenomenon was already discovered by
Cheriyan and Thurimella~\cite{CheriyanThurimella}, who gave algorithms with a weaker approximation guarantee.
The more general Generalized Steiner Network problem admits a polynomial $2$-approximation algorithm due to
Jain~\cite{Jain}.

We review next some of the related research in robust combinatorial optimization.
Liebchen et al.~\cite{RRFirst} proposed the notion of \emph{recoverable robustness} as a tractable
framework for robustness in railway optimization. The authors present
a general recipe for obtaining two-stage robust problems from any combinatorial optimization
problem. In~\cite{AdjRC} Ben-Tal et al.~proposed a method to delay some decisions in an
uncertain convex optimization problem to the stage at which the real data is revealed. Their
approach does not force some decision variables $X$ to be fixed as first-stage
decision variables. Instead, these variables are allowed to be associated in the first stage decision
with affine functions $X(\zeta)$ of the uncertain input vector $\zeta$. The resulting formulation
turns out to be theoretically and practically tractable in many cases. The framework of Bertsimas and
Caramanis~\cite{FiniteadAptability} allows to deal with discrete changes in the
second stage decision in the special case of linear optimization.

Kouvalis and Yu~\cite{RDOBook} provide an extensive study of complexity and algorithms for robust
counterparts of many classical discrete optimization. Instead of resource removal, these models
assume uncertainty in final cost of the resources.
Bertsimas and Sim~\cite{RDO} proposed a framework for incorporating robustness in discrete optimization.
They show that robust counterparts of mixed integer linear programs with uncertainty in the matrix coefficients
can be solved using moderately larger mixed integer linear programs with no uncertainty.

A two-stage robust model called\textit{demand-robust optimization} was introduced by
Dhamdhere et al.~\cite{DemandR}. In this model a scenario corresponds to a subset
of the initially given constraints that need to be satisfied. For shortest
path this means that the source node and the target node are not fixed, but are rather
revealed in the second stage. The authors provide approximation
algorithms for some problems such as Steiner Tree, Multi-Cut, facility Location etc.~in the
demand-robust setting. Golovin et al.~\cite{DemandR2} later improved some of those results,
including a constant factor approximation algorithm for the robust
shortest path problem. Feige et al.~\cite{ExpSce1} and Khandekar et al.~\cite{ExpSce2} studied
an extension of the demand-robust model that allows exponentially many scenarios.

\section{Complexity of FTP}\label{sec:complexity}

Our first observation is that FTP generalizes the \textit{Directed $m$-Steiner Tree Problem} ($m$-DST). The input
to $m$-DST is a weighted directed graph $G = (V,E)$, a \textit{source} node $s\in V$, a collections of \textit{terminals}
$T\subset V$ and an integer $m \leq |T|$. The goal is to find a minimum-cost arboresence $A \subset E$ rooted at $s$, that contains a
directed path from $s$ to some subset of $m$ terminal.

The $m$-DST is seen to be a special case of FTP as follows. Given an instance $I = (G, s,T,m)$ of $m$-DST
define the following instance of FTP. The graph $G$ is augmented by $|T|$ new zero-cost directed edges $E'$
connecting every terminal $u\in T$ to a new node $t$. Finally, we set $M = E'$ and $k = m-1$.
The goal is to find a fault-tolerant path from $s$ to $t$ in the new graph. It is now straightforward to see
that a solution $S$ to the FTP instance is feasible if and only if $S\cap E$ contains a feasible solution
to the $k$-DST problem (we can assume that all edges in $E'$ are in any solution to the FTP instance).

The latter observation implies an immediate conditional lower bound on the approximability of FTP. Halperin
and Krauthgamer~\cite{HalperinKrauthgamer} showed that $m$-DST cannot be approximated within a factor
$\log ^{2-\epsilon} m$ for every $\epsilon > 0$, unless NP $\subset$ ZTIME($n^{polylog(n)}$). As a result
we obtain the following.

\begin{proposition}\label{prop:complexity}
For every $\epsilon > 0$ FTP admits no polynomial approximation algorithms with ratio $\log ^{2-\epsilon} k$,
unless NP $\subset$ ZTIME($n^{polylog(n)}$).
\end{proposition}

\begin{remark}\label{rem:alg:mneart}
The reduction above can be easily adapted to obtain a $k^\epsilon$-approximation algorithm for
FTP for the special case that $M \subset \{e\in E: t\in e\}$ using the algorithm of Charikar et. al.~\cite{CharikarChekuriETC}.
\end{remark}

In fact, any approximation algorithm with factor $\rho(k)$ for FTP is an approximation algorithm with factor $\rho(m)$
for $m$-DST. The best known algorithm for the latter problem is due to Charikar et. al.~\cite{CharikarChekuriETC}.
Their result is an approximation scheme attaining the approximation factor of $m^\epsilon$ for every $\epsilon > 0$.

In light of the previous discussion, it seems that obtaining an approximation algorithm for FTP
with a poly-logarithmic approximation guarantee is a challenging task. In fact, although we have
no concrete result to support this possibility, such an algorithm might not exist.
We end this discussion with another
special case of FTP, which we denote by \textit{Simultaneous Directed $m$-Steiner Tree} ($m$-SDST). An
input to $m$-SDST specifies two edge-weighted graphs $G_1 = (V,E_1,w_1)$ and $G_2=(V,E_2,w_2)$ on the same
set of vertices $V$, a source $s$, a set of terminals $T\subset V$ and an integer $m \leq |T|$. The goal
is to find a subset $U\subset T$ of $m$ terminals and two arboresences $S_1\subset E_1$ and $S_2\subset E_2$ connecting
$s$ to $U$ in the respective graphs, so as to minimize $w_1(S_1) + w_2(S_2)$. $m$-SDST is seen to be a special case
of FTP via the following reduction. Given an instance of $m$-SDST, construct a graph $G = (V',E)$ as follows.
Take a disjoint union of $G_1$ and $G_2$, where
the direction of every edge in $G_2$ is reversed. Connect every copy of a terminal $u\in T$ in $G_1$ to its
corresponding copy in $G_2$ with an additional zero-cost edge $e_u$. Finally, set $M = \{e_u : u\in T\}$ and
$k = m-1$. A fault-tolerant path connecting the copy of $s$ in $G_1$ to the copy of $s$ in $G_2$ corresponds
to a feasible solution to the $m$-SDST instance with the same cost, and vise-verse.

\section{Polynomial Special Cases}\label{sec:exact}

This section is concerned with tractable restrictions of FTP. Concretely we give polynomial
algorithms for arbitrary graphs and $k=1$, directed acyclic graphs (DAGs) and fixed $k$ and
for Series-parallel graphs. We denote the problem FTP restricted to instances with some
fixed $k$ by $k$-FTP.

\subsection{$1$-FTP}\label{subsec:kisone}

For simplicity of the exposition we assume that $G$ is undirected. For the adaptation
to directed graphs see Remark~\ref{rem:dadskisone}.
We start with the following useful structural lemma.

\begin{lemma}\label{lem:disjointpaths}
 Let $X^*$ be an optimal solution to FTP on the instance $(G,M,k)$. Let $H' = (V', X')$ be the
graph obtained by contracting all edges of $X^* \setminus M$ in $H^* = (V, X^*)$. Let $s'$ and
$t'$ be the vertices in $V'$, to which $s$ and $t$ were contracted, respectively.
Then either $s' = t'$, or there are $k+1$ edge disjoint $s'$-$t'$ paths in $H'$.
\end{lemma}

\begin{proof}
If $X^* \subset E\setminus M$ then $s' = t'$. Assume this is not the case, namely that
$X^* \cap M \neq \emptyset$. Observe that $H'$ must contain at least $k+1$ edge
disjoint $s'$-$t'$ paths: indeed if there was a cut of cardinality $k$ or less in $H$,
this would also be a cut in $H^*$. Since $X' \subset M$, it corresponds to a failure scenario,
namely $X' \in \Omega(M,k)$.
\end{proof}

\begin{definition}\label{def:ercc:unionpaths}
An $s$-$t$ \textit{bipath} in the graph $G =(V,E)$ is a union of two $s$-$t$ paths $P_1, P_2 \subset E$, such that
for every two nodes $u,v$ incident to both $P_1$ and $P_2$, the order in which they appear on $P_1$ and $P_2$, respectively,
when traversed from $s$ to $t$ is the same.
\end{definition}

In the context of $1$-FTP, we call a bipath $Q=P_1 \cup P_2$ \textit{robust}, if it holds that $P_1 \cap P_2 \cap M = \emptyset$.
Note that every robust $s$-$t$ bipath $Q$ in $G$ is a feasible solution to the $1$-FTP instance.
Indeed, consider any failure edge $e\in M$. Since $e\not\in P_1\cap P_2$ it holds that either $P_1 \subset Q-e$,
or $P_2 \subset Q-e$. It follows that $Q-e$ contains some $s$-$t$ path.
The next lemma shows that every feasible solution of the $1$-FTP instance contains a robust $s$-$t$ bipath.

\begin{lemma}\label{lem:robustbipath_contained}
Every feasible solution $S^*$ to the $1$-FTP contains a robust $s$-$t$ bipath.
\end{lemma}

\begin{proof}
We assume without loss of generality that $S^*$ is a minimal feasible solution with respect
to inclusion. Let $Y \subset S^*$ be the set of bridges in $(V, S^*)$. From feasibility of
$S^*$, we have $Y \cap M = \emptyset$. Consider any $s$-$t$ path $P$ in $S^*$. Let $u_1, \cdots, u_r$ be
be the set of vertices incident to $Y = P\cap Y$. Let $u_i$ and $u_{i+1}$ be such that $u_iu_{i+1} \not\in Y$.
(if such an edge does not exist, we have $Y = P$, which means that $P$ is a robust $s$-$t$ bipath).
Note that $S^*$ must contain two edge-disjoint $u_i$-$u_{i+1}$ paths $L_1, L_2$. Taking as
the set $Y$ together with all such pairs of paths $L_1, L_2$ results in a robust bipath.
\end{proof}

We can conclude from the previous discussion and Lemma~\ref{lem:robustbipath_contained} that all
minimal feasible solutions to the $1$-FTP instance are robust bipaths. This observations leads
to the simple algorithm, which is formally given as Algorithm~\ref{alg:kisone}. We prove its
correctness in the following theorem.

\begin{theorem}\label{thm:kisone}
There is a polynomial algorithm for $1$-FTP.
\end{theorem}

\begin{proof}
To solve $1$-FTP we need to find the minimum cost robust $s$-$t$ bipath. To this end
let us define two length functions $\ell_1, \ell_2 : V\times V \rightarrow \mathbb{R}_+$. For two
vertices $u,v\in V$ let $\ell_1(u,v)$ denote the shortest path distance from $u$ to $v$ in
the graph $(V,E\setminus M)$, and let $\ell_2(u,v)$ denote the cost of the shortest pair of edge-disjoint
$u$-$v$ paths in $G$.
Clearly, both length functions can be computed in polynomial time (e.g. using
flow techniques). Finally, set $\ell(u,v) = \min\{\ell_1(u,v), \ell_2(u,v)\}$.
Construct the complete graph on the vertex set $V$ and associate the length function $\ell$ with it.
Observe that by definition of $\ell$, any $s$-$t$ path in this graph corresponds to a robust
$s$-$t$ bipath with the same cost, and vice versa. It remains to find the shortest $s$-$t$ bipath
by performing a single shortest $s$-$t$ path in the new graph. For every edge $uv$ in this
shortest path, the optimal bipath contains the shortest $u$-$v$ path in $(V,E\setminus M)$ if
$\ell(u,v) = \ell_1(u,v)$, and the shortest pair of $u$-$v$ paths in $G$, otherwise.
\end{proof}

\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}

\begin{algorithm}
\caption{\textbf{: FindBipath($G, s, t, M$)}}\label{alg:kisone}
\begin{algorithmic}[1]
\REQUIRE{$G=(V,E)$ a directed graph, $s,t\in V$ and $M\subset E$.}
\ENSURE{A minimum-cost $s$-$t$ bipath $Q$.}
\STATE{$Q \gets \emptyset.$}
\STATE{$G_{SAFE} \gets (V,E\setminus M).$}
\FORALL{$u,v \in V\times V$}
  \STATE{Compute $\ell_1(u,v)$, the shortest $u$-$v$ path distance in $G_{SAFE}$.}
  \STATE{Compute $\ell_2(u,v)$, the cost of the minimum $u$-$v$ $2$-flow in $G$ with unit capacities.}
  \STATE{$\ell(u,v) \gets \min\{\ell_1(u,v), \ell_2(u,v)\}$.}
\ENDFOR
\STATE{Compute the shortest $s$-$t$ path $P$ in $(V, V\times V)$ and length function $\ell$.}
\FORALL{$uv \in P$}
  \IF{$\ell(u,v) = \ell_1(u,v)$}
    \STATE{Add to $Q$ a shortest $u$-$v$ path in $G_{SAFE}$.}
  \ELSE
    \STATE{Add to $Q$ a shortest edge-disjoint pair of $u$-$v$ paths in $G$.}
  \ENDIF
\ENDFOR
\RETURN{$Q.$}
\end{algorithmic}
\end{algorithm}

\begin{remark}\label{rem:dadskisone}
Let us remark that Algorithm~\ref{alg:kisone} easily extends to work on directed graphs as follows.
Lemma~\ref{lem:disjointpaths} is replaced with the more general statement that the minimum
$s$-$t$ flow in the graph $(V, X^*)$ capacitated by the vector $c_e = 1$ if $e\in M$ and $c_e = \infty$,
otherwise is at least $k+1$. The rest of the analysis remains the same.
\end{remark}

\subsection{$k$-FTP and Directed Acyclic Graphs}\label{subsec:dagfixedk}

Let us first consider the case of a layered graph. The generalization to a
directed acyclic graph is done via a standard transformation, which we describe later.
Recall that a layered graph
$G = (V, E)$ is a graph with a partitioned vertex set $V = V_1 \cup \cdots\cup V_r$
and a set of edges satisfying $E \subset \bigcup_{i\in [r-1]} V_i \times V_{i+1}$.
We assume without loss of generality that $V_1 = \{s\}$ and $V_{r} = \{t\}$. For every $i\in [r-1]$
we let $E_i = E \cap V_i \times V_{i+1}$.

Analogously to the algorithm in the previous section, we reduce $k$-FTP to a shortest
path problem in a larger graph. The following definition sets the stage for the
algorithm.

\begin{definition}\label{def:conf}
 An \emph{$i$-configuration} is a vector $d \in \{0,1,\cdots,k+1\}^{V_i}$
satisfying $\sum_{v\in V_i} d_v = k+1$. We let $supp(d) = \{v\in V_i : d_v > 0\}$.
For an $i$-configuration $d^1$ and an $(i+1)$-configuration $d^2$ we let
$$
V(d^1, d^2) = supp(d^1) \cup supp(d^2) \,\, \text{ and } \,\, E(d^1, d^2) = E[V(d^1, d^2)].
$$

We say that an $i$-configuration $d^1$ \emph{precedes} an $(i+1)$-configuration $d^2$ if the
following flow problem is feasible. The graph is defined as $H(d^1, d^2) = (V(d^1, d^2), E(d^1, d^2))$.
The demand vector $\nu$ and the capacity vector $c$ are given by
\begin{equation*}
\nu_u =
\begin{cases}
-d^1_u    & \mbox{if }  u\in supp(d^1)\\
d^2_u      & \mbox{if }  u\in supp(d^2)
\end{cases} \,\, \text{and} \,\,\,\,
c_e =
\begin{cases}
1    	    & \mbox{if }  e\in M\\
\infty      & \mbox{if } e\in E\setminus M,
\end{cases}
\end{equation*}
respectively. If $d^1$ precedes $d^2$ we say that the \emph{link} $(d^1,d^2)$ exists. Finally, the \emph{cost}
$\ell(d^1,d^2)$ of this link is set to be minimum value $w(E')$ over all $E'\subset E(d^1, d^2)$,
for which the previous flow problem is feasible, when restricted to the set of edges $E'$.
\end{definition}

The algorithm constructs a layered graph $\mathcal{H} = (\mathcal{V}, \mathcal{E})$ with $r$ layers
$\mathcal{V}_1, \cdots, \mathcal{V}_r$. For every $i\in[r]$ the set of
vertices $\mathcal{V}_i$ contains all $i$-configurations. Observe that $\mathcal{V}_1$ and $\mathcal{V}_r$
contain one vertex each, denoted by $c^s$ and $c^t$, respectively. The edges correspond to links between
configurations. Every edge is directed from the configuration with the lower index to the one with the higher index.
The cost is set according to Definition~\ref{def:conf}.
The following lemma provides the required observation, which immediately leads to a polynomial algorithm.

\begin{lemma}
Every $c^s$-$c^t$ path $P$ in $H$ corresponds to a fault-tolerant path $S$ with $w(S) \leq \ell(P)$, and vise-versa.
\end{lemma}

\begin{proof}
Consider first a fault-tolerant path $S \subset E$. We construct a corresponding $c^s$-$c^t$ path in $H$
as follows. Consider any $k+1$ $s$-$t$ flow $f^S$, induced by $S$. Let $p^1, \cdots, p^l$ be a path
decomposition of $f^S$ and let $1 \leq \rho_1, \cdots, \rho_l \leq k+1$ (with $\sum_{i\in [l]} \rho_i = k+1$) be the
corresponding flow values.

Since $G$ is layered, the path $p^j$ contains exactly one vertex $v^j_i$ from $V_i$ and one edge $e^j_i$
from $E_i$ for every $j\in [l]$ and $i\in [r]$. For every $i\in[r]$ define the $i$-configuration $d^i$ with
$$
d^i_v = \sum_{j\in[l] : v = v^i_j} \rho_i,
$$
if some path $p^j$ contains $v$, and $d^i_v = 0$, otherwise. The fact that $d^i$ is an $i$-configuration follows
immediately from the fact that $f^S$ is a $(k+1)$-flow. In addition, for the same reason
$d^i$ precedes $d_{i+1}$ for every $i\in [r-1]$. From the latter observations and the fact that $d^1 = c^s$
and $d^r =  c^t$ it follows that $P = d^1, d^2, \cdots, d^r$ is a $c^s-c^t$ path in $H$ with cost $\ell(P) \leq w(S)$.

Consider next an $c^s-c^t$ path $P = d^1, \cdots, d^r$ with cost $\ell(P) = \sum_{i=1}^{r-1} \ell(d^i, d^{i+1})$.
The cost $\ell(d^i, d^{i+1})$ is realized by some set of edges $R_i \subset E(d^i, d^{i+1})$ for every $i\in [r-1]$.
From Definition~\ref{def:conf}, the maximal $s$-$t$ flow in the graph $G' = (V, R)$ is at least $k+1$, where
$R = \cup_{i\in [r-1]} R_i$. Next, Lemma~\ref{lem:disjointpaths} guarantees that there exists some feasible solution
$S \subset R$, the cost of which is at most $\ell(P)$. In the latter claim we used the disjointness of the sets
$R_i$, which is due the layered structure of the graph $G$. This concludes the proof of the lemma.
\end{proof}

Finally, we observe that the number of configurations is bounded by $O(n^{k+1})$, which implies that
$k$-FTP can be solved in polynomial time on layered graphs.

To obtain the same result for directed
acyclic graphs we perform the following transformation of the graph. Let $v_1, \cdots, v_n$
be a topological sorting of the vertices in $G$. Replace every edge $e = v_iv_j$ ($i<j$) with a path
$p_e = v_i, u^e_{i+1}, \cdots, u^e_{j-1}, v_j$ of length $j-i+1$ by subdividing it sufficiently many times. Set
the cost of the first edge on the path to $w'(v_iu^e_{i+1}) = w(v_iv_j)$ and set the costs of all other edges on the path
to zero. In addition, create a new set of faulty edges $M'$, which contains all edges in a path $p_e$ if $e\in M$.

It is straightforward to see that the new instance of FTP is equivalent to the original one, while the obtained
graph after the transformation is layered. We summarize with the following theorem.

\begin{theorem}\label{thm:dagpoly}
There is a polynomial algorithm for $k$-FTP restricted to instances with a directed acyclic graph.
\end{theorem}

\subsection{Series-Parallel Graphs}\label{subsec:srp}

Recall that a graph is called \emph{series-parallel (SRP)} with terminal $s$ and $t$ if
it can be composed from a collection of disjoint edges using the \emph{series} and \emph{parallel} compositions.
The series composition of two SRP graphs with terminals $s$, $t$ and $s',t'$ respectively, takes
the disjoint union of the two graphs, and identifies $t$ with $s'$. The parallel composition
takes the disjoint union of the two graphs and identifies $s$ with $s'$ and $t$ with $t'$.
Given a SRP graph it is easy to obtain the aforementioned decomposition.

The algorithm we present has linear running time whenever the robustness parameter $k$ is fixed.
The algorithm is given as Algorithm~\ref{alg:ercc:sircc:srp}.
In fact, the algorithms computes the optimal solutions $S_{k'}$ for all parameters
$0 \leq k' \leq k$. The symbol $\perp$ is returned if the problem is infeasible.

\begin{algorithm}
\caption{\textbf{: FTP-SeriesParallel($G,s,t,k$)}}\label{alg:ercc:sircc:srp}
\begin{algorithmic}[1]
\REQUIRE{$G=(V,E)$ a series-parallel graph, $s,t\in V$ and $M\subset E$, $k\in \mathbb{Z}_+$.}
\ENSURE{Optimal solution to FTP for parameters $0,1,\cdots, k$.}
\IF{$E = \{e\} \wedge e\in M$}
  \STATE Return $(\{e\}, \perp, \cdots, \perp)$
\ENDIF
\IF{$E = \{e\} \wedge e\not\in M$}
  \STATE Return $(\{e\}, \cdots, \{e\})$
\ENDIF

\vspace{2mm}

$\Rightarrow$ $G$ is a composition of $H_1, H_2$.
\vspace{2mm}

\STATE $(S_0^1, \cdots, S_k^1) \leftarrow $ \textbf{FTP-SeriesParallel}$(H_1, M\cap E[H_1], k)$
\STATE $(S_0^2, \cdots, S_k^2) \leftarrow $ \textbf{FTP-SeriesParallel}$(H_2, M\cap E[H_2], k)$

\IF{$G$ \text{is a series composition of} $H_1, H_2$}

\FOR{$i=0,\cdots,k$}
  \IF{$S_i^1 = \perp \vee \, S_i^2 = \perp$}
    \STATE $S_i \leftarrow \perp$
  \ELSE
    \STATE $S_i \leftarrow S_i^1 \cup S_i^2$
  \ENDIF
\ENDFOR

\ENDIF

\IF{$G$ \text{is a parallel composition of} $H_1, H_2$}
  \STATE $m_1 \leftarrow \max\{i : S_i^1 \neq \perp\}$
  \STATE $m_2 \leftarrow \max\{i : S_i^2 \neq \perp\}$

  \FOR{$i=0,\cdots,k$}
    \IF{$i > m_1 + m_2 + 1$}
      \STATE $S_i \leftarrow \perp$
    \ELSE
      \STATE $r \leftarrow argmin_{-1\leq j\leq i}\{w(S_j^1) + w(S_{i-j-1}^2)\}$ \hspace{20mm} $// \, S^1_{-1} = S^2_{-1} := \emptyset$
      \STATE $S_i \leftarrow S_r^1 \cup S_{i-r-1}^2$
    \ENDIF
  \ENDFOR
\ENDIF

\STATE Return $(S_0, \cdots, S_k)$
\end{algorithmic}
\end{algorithm}

\begin{theorem}\label{thm:fastalgSRP}
Algorithm~\ref{alg:ercc:sircc:srp} returns an optimal solution to the FTP problem on SRP
graphs. The running time of Algorithm~\ref{alg:ercc:sircc:srp} is $O(nk)$.
\end{theorem}

\begin{proof}
 The proof of correctness is by induction on the depth of the recursion in Algorithm~\ref{alg:ercc:sircc:srp}. Clearly
the result returned by Algorithm~\ref{alg:ercc:sircc:srp} in lines $1$-$6$ is optimal.
Assume next that the algorithm computed correctly all optimal solutions for the subgraphs $H_1, H_2$,
namely that for every $i\in [2]$ and $j\in [k]$, the set $S_j^i$ computed in lines $7$-$8$ is an
optimal solution to the problem on instance $\mathcal{I}_i^j = (H_i, M \cap E[H_i], j)$.

Assume first that $G$ is a series composition of $H_1$ and $H_2$, and let $0 \leq i \leq k$. If
either $S_i^1 = \perp$ or $S_i^2 = \perp$ the problem with parameter $i$ is clearly also infeasible,
hence the algorithm works correctly in this case. Furthermore, since $G$ contains a cut vertex (the
terminal node, which is in common to $H_1$ and $H_2$), a solution $S$ to the problem is feasible for
$G$ if and only if it is a union of two feasible solutions for $H_1$ and $H_2$. From the inductive
hypothesis it follows that $S_i$ is computed correctly in line $14$.

Assume next that $G$ is a series composition of $H_1$ and $H_2$. Consider any feasible solution $S'$
to the problem on $G$ with parameter $i$. Let $S'_1$ and $S'_2$ be the restrictions of $S'$ to edges
of $H_1$ and $H_2$ respectively, and let $n_1$ and $n_2$ be the maximal integers such that
$S'_1$ and $S'_2$ are robust paths for $H_1$ and $H_2$ with parameters $n_1$ and $n_2$, respectively.
Observe that $i \leq n_1 + n_2 + 1$ must hold. Indeed if this would not be the case, then taking
any cut with $n_1 + 1$ edges in $S'_1$ and another cut with $n_2 + 1$ edges in $S'_2$ yields a cut
with $n_1 + n_2 + 2$ edges in $G$, contradicting the fact that $S'$ is a robust path with parameter $i$.
We conclude that the algorithm computes $S_i$ correctly in line $23$. Finally note
that the union any two robust paths for the graphs $H_1$ and $H_2$ with parameters $n_1$ and $n_2$ with
$i \leq n_1 + n_2 + 1$ yield a feasible solution $S_i$. It follows that the minimum cost such robust
path is obtained as a minimum cost of a union of two solutions for $H_1$ and $H_2$, with robustness
parameters $j$ and $i-j-1$ for some value of $j$. To allow $S_i = S_i^1$ or $S_i = S_i^2$ we let
$j$ range from $-1$ to $k$ and set $S^1_{-1} = S^2_{-1} = \emptyset$. This completes the proof of correctness.

To prove the bound on the running time, let $T(m,k)$ denote the running time of the algorithm on
a graph with $m$ edges and robustness parameter $k$. We assume that the graph is given by a
hierarchical description, according to its decomposition into single edges.
The base case obviously takes $O(k)$ time. Furthermore we assume that the solution $(S_0, \cdots, S_k)$
is stored in a data structure for sets, which uses $O(1)$ time for generating empty sets and for
performing union operations.
If the graph is a series composition then the running time satisfies
$T(m,k) \leq T(m',k) + T(m-m',k) + O(k)$ for some $m' < m$. If the graph is a parallel composition, then $T(m,k)$
satisfied the same inequality. We assume that the data structure, which stores the sets $S_i$
also contains the cost of the edges in the set. This value can be easily updates in time $O(1)$
when the assignment into $S_i$ is performed. It follows that $T(m,k) = O(mk) = O(nk)$ as required.
\end{proof}

\section{Integrality Gap and Approximation Algorithms}\label{sec:intgap}

In this section we study the natural fractional relaxation of FTP. We prove a tight
bound on the integrality gap of this relaxation. This results also suggests a simple
approximation algorithm for FTP with ratio $k+1$. We later combine this algorithm
with Algorithm~\ref{alg:kisone} to obtain a $k$-approximation algorithm.

\subsection{Fractional FTP and Integrality Gap}

Let us start by introducing the fractional relaxation of FTP, which we denote by FRAC-FTP. The
input to FRAC-FTP is identical to the input to FTP. The goal in FRAC-FTP is to find a capacity
vector $x:E \rightarrow [0,1]$ of minimum cost $w(x) = \sum_{e\in E} w_e x_e$ such that for
every $F\in \Omega(M,k)$, the maximum $s$-$t$ flow in $G-F$, capacitated by $x$ is at least one.
Note that by the Max-Flow Min-Cut Theorem, the latter condition is equivalent to requiring that
the minimum $s$-$t$ cut in $G-F$ has capacity of at least one. We will use this fact in the proof
of the main theorem in this section.

Observe that by requiring $x\in \{0,1\}^E$ we obtain FTP, hence FRAC-FTP is indeed a fractional relaxation
of FTP.

In the following theorem by 'integrality gap' we mean the maximum ratio between the optimal solution
value to an FTP instance, and the optimal value of the corresponding FRAC-FTP instance.

\begin{theorem}\label{thm:integralitygap}
The integrality gap of FTP is bounded by $k+1$. Furthermore, there exists an infinite family of
instances of FTP with integrality gap arbitrarily close to $k+1$.
\end{theorem}

\begin{proof}
Consider an instance $I = (G,M,k)$ of FTP. Let $x^*$ denote an optimal solution to the corresponding
FRAC-FTP instance, and let $OPT = w(x^*)$ be its cost. Define a vector $y\in \mathbb{R}^E$ as follows.
\begin{equation}\label{eq:gapproof1}
y_e =
\begin{cases}
(k+1) x_e    & \mbox{if } e\not\in M \\
\min\{1, (k+1)x_e\}  & \mbox{otherwise}.
\end{cases}
\end{equation}
Clearly, it holds that $w(y) \leq (k+1)OPT$.
We claim that every $s$-$t$ cut in $G$ with capacities $y$ has capacity of at least $k+1$.
Consider any such cut $C \subset E$, represented as the set of edges in the cut. Let
$M' = \{e\in M : x^*_e \geq \frac{1}{k+1}\}$ denote the set of faulty edges attaining high fractional
values in $x^*$. Define $C' = C\cap M'$. If $|C'| \geq k+1$ we are clearly done. Otherwise, assume $|C'| \leq k$.
In this case consider the failure scenario $F = C'$. Since $x^*$ is a feasible solution it must hold that
\begin{equation}\label{eq:gapproof2}
\sum_{e\in C\setminus C'} x^*_e \geq 1.
\end{equation}
Since for every edge $e\in C \setminus C'$ it holds that $y_e = (k+1)x^*_e$ we obtain
\begin{equation}\label{eq:gapproof3}
\sum_{e\in C\setminus C'} y_e \geq k+1,
\end{equation}
as desired. From our observations it follows that the maximum flow in $G$ with capacities $y$ is at least
$k+1$. Finally, consider the minimum cost $(k+1)$-flow $z^*$ in $G$ with capacities defined by
\begin{equation}\label{eq:gapproof4}
c_e =
\begin{cases}
k+1    & \mbox{if } e\not\in M \\
1  & \mbox{otherwise}.
\end{cases}
\end{equation}
From integrality of $c$ and the minimum-cost flow problem we can
assume that $z^*$ is integral. Note that $y_e \leq c_e$ for every $e\in E$, hence any feasible $(k+1)$-flow with capacities
$y$ is also a feasible $(k+1)$-flow with capacities $c$. From the previous observation it holds that $w(z^*) \leq w(y) \leq (k+1)OPT$.
From Lemma~\ref{lem:disjointpaths} we know that $z^*$ is a feasible solution to the FTP instance.
This concludes the proof of the upper bound of $k+1$ for the integrality gap.

To prove the same lower bound we provide an infinite family of instances, containing instances with integrality
gap arbitrarily close to $k+1$. Consider a graph with $D \gg k$ parallel edges with unit cost connecting $s$ and $t$, and let $M=E$.
The optimal solution to FTP
on this instance chooses any subset of $k+1$ edges. At the same time, the optimal solution to FRAC-FTP assigns
a capacity of $\frac{1}{D-k}$ to every edge. This solution is feasible, since in every failure scenario, the number
of edges that survive is at least $D-k$, hence the maximum $s$-$t$ flow is at least one. The cost of this solution is $\frac{D}{D-k}$.
Taking $D$ to infinity yields instances with integrality gap arbitrarily close to $k+1$.
\end{proof}

The proof of Theorem~\ref{thm:integralitygap} implies a simple $(k+1)$-approximation algorithm for
FTP. This algorithm simply solves the integer minimum-cost flow problem, defined in proof of the theorem, and returns
the set of edges corresponding to the optimal integral flow $z^*$ as the solution.
This result is summarized in the following corollary.

\begin{corollary}\label{cor:kplusone_pprox}
There is a polynomial $(k+1)$-approximation algorithm for FTP.
\end{corollary}

\subsection{A $k$-Approximation Algorithm}\label{subsec:kapprox}

In this section we improve the approximation algorithm from the previous section.
The new algorithm can be seen as a generalization of Algorithm~\ref{alg:kisone} to arbitrary FTP instances. The main
observation is the following. The reason why the approximation algorithm implied by Theorem~\ref{thm:integralitygap}
gives an approximation ratio of $k+1$ is that the capacity of edges in $E\setminus M$ is set to $k+1$, hence,
if the flow $z^*$ uses such edges to their full capacity, the cost incurred is $k+1$ times the cost of these
edges. This implies that the best possible lower bound on the cost $w(z^*)$ is $(k+1)OPT_{FRAC}$, where
$OPT_{FRAC}$ denotes the optimal solution value of the corresponding FRAC-FTP instance.
To improve the
algorithm we observe that the edges in $z^*$, which carry a flow of $k+1$ are cut-edges in the obtained solution.

To conveniently analyze our new algorithm let us consider a certain canonical flow defined
by minimal feasible solutions.

\begin{definition}\label{def:canonicalflow}
Consider an inclusion-wise minimal feasible solution $S\subset E$ of an instance $I = (G,s,t,M)$ of FTP.
A \emph{flow $f^S$ induced by $S$} is any integral $s$-$t$ $(k+1)$-flow in $G$ respecting the capacity vector
$$
c^S_e =
\begin{cases}
1    & \mbox{if }  e \in S\cap M\\
k+1  & \mbox{if }  e \in S \setminus M\\
0    & \mbox{if }   e\in E\setminus S.
\end{cases}
$$
\end{definition}

To this end consider an optimal solution $X^* \subset E$ to the FTP instance and consider any corresponding
induced flow $f^*$. Define
$$X_{PAR} = \{e\in X^* : f^*(e) \leq k\} \,\,\, \text{and} \,\,\,  X_{BRIDGE} = \{e\in X^* : f^*(e) = k+1\}.$$
As we argued before, every edge in $X_{BRIDGE}$ must be a bridge in $H = (V,X^*)$ disconnecting $s$ and $t$.
Let $u_e$ denote the tail vertex of an edge $e\in E$. Since every edge $e\in X_{BRIDGE}$ constitutes
an $s$-$t$ cut in $H$, it follows that the vertices in $U = \{e_u : e\in X_{BRIDGE}\} \cup \{s,t\}$ can be unambigously
ordered according to the order in which they appear on any $s$-$t$ path in $H$, traversed from $s$ to $t$.
Let $s = u_1, \cdots, u_q = t$ be this order. Except for $s$ and $t$, every vertex in $U$ constitutes
a cut-vertex in $H$. Divide $H$ into $q-1$ subgraphs $H^1, \cdots, H^{q-1}$ by letting $H^i = (V, Y_i)$ contain the
union of all $u_i$-$u_{i+1}$ paths in $H$. We observe the following property.

\begin{proposition}\label{prop:decomp}
For every $i\in [q-1]$ the set $Y_i\subset E$ is an optimal solution to the FTP instance $I_i = (G,u_i, u_{i+1}, M)$.
\end{proposition}

Consider some $i\in[q-1]$ and let $f^*_i$ denote the flow $f^*$, restricted to edges in $H^i$. Note that
$f^*_i$ can be viewed as a $u_i$-$u_{i+1}$ $(k+1)$-flow.
Exactly one of the following cases can occur. Either $H^i$ contains a single edge $e\in E\setminus M$, or
$$
\max_{e\in Y_i}{f^*_i(e)} \leq k.
$$
In the former case, the edge $e$ is the shortest $u_i$-$u_{i+1}$ path in $(V,E\setminus M)$. In the latter
case we can use a slightly updated variant of the algorithm in Corollary~\ref{cor:kplusone_pprox} to obtain
a $k$-approximation of the optimal FTP solution on instance $I_i$. Concretely, the algorithm defines
the capacity vector
$$
c'_e =
\begin{cases}
k    & \mbox{if } e\not\in M \\
1  & \mbox{otherwise},
\end{cases}
$$
and finds an integral minimum-cost $u_i$-$u_{i+1}$ $(k+1)$-flow $Y^*$ in $G$, and returns the support $Y\subset E$
of the flow as the solution. The existence of the flow $f^*_i$ guarantees that $w(y^*) \leq w(f^*_i)$,
while the fact that the maximum capacity in the flow problem is bounded by $k$ gives $w(Y) \leq kw(y^*)$.
It follows that this algorithm approximates the optimal solution to the FTP instance $I_i$ to within a factor $k$.

To describe the final algorithm it remains use the blueprint of Algorithm~\ref{alg:kisone}. The only difference
is in the computation performed in step $5$. Instead of finding two edge-disjoint $u$-$v$ paths, the new
algorithm solves the aforementioned flow problem. An appropriate adaptation is also made to step $12$.
We summarize the main result of this section in the following theorem. The proof is omitted, as it is identical
to that of Theorem~\ref{thm:kisone}, with the exception of the preceding discussion.

\begin{theorem}\label{thm:kapprox}
There is a polynomial $k$-approximation algorithm for FTP.
\end{theorem}

\section{Conclusions and Future Work}\label{sec:conclusions}

This paper presents FTP, a simple natural extension of the minimum $k$ edge-disjoint path problem,
interpreted as a fault-tolerant connection problem. In our model not all subsets of $k$ edges
can be removed from the graph after a solution is chosen, but rather any subset of \emph{faulty edges},
a subset provided as part of the input. Such an adaptation is natural from the point of view of
many application domains.

We observed a dramatic increase in the computational complexity of FTP with respect EDP.
At the same time we identified several classes of instances admitting a polynomial exact algorithm.
These classes include the case $k=1$, directed acyclic graphs and fixed $k$ and series-parallel graphs.
Next, we defined a fractional counterpart of FTP and proved a tight bound on the corresponding integrality
gap. This result lead to a $k$-approximation algorithm for FTP. We conclude by stating a number
of promising directions for future research.

Perhaps the main problem remains to understanding the approximability of FTP. In particular,
it is interesting to see if the approximation guarantee for FTP can be decreased to the
approximation guarantees of the best known algorithms for the Steiner Tree problem.
It is also interesting to relate FTP to more general problems such a Minimum-Cost Fixed-Charge
Network Flow and special cases thereof.
The complexity of $k$-FTP in still unknown. It is interesting to see if the methods
employed in the current paper for $1$-FTP and $k$-FTP on directed acyclic graphs can be
extended to $k$-FTP on general graphs.
It is also natural to study other classical combinatorial optimization problems in the
model proposed in this paper. In particular, the counterpart of the Minimum Spanning Tree
problem seems to be a promising next step. We remark that all the results about integrality
gaps and approximability of FTP can be carried over (with some modifications) to this problem.
For example, the proof of Theorem~\ref{thm:integralitygap} can be adapted to the latter
problem to give an integrality gap of $k+3$, in combination with the integrality gap result
for the Minimum-Cost $k$-Edge Connected Spanning Subgraph problem of Gabow et al.~\cite{kEdgeConnected1}.

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