text:
Youth, Education, & Community
Governance & Volunteerism
Conducting, Composing & Performing
Symphony Magazine
Advancing the orchestral experience for all
Knowledge, Research & Innovation
Jobs Center
Advocacy & Government
Learning & Leadership Development
American Orchestras' Futures Fund
The Catalyst Fund
Mergers and Creative Alliances
Music Director Search Handbook
Professional Development and Grants
State Funding Resources
Essentials of Orchestra Management
Essentials of Orchestra Management Testimonials
Sexual Harassment Prevention
The Noteboom Governance Center
Visa, Tax and Travel Information
The League of American Orchestras
The University of Southern California Arts Leadership Program
Hosted by the USC Thornton School of Music
Congratulations to our 2019 cohort!
Read the press announcement here.
Read about this year's seminar here.
Essentials of Orchestra Management, the premier orchestra management course in the United States, prepares orchestra professionals to be effective administrators and leaders by giving them the knowledge and tools they need. Taking place over the course of ten days on the University of Southern California campus in Los Angeles, the Essentials seminar provides participants with an in-depth overview of orchestra management, builds career networks, and offers a unique opportunity to obtain what's necessary to become an innovative and successful leader in the arts. To many, it's the gold standard of orchestra management training.
Led by a faculty of high-profile orchestra executives, leadership experts, artists, and arts activists with diverse backgrounds and viewpoints, the seminar is designed to address both theory and practice, integrating the faculty's extensive practical experience and professional knowledge into a course of study that not only explores the challenges faced by orchestra managers today, but also envisions how orchestras must adapt and change to be sustainable in the future.
Since its inception in 2000, over 500 individuals have participated in Essentials. Many Essentials alumni are now professionals excelling at an array of highly-regarded music organizations including the National Symphony Orchestra; Silkroad; the New York Philharmonic; and El Sistema USA.
The purpose of Essentials is to:
Serve as a platform for the development of future orchestra leaders
Envision orchestras as they look to the future
Deepen the knowledge and skills of the people in the field
Connect orchestra professionals to one another
Inspire life-long careers in, and dedication to, orchestras, and
Strengthen orchestras as they carry out their missions
Through the immersive Essentials experience, and in addition to cultivating a network of peers and mentors, participants who complete the seminar will have the critical thinking skills required to be able to:
Identify, diagnose, and resolve the strategic challenges of the orchestral field, of both today and tomorrow.
Advance orchestras as diverse and inclusive organizations and fulfill their unique role in the cultural context of their communities.
Develop leadership abilities to work productively with competing imperatives and within a wide range of constituent groups including audience members, musicians, staff, boards, and partner organizations.
Demonstrate basic knowledge of core business and operating practices in American orchestras.
Deepen their knowledge of orchestral performances and learn to speak about orchestral music with understanding and insight.
Identify key resources for continued professional development.
Receive a one-year membership to the League of American Orchestras
Who Was Eligible to Apply?
Early career professionals, musicians, graduate students, career changers, and experienced administrators who wish to expand and deepen their knowledge base in all aspects of orchestral management.
Applications are now closed.
Dates: July 8-18, 2019
Location: USC Thornton School of Music, Los Angeles, California
Essentials participants may reside on or commute to the USC campus during the duration of the seminar. Excursions to Walt Disney Concert Hall, the Hollywood Bowl, and community-based programs will make the most of the Los Angeles setting.
Class size: Limited to 30
Essentials participants in a group activity
photo credit: Agibail R. Collins Photography
Simon Woods, chief executive officer, Los Angeles Philharmonic, Essentials director
Scott Faulkner, bassist, Reno Philharmonic; former executive director, Reno Chamber Orchestra, Essentials associate director
Jennifer Barlament, executive director, Atlanta Symphony Orchestra
Alexander Laing, principal clarinetist, Phoenix Symphony
Jesse Rosen, president and CEO, League of American Orchestras
John-Morgan Bush, director of learning and leadership programs, League of American Orchestras
Guest Faculty
Tanya Kalmanovitch, assistant professor, the New School University and Faculty, New England Conservatory
Laura Reynolds, vice president of education and community engagement, Seattle Symphony
Ken Foster, director, arts leadership program, USC Thornton School of Music
Faith Raiguel, chief financial officer, LA Opera (Retired)
Dan Song, vice president, philharmonic and production, Los Angeles Philharmonic
Dan Coleman, senior development advisor, Cleveland Institute of Music
Amy Fogerson, director, Street Symphony Chamber Singers
Reveta Bowers, head of school, the Center for Early Education, board member, Los Angeles Philharmonic
Meredith Snow, violist, Los Angeles Philharmonic, chairperson, International Conference of Symphony and Opera Musicians
Vince Ford, vice president, audiences and innovation, New York Philharmonic
Rachel Roberts, associate professor of music leadership, Eastman School of Music
Ed Yim, President, American Composers Orchestra
**Additional guest faculty will be announced this Spring**
And prominent musicians and artists from the Greater L.A. area will supplement the seminar experience.
Program Fees*
Application fee - $60
$3,495 Includes housing** on the USC campus
$2,810 Campus housing not included.
*The application fee is non-refundable. If accepted into the Essentials Class of 2019, a non-refundable tuition deposit of $1,000 will be due on or before April 23, 2019. The balance will be due on or before July 1, 2019.
**Housing covers ten nights accommodation in a single room in a two-bedroom suite with a shared bathroom in the University Village complex on the USC campus. Each participant will be responsible for covering their own costs of travel, and most meals.
Financial Aid Available
Scholarships are available. Those wishing to apply for financial aid may do so in the application module.
Links to More Essentials Information
See a photo story of last year's class
Read what faculty and alumni say about Essentials
See a recent Essentials ad in Symphony magazine
Article in Winter 2019 Symphony magazine on Essentials
For any questions, please contact John-Morgan Bush, Director, Learning and Leadership Programs at This email address is being protected from spambots. You need JavaScript enabled to view it. .
Essentials of Orchestra Management is made possible by generous grants from
The Andrew W. Mellon Foundation and the National Endowment for the Arts.
photo credits of faculty: Brandon Patoc, Elizabeth Harraman, Jeff Roffman, Jared Platt, Dan Rest, Greg Mahan
520 8th Avenue, Suite 2005, 20th Floor, New York, NY 10018 t 212 262 5161
© 2020 League of American Orchestras
Concept, design, production: Sametz Blackstone Associates
Fonteva Status: Not Logged In


text:
Joaquima tricolora is a species of moth of the family Tortricidae. It is found in Santa Catarina, Brazil.

References

Moths described in 1999
Euliini
Moths of South America
Taxa named by Józef Razowski

redpajama_set_name: RedPajamaWikipedia


The easiest way to get to Eugene Yoga South from Downtown is via Willamette St (going South). Willamette will turn into Donald at 32nd St. Follow Donald up the hill, look on your left, and turn in on Pearl (between 34th and 36th St) to find the Tamarack Wellness Center.
If you park in the upper lot near the intersection of Pearl and Donald, come through the main doors. Our office will be directly in front of you.
If you park in the lower pool/overflow lot, use the door under the covered walkway that goes into the main/upper building (i.e. not the pool). Follow the hallway upstairs, take a left at the top of the stairs, continue down the hallway, and you will find Eugene Yoga's office on your left.
Please plan on arriving at least 10 minutes before class begins. Students arriving more than 10 minutes after the start of class may find the class closed to late arrivals.
We are on the corner of E. Broadway and Pearl St, next to the Interstate Bank and across the street from the Eugene Hotel.
We do not have our own parking lot; however, there are many parking options in addition to bike parking.
We validate for 2 hours at the Overpark (10th & Oak) and Parcade (8th & Willamette) parking garages, which are also free on weekends.
Parking is $1/hr at the city lot behind our building (8th & Pearl) and free if you enter after 6 pm or on Sundays.
Two-hour free on-street parking between 7th to 11th Ave from Lincoln St to Willamette St (Monday through Saturday from 7:00 am to 6:00 pm).
After 5 pm and on Sunday, you can park in the bank parking lot adjacent to our building.


I'm sorry, this chapter was long, with a lot going on. Today, Tim goes to the Stock exchange and the Director carries out the plot. Are the mermaids safe?
A long time ago the Nagoya Railroad Company(Meitetsu) decided that they needed an attraction to drum up business on weekends. Rather than build the traditional amusement park, the company created a wonderful outdoor museum of buildings from the Meiji era that were facing demolition in the face of development in the 1960's.
Akihabara News is posting short videos of the Yamanote line station chimes along with short video tours of the Neighborhoods.
I'm doing a larger post of the stations, but it's likely that doing all the stations will take some time and the people at Akihabara New provided this great link about the history of Akihabara.
I didn't know that there was an elevated freight station at Akihabara.
Here's my main Akihabara post.
Now I've posted in the past about Akihabara. I've discussed it's influence as a business cluster, posted about Otaku videoing the screen on a building, a livestream from a hobbyshop window and posted a 4K video. I've never delved into the place in detail, until now.
Miniatur Wunderland in Hamburg Germany is the largest model railroad in the world. Since it was opened it's become a well known attraction in Hamburg and amongst the railfan and model railroader community. But Hamburg may be off your driven path. Well never fear, now you can travel the railroad via the Google Streetview cam.
It's hard to understand how commercial air travel has changed things until you see maps like this that show how long it took to travel from place to place.
This map is just 100 years old, yet look how long it took to go from London to anyplace beyond Europe. Any trip outside Europe and maybe the east coast of the US was a serious voyage and one wouldn't undertake it unless one expected to spend months and perhaps years at their destinations. There is a reason that steamer trunks are so large. You weren't packing for an overnight, you packing for living at the other end.
I recently read you Voyages issue Of the NYT Sunday Magazine.
To show Mr. Soth a little of what he missed, well here's some pics I've shot in Tokyo.
I was spending my own money, so I didn't stay as fancy as the Park Hyatt. The View from my hotel room was rather prosaic, though the love hotel did have it's romanticism.
Perhaps that was good reason to leave the room. Tokyo has a lot of the unexpected.
Such as little French cars. It pays though to go off the beaten path and try to find perhaps the prosaic and unloved, well at least by it's owners, photographers certainly love this little rusty locomotive.
It may be the most photographed train in Japan. Of course it's not hard to find rust and steel in the industrial parts of Tokyo, far from the usual tourist traps.
Still, even in the industrial area a grandfather can share with his grandson.
or get up early and join the morning commute.
You can visit markets not yet busy.
Or watch a park being prepared for the day.
Guarded by stalwart giant guardians.
And some not so giant.
Before catching the evening commute.
And daring the night again.
And wandering the underground warrens of Tokyo Station.
Before going out to play the obligatory video game.
Perhaps it may be better to take a speedy journey.
Before having a quiet drink in a sublime location.
While I didn't eat such exotic things as bug sushi, I did have a series of rather strange to the US hotel breakfasts before setting out to explore.
Finding out that perhaps the culture was a bit loser to home, or at least the advertising was.
One thing about being behind the camera is that you never have to feel as if you don't belong there.
Which can help if you are in a place where normally you would feel out of place.
No matter how attractive the scenery.
Though it's still easy to find people in touch with tradition.
Though the strange and exotic does have it's charms.
Still there's always time for eats, even takiyoki, maybe especially takiyoki.
Isn't the point of the voyage, the experience? If I want room service and pizza delivery, all I have to do is stay home. Tokyo has nothing that should truly scare you and it's a great place to shoot. There's always something to find, the unexpected the strange the new. And it's so easy to explore. Why stay in your room?


What's New for the 2018 Nissan Rogue?
Although already popular because of its many amenities, the 2018 Nissan Rogue features updates in the current year that make it even more desirable. Color additions include Scarlet Amber and Midnight Pine for the exterior. The Midnight Edition package for the SV trim adds black as an accent. On the infotainment side, all trims now come with a 7-inch screen that integrates Apple CarPlay and Android Auto so you can use a compatible smartphone with the system. The motion-activated power liftgate, which updates the SV trim, opens and closes when you kick under the rear bumper. The SL gains Adaptive Cruise Control and Lane Departure Warning as standard driver assist features. However, also available as part of the SL Platinum package is an advanced system called ProPilot Assist that comes with Stop-and-Go Adaptive Cruise Control and Lane Keeping Assist.
With the S trim, you can run the engine more confidently on any incline with Hill Start Assist, which holds brakes momentarily to give you time to transfer your foot from the brake to the accelerator pedal to avoid rolling back hills. LED Daytime Running Lights maximize your visibility even when the sun is out.
The SV trim builds on the S but add such features as Dual Zone Automatic Temperature Control, which allows you and your front passenger to create your own climate levels that are then automatically maintained despite what the weather is doing outside. With the Nissan Intelligent Key, you can unlock or lock the door, or start the engine with a push of a button by using the switches on the key fob.
For the style conscious, the Midnight Edition™ trim clothes the crossover in black exterior accents such as on the outside mirrors that also have LED turn signals. A distinctive Black Midnight Edition badge announces another exclusive feature: 17-inch Midnight Edition alloy wheels.
The highest level trim, the SL trim, covers all the seats in full leather while adding Door-to-Door Navigation that relies on Premium Traffic to steer you around congested roads using turn by turn directions. Automatic Emergency Braking with Pedestrian Detection can stop your vehicle if it discovers a person crossing the path up ahead.
The 2.5L 4-cylinder that comes with Dual Overhead Cams, Front Wheel Drive and 16 valves to generate up to 170 hp and 175 lb.-ft. of torque. Fuel economy achieves an EPA estimated 33 miles per gallon on the highway and 26 miles per gallon in the city through the use of the Xtronic Continuously Variable Transmission. Define your drive with Eco mode to maximize fuel efficiency or Sport mode when you demand more responsiveness for an aggressive drive.
The 2018 Nissan Rogue comes with multiple safety advantages. The available Intelligent Around View Monitor offers a top-down view of your immediate surroundings, which is created from the front, rear and side external cameras. This virtual display eases maneuvers in tight situations. To make changing lanes more secure, Blind Spot Warning informs you of any cars that are just beyond your peripheral vision in the next lane. Giving you sure footing on uncertain or slippery terrain, the Traction Control System reduces your speed to improve your control. Finally, vehicle dynamic control brakes individual wheels or reduces your speed to prevent oversteering and understeering.
Learn more about the 2018 Nissan Rogue by either scheduling an appointment with or coming to our dealership for a test drive.


An instructor at our college would like to see what the average score is in columns where some students have zeroes - but doesn't want the zeroes to skew the average for everyone that did take the test/assignment/etc.
He could just not enter zeroes but then a running total is off. If it's not a running total, then columns for grades not entered yet skew things there and that's not preferable either.


text:
Why the Need ?
Truths About Noyes Home
Plan a Tour or Visit
Noyes Home in the Media
Crisis/Emergency Numbers
Admission to Noyes Home
Support Our Home
"Champions for Children" Tax-Credit Program
Memorials & Honorariums
Many adults who grew up in the embrace of the Noyes Home walls recall some wonderful memories that were created here. Not only is a sense of comradery and a "sibling" atmosphere created at the Noyes Home, but volunteer mentors, host families, and many other community partners have all brought smiles and educational opportunities to the children.
Many children at the Noyes Home await the day they can return home with their parents, but while they wait, they have a loving and caring family around them at the Noyes Home. Alice Fern Francis, for example,who lived at the Noyes Home between 1925 and 1928, recalled anxiously waiting at the window each week for her mother to visit. While she wanted to be home again with her mother, she was so grateful to the Noyes Home for helping her mother and herself to grow to become stronger individuals and a stronger family.
Josh's Story…
At this year's Take A Seat with The Wizard of Oz event, many of us had the opportunity to meet a young man who shared that he had lived at the Noyes Home when he was 7 years old. Josh Johnson and his younger brother lived at the Noyes Home for approximately one month while their mother sorted out some personal matters and dealt with major depression.
Josh shared that, "Having the Noyes Home as an option gave our family the chance for my mom to adjust and deal with the things in her life so she could properly care for us again." He also said, "Our family may not have had the opportunity to fully recover and rebuild" without the safe haven of the Noyes Home.
As a child, Josh went through many hardships and was not certain what the future held for him. As an adult, however, he was able to turn things around and made a great deal of progress in his life. "I am blessed to have a great career and life which I have built with my wife and three children." Josh said, "I feel that the Noyes Home helped shape me in many ways as I now work in the social service field." Today, Josh not only serves those in need in the community but continues to support the mission and the children of the Noyes Home.
Josh's story reminds us that the Noyes Home does indeed "foster a brighter future" for our children. Like Josh, our donors and volunteers not only make an immediate difference today, but their impact continues as our residents move forward outside of the Noyes Home.
As Josh shared, "I have been very blessed in my life and am proud to be able to support such an excellent organization and their mission."
"I Can't Express How Much it Meant for Me" Ralph Hicks …
In 1933 Ralph Hicks was placed at birth in the Home for Little Wanderers (now Noyes Home). As he says, he had "no home, no parents, no food, no clothing." At that time, he was a ward of the St. Joseph and Buchanan County Humane Society!
A few months later, Ralph was placed with his new parents. He is now 83, living in Wisconsin with his lovely wife Bonnie. For over 20 years they have portrayed Mr. and Mrs. Santa Claus in order to give back to children in gratitude for Noyes Home. According to Ralph, "My wonderful life is because I was taken in and given a great start. Without Noyes Home, where would I be? Who would I have become? It changed my life."
Meet Howard, another example …
In 1900 a two-year-old toddler named Howard Engleman was brought to the Home for Little Wanderers (Noyes Home) by his mother. She never returned. He remained at Noyes Home as part of our family until the age of 13 when he left for high adventure and became a sailor on an ocean-going ship. Two years later while in South America, he contracted tuberculosis. Realizing that he was going to die, he visited the United States Consul and asked that all of his belongings be sold and that the money be sent to the Home for Little Wanderers, the only home he had ever known, so that he might be remembered. The money was used to buy a chime clock which hangs in our entrance hall today and bears the inscription "In loving memory of Howard Engleman who died in Santos, Brazil, July 17, 1913 age 15 years. I die trusting in the Lord."
Noyes Home has been taking care of "good kids with bad breaks" since 1894. Countless children like Howard have walked through our doors and have found not just a warm bed and food but a safe place to call home. With the support of our dedicated staff and our network of community friends, we have been able to give the children the guidance and encouragement they need to cope with their life challenges.
Every story at the Noyes Home is different, but they all have one thing in common: for a season, the Noyes Home is home to children from diverse life journeys.
Watch Howard's Story…
Become A Champion Of Hope

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: MySQL Connecting I have a MySQL db on A2 hosting. I can connect via terminal with no issues. I can also connect using Oracle workbench. But when I use the exact same creds on Google Data Studio and it says "Access denied, please check your username and password."
I even tried changing the password to 12345 just to test and it says the same thing.
I have tried using both the ip and the host name with no luck.
It is only on google data studio that I can not connect.

A: So it ended up being a firewall setting in the hosting site that I did have to update.

redpajama_set_name: RedPajamaStackExchange


text:
Q: How to determine if website using SharePoint I have a list of websites and I need to determine which ones are using SharePoint (and preferably what version). Is there any way to find it out from headers, HTML code or something else?
Currently I'm checking for '/_layouts/1033/' string in HTML, but I noticed that some SharePoint sites don't have it.
Thanks!

A: It seems that your question may be resolved by the answer to this one : SO question
Hope that helped.

A: Look at the response headers.
SP 2007: microsoftsharepointteamservices: 12
SP 2010: microsoftsharepointteamservices: 14
SP 2013: microsoftsharepointteamservices: 15


6 x 4 inchcolour photograph seen at Rawlings Way, damaged: new item, but damaged in transit, wing mirror broken as shown on photo box is also badly damaged.
Check out these interesting ads related to "enviro 400"
Mint condition model. selling my collection nice model, good conditioned box. in excellent condition, with certificate, slight handling wear to outer box, will be well packed and posted recorded delivery.
NORTHCORD UKBUS6019 ADL Enviro 400 - Quality creative master northcord hedingham bus 1:76 alexander dennis enviro400 save time & money with free auctiva image hosting. i will be adding addition bubblewrap to protect model inside the case as well.
Cmnl ukbus6015 alexander dennis enviro400. In excellent condition with certificate, slight mark on front of box. missing the lh mirror, otherwise in very good unboxed condition.
3714. birmingham. uk. buses. january 2018. recent. From my own collection buy it now may be available.
This PLA 3D Pen Filament Refills is in good working order. It comes complete with original box + instructions etc,. Bargain price 9.27 .


text:
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Canadian mortgage rates,
mortgage calculator & news.
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Construction Set to Begin on New Affordable Housing in Windsor
WINDSOR, Ontario, July 10, 2009 — The Government of Canada, the Province of Ontario and the City of Windsor today announced $1.9 million to fund six construction-ready projects.
The funding was made available as a result of a $1.2 billion joint investment under the amended Canada – Ontario Affordable Housing Program Agreement, which includes funding through Canada's Economic Action Plan and by the Government of Ontario. The federal and provincial governments are contributing equally to this overall investment.
Today's announcement recognized the successful applications by Community Living Windsor and Community Living Essex County, to create 24 units of affordable housing for low-income individuals with disabilities.
Jeff Watson, Member of Parliament forEssex, on behalf of the Honourable Diane Finley, Minister of Human Resources and Skills Development Canada and Minister Responsible for Canada Mortgage and Housing Corporation (CMHC); Bruce Crozier, Member of Provincial Parliament for Essex, on behalf of the Honourable Jim Watson, Ontario Minister of Municipal Affairs and Housing; Eddie Francis, Mayor of Windsor; along with Nelson Santos, Essex County Warden, made the announcement today.
"This Government is working hard to support Canadians during these challenging economic times and we are moving aggressively to ensure Canada's Economic Action Plan is implemented rapidly," said MP Watson. "With these investments in Ontario, we are making an important difference in the lives of some of our community's most vulnerable citizens. Low-income seniors, families and persons with disabilities will soon have access to suitable, affordable housing."
"New housing initiatives add significant support to the McGuinty government's Poverty Reduction Strategy," said MPP Crozier. "This first wave of funding will lead to construction by September and we will continue to work with our municipal partners to ensure more units are built during the life of this program."
"We are very pleased to be part of this significant contribution in support of affordable housing in our community. We are pleased to partner with our local Community Living Associations to help people with developmental challenges live in local communities," said Mayor Francis "Shelter is a fundamental human need and the availability of affordable housing is at the heart of a community's strength and resiliency. This funding is an important investment in the people and the future of our community for generations to come."
"These investments in affordable housing represent a significant step forward in helping our region meet the goals of our poverty reduction strategy for Windsor and Essex County," said Warden Santos. "It is important to recognize that this project is being achieved only because of the tremendous spirit of partnership with our local Community Living Associations and each level of government. Together, this project provides an important element of inclusion for persons of all abilities."
Ontario is moving quickly to implement this new funding, which increases the number of "quick start" projects to 33 totalling $54.7 million and which will improve access to affordable housing for low-income families, seniors and persons with disabilities. It will also create jobs and strengthen local economies. To find out more about affordable housing in Ontario, visit www.mah.gov.on.ca.
Last fall, the Government of Canada committed more than $1.9 billion over the next five years to improve and build new affordable housing and to help the homeless. Canada's Economic Action Plan builds on this with an additional one-time investment of more than $2 billion over two years in new and existing social housing and lending of up to another $2 billion to municipalities for housing-related infrastructure.
To find out more about how the Government of Canada and CMHC are working to build stronger homes and communities for all Canadians, call CMHC at 1-800-668-2642 or visit www.cmhc.ca/housingactionplan. For more information on Canada's Economic Action Plan, call 1-800-O Canada or visit www.actionplan.gc.ca.
Michelle Bakos
Office of Minister Finley
Nicole Burgers
Adam Grachnik
Office of Minister Watson
News source: Canada Mortgage and Housing Corporation (CMHC)
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If you come across another site or company that is offering a lower rate for a comparable term, please let us know.
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All trademarks and logos are the property of their respective owners. All rights reserved.


BIO: Jan Janas has a degree in art education and has been teaching art to adults for 25 years. She is co-author of three art books on silk painting published by North Light Publishing Co. Janas' artwork has been displayed in museums, in owned by corporations, collected by private individuals, and has been hung in many national and international juried art shows. Janas resides in Truchas, New Mexico and opens her studio to the public for sales of her work and to teach private lessons. Jan travels and teaches art throughout the United States.


In 2016 Congatec had an internal workshop in their company and based on it they created a simple tool to calculate the possible lifetime of a copper plated through hole in dependence of the laminate and thermal stress situations.
It's still a beta version, but it is stable and the algorithms behind should be correct.
EIPC have been asked to convey this information to their members and ask if they would like to test it, and report back on their findings, so that Congatec can collect some opinions for improvements in this invaluable tool.
On the 16th of January 2017 the name change will be effective. The name change has no influence on the relationships and the way of working. The contact persons and the interaction with our legal entities will be the same, we also keep the same VAT number and bank accounts and all contracts will remain in place.
We are looking forward to our continued collaboration in the future. Please feel free to reach out to your usual contact person in case of any questions and we will be happy to help wherever we can.


text:
Q: How do i get a value of textField in an UIAlertController? I'm trying to get a textField value from UIAlertController, but whenever I try to enter the value and try to print it out, the output doesn't show anything.
code:
var alert = UIAlertController(title: "Enter the password", message: "Message", preferredStyle: UIAlertControllerStyle.Alert)
alert.addAction(UIAlertAction(title: "Click", style: UIAlertActionStyle.Default, handler: nil))
alert.addTextFieldWithConfigurationHandler({(textField: UITextField!) in
                textField.placeholder = "Enter text:"
                textField.secureTextEntry = true
                println(textField.text)

      })
self.presentViewController(alert, animated: true, completion: nil

A: Try this code :
var inputTextField: UITextField?
let alertController = UIAlertController(title: "Title", message: "Message", preferredStyle: .Alert)
let ok = UIAlertAction(title: "OK", style: .Default, handler: { (action) -> Void in
    // Do whatever you want with inputTextField?.text
    println("\(inputTextField?.text)")
})

let cancel = UIAlertAction(title: "Cancel", style: .Cancel) { (action) -> Void in
}
alertController.addAction(ok)
alertController.addAction(cancel)
alertController.addTextFieldWithConfigurationHandler { (textField) -> Void in
    inputTextField = textField
}
presentViewController(alertController, animated: true, completion: nil)


text:
A propos de Mayer Brown
© 2023 Mayer Brown. Tous droits réservés.
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Stephen Lilley
slilley@mayerbrown.com
Washington DC +1 202 263 3865
Stephen Lilley is a partner in the Washington DC office of Mayer Brown. He focuses his practice on helping clients navigate cutting-edge and interrelated litigation, regulatory, and policy challenges. A member of the firm's Litigation and Cybersecurity & Data Privacy practices, Stephen develops strategies to manage legal risks and to shape regulatory policy across a broad range of substantive areas.
Stephen has significant experience working with clients to identify, evaluate, and manage cybersecurity and data privacy risks; responding to cyber incidents and vulnerability disclosures; and defending businesses in related litigation. Stephen is regularly called upon to advise senior executives and board members on their most challenging cybersecurity risks, to help companies develop governance programs to mitigate those risks, and to lead training exercises to implement and refine those programs. Stephen has particular experience advising on cybersecurity and national security issues relating to the Internet of Things, including vehicles and medical devices, and to manufacturing, critical infrastructure, and other industrial systems. Widely recognized for his cybersecurity law and policy experience, Stephen previously served as Chief Counsel to the Senate Judiciary Committee's Subcommittee on Crime and Terrorism, where he focused on cybersecurity issues.
Stephen's representative cybersecurity and data privacy experience includes:
Counseling businesses across sectors on responding to ransomware and other cybersecurity incidents.
Advising financial institutions on cybersecurity regulatory compliance, including with respect to new regulatory requirements or amended guidance.
Advising manufacturing, energy, chemicals, and other critical infrastructure businesses on legal questions relating to the security of operational technology and other industrial systems.
Counseling businesses on the development of coordinated disclosure programs and the management of vulnerability disclosures.
Providing strategic counsel to companies in a wide range of industries as they assess their cybersecurity posture and engage with their boards of directors.
Drafting comment letters in response to cybersecurity rulemakings and advising companies on legislative developments.
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\section{Introduction}
Linear programming is a fundamental tool in both the theory and applications of combinatorial optimization: We are given a system $A\vex = \veb, \, B\vex \leq \ved$ with $A \in \mathbb R^{m_A \times n}, B \in \mathbb R^{m_B \times n}, \veb \in \mathbb R^{m_A}$ and $\ved \in \mathbb R^{m_B}$ and a cost vector $\vecc \in \mathbb R^n$.
We call an assignment $\vex \in \mathbb R^n$ to the variables \emph{feasible} if it satisfies the system of equalities and inequalities, and the set of these feasible assignments is a \emph{polyhedron}, which will be denoted as $P$ throughout. The goal is to find a feasible assignment $\vex \in \mathbb R^n$ minimizing $\vecc^T \vex$.

Linear programming has been known to be solvable in weakly-polynomial\footnote{From here on out, whenever we speak of a problem with instances containing numbers as inputs as being solvable in polynomial time, we intend this to mean weakly-polynomial time, unless explicitly stated otherwise.} time since the groundbreaking work of Khachiyan on the ellipsoid method \cite{khachiyan1979polynomial} and Karmarkar on the interior point method \cite{karmarkar1984new}. The existence of a strongly polynomial algorithm for linear programming, that is, an algorithm which makes $\poly(n, m_A, m_B)$ arithmetic operations and finds an optimal solution, is a major open problem.
Exploring methods other than the ellipsoid and interior point methods is a possible pathway for a resolution of this important open problem.

One such family of methods are iterative augmentation methods~\cite{dhl-15} using the circuits of the matrix pair $A,B$, which are defined as follows:
\begin{definition}\label{def:circuits}
	Given matrices $A,B$, the set of \textit{circuits} $\mathcal{C}(A,B)$ consists of all $\veg \in \ker(A) \setminus \{ \ve 0 \}$ normalized to coprime integer components for which $B \veg$ is support-minimal over $\{B \vex \mid \vex \in \ker(A) \setminus \{\ve0 \} \}$.
\end{definition}
The set $\mathcal{C}(A,B)$ is, for instance, the set of all \emph{potential} edge directions, arising from any polyhedron having $A$ and $B$ as their constraint matrices over the varying choices of the right-hand sides $\veb$ and $\ved$. This set contains the set of set of \emph{actual} edge directions appearing on $P$ with $\veb$ and $\ved$ fixed as a subset. To be precise, by an \emph{edge direction}, we mean any (normalized) vector in a one-dimensional subspace spanned by the set of optimal points with respect to some cost vector.
This means that considering all circuits in each iteration gives a potentially larger improvement than by only
considering the edge directions, as is the case in the Simplex method.

The set of circuits $\mathcal{C(P)}$ of the polyhedron $P = \{ \vex \in \mathbb R^n \mid A \vex = \veb, B \vex \leq \ved \}$ satisfies $\mathcal{C(P)}=\mathcal{C}(A,B)$. A generic iterative augmentation method for linear programming over $P$ starts from some initial
feasible
solution $\vex^0\in P$ and then, in the $i$-th
iteration with $i=0, 1, \dots$, finds a circuit $\veg^i \in \mathcal{C}(A,B)$ and a
step-length $\lambda^i \in \mathbb R_+$ such that $\vex^{i+1} = \vex^i + \lambda^i \veg^i$ is
feasible and $\vecc^T \veg^i < 0$. The specific choice of $\lambda^i$ and $\veg^i$ distinguishes the individual methods.
For example, a \emph{steepest-descent step} is one which minimizes $\vecc^T \veg^i / \|\veg^i\|_1$, where $\|\cdot\|_1$ is the $1$-norm, and then chooses the largest feasible step-length $\lambda^i$ with respect to $\veg^i$. A \emph{deepest-descent step} is one which minimizes $\lambda^i \vecc^T \veg^i$.

This gives rise to the circuit diameter conjecture \cite{bfh-14}, which states that for any $d$-dimensional polyhedron with $f$ facets, the circuit diameter is bounded above by $f-d$; the circuit diameter is the smallest number of feasible circuit steps of maximal length between two points of a polyhedron.
The significance of studying the circuit-based iterative augmentation methods is also highlighted by recent success of Graver bases in the design of integer programming algorithms~\cite{EisenbrandHKKLO:2019}, since a Graver basis is essentially the integer programming analogue of the set of circuits.

Throughout this paper, we consider polyhedra in the general form $P = \{ \vex \in \mathbb R^n \mid A \vex = \veb, B \vex \leq \ved \}$, just as we already did up to this point.
We assume that $P$ is pointed, i.e., it has a vertex. This is required for some of our problem statements to be well-defined. A check whether $P$ is pointed can be done efficiently through elementary linear algebra.

Let us formally define a deepest-descent step:
\begin{definition}
	Let $P = \{ \vex \in \mathbb R^n \mid A \vex = \veb, B \vex \leq \ved \}$, let $\vecc\in \mathbb{R}^n$ and $\vex^0\in P$, and consider the LP $\min\{\vecc^T\vex: \vex\in P\}$. A {\em $\vecc$-deepest-descent step $\vey$ from $\vex^0$} is a vector $\vey=\alpha \cdot \veg$ for some circuit $\veg \in \mathcal{C}(A,B)$ that maximizes the objective function improvement $-\vecc^T(\alpha \veg)$ among all circuits $\veg \in \mathcal{C}(A,B)$ and all $\alpha > 0$ with $\vex^0+\alpha\veg \in P$.
\end{definition}
When the context is clear, we simply refer to a deepest-descent step $\vey$ (dd-step), dropping information about $\vecc$, $P$, or $\vex^0$. We call the term $c_{\vey} = -\vecc^T\vey$ the {\em deepest-descent improvement} (dd-improvement).
It is known that repeatedly taking deepest-descent steps converges to an optimal solution in $\mathcal{O}(n \log(\veb, \vecc, \ved))$ iterations~\cite{dhl-15}.
A \emph{$k$-approximate dd-step} $\vez$ is a circuit step whose improvement is at least $1/k$ of the improvement of a dd-step, as measured by the objective value $c_{\vey}$ of a dd-step versus $\vecc^T\vez$.
It is known~\cite{AltmanovaKK:2018,dks-19} that iteratively augmenting $k$-approximate dd-steps takes at most $k$-times more iterations to converge to an optimum.
Thus, we are interested in exact and approximate computations of a deepest-descent step. We formally denote this search as follows.

\defProblemFind{Deepest-Descent Step Problem (dd-SP)}
{$\vecc\in \mathbb{R}^n$, polyhedron $P\subset\mathbb{R}^n$, $\vex^0\in P$}
{$\vecc$-deepest-descent circuit step $\vey$ in $P$ from $\vex^0$.}
The natural question leading to our results is then: How hard is it to compute a dd-step?

\subsection{Our Contribution}
Our first positive result with respect to this question pertains to the efficient approximability of \textsc{dd-SP}:
\begin{theorem}\label{thm:deepest-descent-approximation}
	\textsc{dd-SP} can be approximated within a factor of $n$ in polynomial time.
\end{theorem}
This follows by an averaging argument on well-behaved decompositions of the difference of two solutions to an LP as a set of (scaled) circuits

The obvious follow-up question is whether an $n$-approximation can be significantly improved.
We answer this negatively, even for a fairly restricted family of LPs:
\begin{theorem}\label{thm:deepest-descent-unique}
	Even for LPs over $0/1$-polytopes defined by a totally unimodular matrix and with unique optima,
	\textsc{dd-SP} cannot be approximated
	within $O(n^{1-\epsilon})$ for any $\epsilon > 0$ in polynomial time, unless ${\sf P}\xspace
	= {\sf NP}\xspace$.
\end{theorem}
In particular, this demonstrates that to obtain a better approximation ratio or even polynomial tractability, one would need to consider an even more restricted family of LPs.

Further, we turn to the complexity of computing dd-steps exactly. De Loera et al.~\cite{dks-19} have recently shown that \textsc{dd-SP} is \hbox{{\sf NP}-hard}\xspace.
However, a closer look at their construction reveals that they in fact show hardness of detecting whether it is possible to get to some optimum in one circuit step from a given initial point $\vex^0$. We call this problem OCNP:

\defProblemDecide{Optimal Circuit-Neighbor Problem (OCNP)}
{$\vecc\in \mathbb{R}^n$, polyhedron $P\subset\mathbb{R}^n$, $\vex^0\in P$ }
{Is there an optimum $\vex^*$ with respect to $\min\{\vecc^T\vex: \vex\in P\}$ such that $\vex^*-\vex^0$ is a circuit direction?}
Somewhat surprisingly, we show:
\begin{theorem}\label{thm:circuit-neighbor-efficiency}
	OCNP is solvable in polynomial time for LPs with a unique optimum.
\end{theorem}
The standard trick of slightly perturbing the objective $\vecc$ of an LP makes some optimum unique, and the set of objectives with non-unique optima has volume $0$, so in a sense OCNP is easy ``almost always.'' This is contrasted by De Loera et al.~\cite{dks-19} showing the $\mathsf{NP}$-hardness of general OCNP.

This raises the following question: What is the complexity of \textsc{dd-SP} for LPs with a unique optimum, given that OCNP is easy?
Despite the encouraging polynomial-time solvability of OCNP for this special case,
we obtain as a byproduct of  Theorem \ref{thm:deepest-descent-unique} that, unlike OCNP, \textsc{dd-SP} remains hard, even for the same, restricted family of LPs:
\begin{theorem}\label{thm:deepest-descent-hardness}
	\textsc{dd-SP} is
	\hbox{{\sf NP}-hard}\xspace, even for LPs over $0/1$-polytopes defined by a totally unimodular matrix and with unique optima.
\end{theorem}

\subsection{Connections to Previous Work}\label{sec:related}
There are two papers in the literature with an especially strong connection to ours.
We detail this connection separately, and discuss other related work hereafter.

Firstly, and most importantly, an inspiration for this note is the recent paper of De Loera et al.~\cite{dks-19}. Our polynomial-time algorithm for OCNP (Theorem~\ref{thm:circuit-neighbor-efficiency}) stands in contrast to the results of De Loera et al.~\cite{dks-19},
where it is shown that finding optimal circuit-neighbors is \hbox{{\sf NP}-hard}\xspace in general.
Hence, the hardness of OCNP hinges on the existence of multiple optima.
At this point, a flawed line of reasoning might become appealing:

The reduction in \cite{dks-19} comes from the directed Hamiltonian path problem.
By introducing a negligible probability for one-sided error through the Isolation Lemma \cite{isolation},
we may assume that the reduction source instance $D = (N,F,s,t)$ on $n=|N|$ nodes,
has a unique solution---that is, a unique Hamiltonian path from $s$ to $t$.
It is tempting to apply the reduction of \cite{dks-19},
and use the above algorithm for OCNP to solve the produced instance.
This is an optimization problem on the matching polytope $P_M(H)$ of some undirected bipartite graph $H$ on $2n+1$ vertices.
We then have also solved the original instance of the (unique) Hamiltonian path problem in polynomial time.
This fails, however, since the optima of the instance of OCNP are not in one-to-one correspondence with
Hamiltonian paths in the input instance.
Namely, the set of optima in the instance of OCNP is the set of matchings of size $n-1$ in a graph $H'$ obtained from $H$ through the deletion of some edges.
In particular, this set is not necessarily a singleton if the original graph $D$ had a unique Hamiltonian path.

To save this approach, one might apply a perturbation to the cost vector of the produced LP on $P_M(H)$,
to ensure uniqueness of solutions nonetheless (as remarked, uniqueness holds with probability 1).
This perturbation, however, would have to retain precisely all optimal circuit neighbors, and not one of the other optima. Producing this perturbation would therefore require us to have at hand an optimal circuit neighbor in the first place.

To avoid confusion, we stress that ``uniqueness'' refers not to the solutions of OCNP itself,
but to the LP that constitutes part of the input of OCNP (which implies uniqueness of the solution for OCNP).
In other words, there might be a unique optimal circuit neighbor, while the LP has several optima.

Also note that while~\cite{dks-19} discusses approximability, it does \emph{not} concern \textsc{dd-SP} but a different problem: deciding what is the shortest path between two vertices of a polytope, either using the edges of the $1$-skeleton, or using circuit steps.
It is not clear to us whether any inapproximability of \textsc{dd-SP} follows from their construction.

Secondly, we make use of \cite{bv-18} for our positive results on OCNP and the $n$-approximability of \textsc{dd-SP}. Most importantly, the set of circuit directions appear as a subset of the extreme rays of a polyhedral cone constructed from the original input~\cite[Theorem 3]{bv-18}. Recall that extreme rays are those not in the conic hull of any other rays in the object at hand.

\begin{proposition}\label{prop:circuits_as_rays}
Let $P = \{ \vex \in \mathbb R^n \mid A \vex = \veb, B \vex \leq \ved \}$ be a pointed polyhedron. The pointed cone
\begin{align*}
C_{A,B} = \{(\vex, \vey^+, \vey^-) \in \mathbb R^{n + 2m_B} \mid A\vex = \ve0, \ B\vex = \vey^+ - \vey^-, \ \vey^+, \vey^- \geq \ve0\}
\end{align*}
is generated by the set of extreme rays $S \cup T'$, where:
\begin{enumerate}
\item The set $S := \{ (\veg, \vey^+, \vey^-) \mid \veg \in \mathcal{C}(A, B), \ y_i^+ = \max\{(B\veg)_i,0\}, \  y_i^- = \max\{-(B\veg)_i,0\}\}$ gives the circuits of $P$.
\item The set $T' \subseteq T := \{ (\ve0, \vey^+, \vey^-) \mid y^+_i = y^-_i = 1 \text{ for some $i \leq m_B$}, \  y^+_j = y^-_j = 0 \text{ for $j \neq i$}\}$ has size at most $m_B$.
\end{enumerate}
\end{proposition}
Informally, all circuits of $P$ can be found as extreme rays of $C_{A,B}$: a projection of a vector in the set $S$ onto its first $n$ components gives the corresponding circuit $\veg$. The `non-circuits' in the set $T$ are trivial to identify, and the corresponding projection just returns $\ve0$. Note further that the length of a bit encoding of $C_{A,B}$ is (in the order of) at most twice the bit encoding length of $P$. This implies that one can efficiently optimize linear objective functions over the set of (one-normed) circuits. Further, this allows the efficient computation of a conformal sum.

\subsection{Related Work}
Apart from the directly related papers mentioned in the previous subsection, there is vast literature revolving around pivoting rules for circuit augmentation algorithms, and circuits of linear programs in general.
Without any pretense of being comprehensive, let us point to a couple of seminal works (below) and refer to~\cite{bv-18} with respect to circuits, and to~\cite{dks-19} for circuit augmentation and the references therein for a more extensive treatment.

The idea of performing augmenting steps in the direction of circuits instead of only edges during an execution of the simplex algorithm goes back at least to Bland's thesis \cite{bland-thesis} and is explored in detail in \cite{dhl-15} and implemented, for example, in \cite{bv-20}. The notion of a circuit itself in turn was conceived only slightly before that by Rockafellar~\cite{r-69}, and quite fruitfully~\cite{h-99,h-02,h-03} adapted to the integral case by Graver~\cite{g-75}.

\subsection{Outline}\label{sec:outline}
Our main contribution is a proof of the inapproximability of the computation of a dd-step within a factor of $O(n^{1-\epsilon})$, even when restricted to special classes of polyhedra. We begin by connecting to and generalizing previous results in the literature, in Sections \ref{sec:optneighbor} and \ref{sec:napprox}. In Section \ref{sec:inapprox}, we prove our main result. In Section \ref{sec:open}, we conclude with some open questions.

\section{Efficiency of OCNP for LPs with unique optima}\label{sec:optneighbor}
We begin by discussing the OCNP problem.
De Loera et al.~\cite{dks-19} showed that OCNP is \hbox{{\sf NP}-hard}\xspace, and this implies that computing an optimal dd-step is \hbox{{\sf NP}-hard}\xspace.
(We call an optimization problem \hbox{{\sf NP}-hard}\xspace if a corresponding decision version---is it possible to meet or exceed a given objective function value?---is \hbox{{\sf NP}-hard}\xspace.) Recall the discussion in Section \ref{sec:related}.

The proof in~\cite{dks-19} is based on the underlying LP having multiple optima. While showing the claim under this assumption clearly is sufficient, note that for a given polyhedron $P$, the set $C_{\text{multi}}\subset \mathbb{R}^n$ of $\vecc$ for which there exist multiple optima has volume $0$ in $\mathbb{R}^n$. Informally, it is enough to slightly perturb the objective function to create a unique optimum.
We now show that this hardness does not hold if the underlying LP has a unique optimum.
\begin{lemma}\label{lem:circuit-neighbor-efficiency}
Let $\min\{\vecc^T\vex: \vex\in P\}$ be an LP over a polyhedron $P$ with a unique optimum $\vex^*$ (that may not be known), and let $\vex^0\in P$. In polynomial time, it can be verified whether $\vex^*-\vex^0$ is a circuit direction.
\end{lemma}

\begin{proof}
Generally, LPs are solvable in polynomial time. As the LP at hand has a unique optimum $\vex^*$, this $\vex^*$ can be found in polynomial time. Let $\ved = \vex^*-\vex^0$. If $\ved = \ve0$, there is nothing to prove: $\vex^0$ itself already is optimal  and we were able to verify so efficiently. Thus $\ved \neq \ve0$ in the following.

Recall that the circuit directions of a polyhedron $P$ appear as a subset $S$ of the extreme rays of a polyhedral cone $C_{A,B}$, as in Proposition \ref{prop:circuits_as_rays} \cite{bv-18}.  We construct $\ved_S:= (\ved, \vey^+, \vey^-)$, where $\vey_i^{\pm} = \max\{\pm(B\ved)_i,0\}$ as in the definition of $S$. The construction of $\ved_S$ is efficient: $\ved$ is copied over and $\vey^{\pm}$ is derived from a matrix-vector product on the original input and component-wise comparisons.

 Note that $\ved_S\in C_{A,B}$, as $\vex^{\ast},\vex^0 \in P$, and that $\ved_S \notin T$ (as $\ved \neq \ve0$). Thus, if $\ved_S$ is an extreme ray of $C_{A,B}$, it can only be a member of $S$, which would imply that $\ved$ is a circuit. A check whether a given $\ved_S\in C_{A,B}$ is an extreme ray is possible in polynomial time: first, identify the set of active constraints of $\ved_S$ with respect to $C_{A,B}$, i.e., check which constraints in the formulation of $C_{A,B}$ given in Proposition \ref{prop:circuits_as_rays} are satisfied with equality, and construct the associated row submatrix of all active constraints; then perform a rank check for this submatrix -- if its rank is precisely $(n+2m_B) -1$, then $\ved_S$ lies in a one-dimensional face of $C_{A,B}$, i.e., in an extreme ray. These steps are possible in polynomial time because the bit encoding length of $C_{A,B}$ is at most twice the bit encoding length of $P$.

Summing up, $\vex^{\ast}$ can be found efficiently, $\ved_S$ can be constructed efficiently, and $\ved_S$ is an extreme ray of  $C_{A,B}$ if and only if $\vex^{\ast}-\vex^0$ is a circuit direction, and the required check is efficient, too. This proves the claim.
\end{proof}
As an immediate consequence, we obtain the following theorem.
\begin{reptheorem}{thm:circuit-neighbor-efficiency}
OCNP is solvable in polynomial time for LPs with a unique optimum.
\end{reptheorem}
Because the set of objective functions for which there exist multiple optima for a given polyhedron has volume $0$, Theorem \ref{thm:circuit-neighbor-efficiency} tells us that OCNP ``almost always'' can be decided efficiently.

\section{$n$-Approximability of \textsc{dd-SP}}\label{sec:napprox}
Next, we show that an efficient approximation of \textsc{dd-SP} with an error equivalent to the dimension of the underlying polyhedron is possible.

\begin{lemma}\label{lem:deepest-descent-approximation}
Let $P = \{ \vex \in \mathbb R^n \mid A \vex = \veb, B \vex \leq \ved \}$, let $\vecc\in \mathbb{R}^n$ and $\vex^0\in P$, and consider the LP $\min\{\vecc^T\vex: \vex\in P\}$.
Then an $\left(n{-}\rank(A)\right)$-approximation of a $\vecc$-deepest-descent circuit step in $P$ from $\vex^0$ can be computed in polynomial time.
\end{lemma}

\begin{proof}
Let $\vey$ be a $\vecc$-deepest-descent step $\vey$ in $P$ from $\vex^0$ and let $\vex^{\ast}$ be an optimum of $\min\{\vecc^T\vex: \vex\in P\}$. LPs generally are polynomial-time solvable, so an optimal $\vex^{\ast}$ can be computed efficiently.

The vector $\vex^{\ast}-\vex^0$ can be written as a so-called {\em conformal sum} $\vex^{\ast}-\vex^0=\sum_{i=1}^{n'} \alpha_i \veg_i$, where $n'\leq n$, $\alpha_i>0$ and $\veg_i$ is a circuit of $P$ for all $i\leq n'$, and all the circuits $\veg_i$ are sign-compatible with each other (and with $\vex^{\ast}-\vex^0$) \cite{g-75,o-10}. In fact, $n'\leq n{-}\rank(A)$, as a lifting of $\vex^{\ast}-\vex^0$ lies in a cone of dimension at most $n{-}\rank(A)$ whose extreme rays correspond to sign-compatible circuits to it~\cite[Theorem 6 and paragraphs after]{bv-18}. Such a conformal sum can be computed in polynomial time, see, e.g., Algorithm 4 in \cite{bv-18}.

Next, note that $\vecc^T\vey \geq \vecc^T(\vex^{\ast}-\vex^0)=\sum_{i=1}^{n'} \vecc^T(\alpha_i \veg_i)$. (Recall that $\vecc^T\vey$ is negative, as LP is a minimization problem.) Thus, at least one of the $\alpha_i \veg_i$ satisfies $\vecc^T(\alpha_i \veg_i) \leq  \frac{1}{{n'}}\vecc^T(\vex^{\ast}-\vex^0) \leq \frac{1}{{n'}}\vecc^T\vey$. For a given conformal sum, it is efficient to find an $\alpha_i \veg_i$ with smallest value $\vecc^T(\alpha_i \veg_i)$.

 By sign-compatibility of the $\veg_i$, for any index set $I\subset \{1,\dots,n'\}$, $\vex^0+\sum_{i\in I} \alpha_i \veg_i$ lies in $P$. In particular, this holds for $|I|=1$: each of the $\veg_i$ allows for a (maximal-length) circuit step $\beta_i \veg_i$ from $\vex^0$ that stays in $P$, and where $\beta_i \geq \alpha_i$. Note $\vecc^T(\beta_i \veg_i)\leq \vecc^T(\alpha_i \veg_i)$

For a given $\veg_i$, it is efficient to compute the maximal $\beta_i$ such that $\vex^0+\beta_i\veg_i \in P$: each facet of the polyhedron provides an upper bound on $\beta_i$ and one picks the smallest from them. Thus a $\beta_i\veg_i$ with $\vecc^T(\beta_i \veg_i)\leq \frac{1}{{n'}}\vecc^T\vey$ can be computed in polynomial time. This proves the claim.
\end{proof}

As an immediate consequence, we obtain the following corollary.

\begin{reptheorem}{thm:deepest-descent-approximation}
\textsc{dd-SP} can be efficiently approximated within a factor of $n$.
\end{reptheorem}

\section{$O(n^{1-\epsilon})$-Inapproximability of \textsc{dd-SP}}\label{sec:inapprox}
The efficiency of OCNP for LPs with a unique optimum (Section \ref{sec:optneighbor}) is one of the reasons for our interest in a proof for the inapproximability (and implied NP-hardness) of \textsc{dd-SP} that does not rely on this restriction. In Section \ref{sec:napprox}, we saw that there is an efficient $n$-approximation. In this section, we show that this is essentially the best one can expect.

We will provide a proof for the claimed inapproximability of \textsc{dd-SP} that holds even when restricted to special classes of polyhedra. We call a polyhedron  $P = \{ \vex \in \mathbb R^n \mid A \vex = \veb, B \vex \leq \ved \}$ with totally unimodular constraint matrices $A$ and $B$ and integral right-hand sides a TU-polyhedron.

To this end, we will perform a reduction from the following problem

\defProblemFind{Directed Weighted Longest Cycle Problem (DWLCP)}
{Directed graph $G=(V,E,c)$ with arc costs $\vecc\in \mathbb{Q}^{|E|}$ }
{Directed cycle of maximum cost}

DWLCP is a generalization of the Directed (Unweighted) Longest Cycle Problem (DLCP), where the number of arcs of a cycle is counted, i.e., $c_{ij}=1$ for all $(i,j)\in E$. Note that $|V|$ is the largest possible cost of a simple cycle for any instance of DLCP. For a graph $G=(V,E,c)$, DLCP cannot be polynomial-time approximated within $|V|^{1-\epsilon}$ for any $\epsilon > 0$, unless $P = NP$ \cite{bhk-04}. This hardness transfers immediately to DWLCP: the cost $c_{\vex}=\vecc^T\vex$ of a longest cycle $\vex$ cannot be polynomial-time approximated within $|c_\vex|^{1-\epsilon}$  for any $\epsilon > 0$.

Through a reduction from DWLCP, we will prove that a dd-step $\vey$ also cannot be polynomial-time approximated within
$|c_{\vey}|^{1-\epsilon}$ for any $\epsilon > 0$. In our construction, we will guarantee that the underlying LP has a unique solution (and, even stronger, that this fact is known), which allows us to obtain inapproximability and hardness even for such LPs. To this end, we begin with the polynomial construction of an instance of DWLCP from DWLP where all cycles have different costs while retaining the original ``hierarchy'' of costs. We denote the length of a bit encoding of a weighted graph $G$ as $\mathcal{I}_G$.

\begin{lemma}\label{lem:unique-cost-cycles}
Let $G=(V,E)$ be a directed graph. It is possible to construct a set of arc
costs $\vecc\in \mathbb{Q}^{|E|}$ in polynomial time such that: all cycles in
$G'=(V,E,\vecc)$ have different cost, the cost of a cycle is at least its number of arcs, and the cost of a cycle exceeds the number of
arcs by strictly less than one. Further, the bit encoding length of $G'$ is
polynomial in the bit encoding length of $G$.
\end{lemma}

\begin{proof}
Let $G=(V,E)$ be an unweighted directed graph. Let $n=|V|$ and $m=|E|$. First,
we endow $G$ with weights to obtain the weighted graph $G'=(V,E,\vecc')$, where $c'_{ij}=1$ for all
$(i,j)\in E$. In this graph, the cost of a cycle is measured through the number
of arcs. Cycles with the same number of arcs have the same cost. To simplify notation, we will refer to a cycle interchangeably either as a subset of $E$ or as a $0/1$-vector $\vex$ with components $1$ precisely for the arcs on the cycle (a unit flow along the cycle); in particular, for two cycles $C_1, C_2 \subseteq E$ represented by vectors $\vex_1, \vex_2$, by $\vex_1 \setminus \vex_2$ we mean the arc set $C_1 \setminus C_2$. Note that
$\mathcal{I}_{G'}$ is polynomial in $\mathcal{I}_G$: for each arc, only a
(constant-size/single-bit) encoding of the number $1$ is needed.

We will prove the claim through a simple perturbation on $\vecc'$ to resolve any ties between cycles. The new, perturbed costs are called $\vecc$. We are going to show that the perturbation is efficient and changes $\mathcal{I}_{G'}$ only polynomially.

Let $\vecc=\vecc' + \bm{\delta}$, where $\bm{\delta} = (\delta_1,\dots,\delta_m)^T$ and $\delta_i = 2^{-i}$. Informally $\delta_1 = \frac{1}{2}$, $\delta_2 = \frac{1}{4}$, $\delta_3 = \frac{1}{8}$, and so on. Each $\delta_i$ can be encoded in at most $m+1$ bits, due to being the inverses of powers of $2$. Thus, each $c_i=c'_i+\delta_i$ can be encoded in at most $m+2$ digits and $\mathcal{I}_{G'} \leq (m+2) \mathcal{I}_{G}$. As $\mathcal{I}_G\geq m$, the change in encoding length is polynomial. Further, $\vecc$ can be constructed in polynomial time.

It remains to prove that all cycles in $G'$ are of different cost with respect to $\vecc$ and that the cost of cycles has increased by less than one. The latter is immediately clear from $\sum_{i=1}^m \delta_i <1$. Note that the number of arcs of a cycle is $\vecc'^T\vex$ and the cost with respect to $\vecc$ is $\vecc^T\vex$. Let $\vex_1$, $\vex_2$ be two cycles and assume $\vecc'^T\vex_1 > \vecc'^T\vex_2$, which in particular implies $\vecc'^T\vex_1 \geq \vecc'^T\vex_2 + 1$. As $\vecc^T\vex_2<\vecc'^T\vex_2+\sum_{i=1}^m \delta_i$ and $\sum_{i=1}^m \delta_i <1$, we have $\vecc^T\vex_1\geq \vecc'^T\vex_1 > \vecc^T\vex_2$.

Finally, consider two cycles $\vex_1\neq\vex_2$ with $\vecc'^T\vex_1 = \vecc'^T\vex_2$.  The cycles have the same number of arcs, so $\vex_1\backslash \vex_2\neq \emptyset$ and $\vex_2 \backslash \vex_1\neq \emptyset$. Let index $k$ be smallest among all arcs in $\vex_1\backslash \vex_2$, and let $l$ be smallest among all arcs used in $\vex_2 \backslash \vex_1$. Without loss of generality, assume $k < l$. Note $\delta_k > \sum_{i=k+1}^m \delta_i$. Thus $\vecc^T\vex_1-\vecc^T\vex_2\geq \delta_k - (\sum_{i=k+1}^m \delta_i) > 0$, i.e., $\vecc^T\vex_1>\vecc^T\vex_2$. This proves the claim.
\end{proof}

\begin{remark}
It is natural to ask whether it is necessary to introduce numbers of exponential size into $\vecc$ in the Lemma above.
In other words, does every integer vector $\vecc$ which preserves exactly one optimum of $\vecc'$ and does not introduce any new optima have some entry of order $2^n$?
This is open, but observe that if we require something stronger, the answer is ``yes.''

We show that every integer $\vecc$ which breaks all ties between cycles of the same length must have exponential entries.
Clearly the number of cycles of length $n$ can be $\Omega(2^n)$.
Denote $c_{\max} = \|\vecc\|_\infty$.
In order to get a distinct value $\vecc^T \vex$ for every cycle $\vex$ of length $n$, $c_{\max} \in \Omega(2^n)$, as otherwise there are not enough distinct values $\vecc^T \vex$ since clearly $0 \leq \vecc^T \vex \leq n \cdot c_{\max}$.
\end{remark}

We are now ready to prove our main claim.

\begin{reptheorem}{thm:deepest-descent-unique}
Let $P = \{ \vex \in \mathbb R^n \mid A \vex = \veb, B \vex \leq \ved \}$ with
$A,B\in \mathbb{R}^{m\times n}$, let $\vecc\in \mathbb{R}^n$ and  $\vex^0\in
P$, and consider the LP $\min\{\vecc^T\vex: \vex\in P\}$. A deepest-descent
circuit step $\vey$ in $P$ from~$\vex^0$ cannot be approximated
within $O(n^{1-\epsilon})$ for any $\epsilon > 0$ in polynomial time, unless ${\sf P}\xspace
= {\sf NP}\xspace$. The hardness holds for LPs with unique optima, over $0/1$-polytopes, TU-polyhedra, or any
combination thereof.
\end{reptheorem}

\begin{proof}
We will prove the claim through a reduction from the Directed Longest Cycle
Problem (DLCP), for which it was shown in ~\cite{bhk-04} that no
$|V|^{1-\epsilon}$-approximation can be computed for any $\epsilon>0$ in
polynomial time, unless ${\sf P}\xspace={\sf NP}\xspace$, even in graphs of constant maximum
out-degree $\Delta^+$. By
\Cref{lem:unique-cost-cycles}, for a given graph $G=(V,E)$ it is possible to
efficiently construct a weighted graph $G'=(V,E,\vecc)$ in which all cycles have a
different cost and their cost lies strictly between the number of arcs of the
cycle and that number plus one. The graph $G'$ can be used as input for a
Directed Weighted Longest Cycle Problem (DWLCP) and also has constant maximum
out-degree $\Delta^+$. If there was an efficient
$|V|^{1-\epsilon}$-approximation for DWLCP on $G'$, then there would be an
efficient $|V|^{1-\epsilon}$-approximation for DLCP on $G$. We will show that if
there exists an algorithm to efficiently $O(n^{1-\epsilon})$-approximate \textsc{dd-SP}, then there exists an efficient
$|V|^{1-\epsilon}$-approximation for DWLCP, and in turn DLCP -- a contradiction
unless $P= NP$. Further, the move from $G$ to $G'$ will allow us to show that we
retain this hardness even for LPs with unique optima.

Let $G=(V,E)$ be a directed graph underlying an instance of DLCP and
$G'=(V,E,\vecc)$ the corresponding weighted directed graph with perturbed costs
constructed as in \Cref{lem:unique-cost-cycles}, in turn an instance of DWLCP.
Next, specify capacities $u_{ij}=1$ for each $(i,j)\in E$ to obtain a network
$G''=(V,E,\vecc,\veu)$. The costs $c_{ij}$ remain unchanged for all $(i,j) \in E$,
i.e., they are the same as in $G'$.

This input can be used to specify a circulation problem. Recall that
a circulation problem is a special case of a minimum-cost-flow problem and has a
natural representation as an LP. Using the negative costs $-c_{ij}$ (recall the $c_{ij}$ are positive), we obtain
\begin{align}
\tag{LP}
\begin{split}\label{LP}
\min \ &  -\vecc^T \vex \\
\text{s.t.} \  & A\vex = \ve0 \\
  & \ve0  \leq\vex \leq \ve 1,
\end{split}
\end{align}
where $A$ is the node-arc incidence matrix of $G'$, and $\ve0$ and $\ve1$ are
vectors of all-zeros and
all-ones of appropriate dimensions, respectively. The all-ones vector gives the
capacity constraints. Let
$P$ refer to the polyhedron forming the feasible region of~\eqref{LP}. As
node-arc incidence matrices are totally unimodular, and as the right-hand side
vectors are the integral $\ve0$ and $\ve1$, $P$ is a $0/1$-polytope in
$\mathbb{R}^n$, with $n=|E|$. There always exists an optimal vertex to
an LP on a bounded polytope, so an optimal objective function value for~\eqref{LP} is defined through
a selection of arcs forming a circulation in $G'$. By the same argument as in
\Cref{lem:unique-cost-cycles}, any subset of arcs sums up to a different total
cost. Thus~\eqref{LP} has a unique optimal solution.

Next, consider a trivial feasible flow $\vex^0$ defined by $x^0_{ij}=0$ for
each $(i,j)\in E$. We are going to show that an efficient approximation of the
dd-step in $P$ from $\vex^0$ would imply an efficient approximation of DLCP.

Recall that the set of circuits of a node-arc incidence matrix $A$ corresponds
precisely to the simple {\em undirected} cycles underlying the network; a corresponding vector $\veg \in \{-1,0,1\}$ would have entry $1$ for each directed arc used in the `correct' direction and $-1$ for each directed arc used in the `wrong', opposite direction~\cite{ykk-84,ort-05,bdf-16b}.  The same holds for the
circuits of $P = \{x\in \mathbb{R}^n\mid A\vex = \ve0, \ve0  \leq\vex \leq \ve 1\}$, as the inequality constraints $\ve0  \leq\vex \leq \ve 1$ are represented through a constraint matrix $B=\binom{I}{-I}$, where $I$ is an identity matrix; recall Definition \ref{def:circuits}.

Since we have $x^0_{ij}=0$ and $u_{ij}=1$ for each $(i,j)\in E$, the step
length~$\alpha$ can always be set to $1$ for any valid circuit, i.e., if there
exists $\alpha>0$ with $\vex^0+\alpha\veg \in P$ for a circuit $\veg$, then
$\vex^0+\veg \in P$ and $\vex^0+\beta\veg \not\in P$ for any $\beta>1$. Further, any circuit $\veg$ with $\vex^0+\veg \in P$ can only have $0,1$ entries, as $x^0_{ij}=0$ for each $(i,j)\in E$. This means that edges can only be used in the correct direction. Therefore, an exact dd-step~$\vey$ for~\eqref{LP} from $\vex^0$ is in one-to-one
correspondence to a simple {\em directed} cycle of maximum length (as~\eqref{LP} minimizes
over negative arc costs).

Recall that by the hardness result in~\cite{bhk-04},  we may assume that the
maximum out-degree of $G'$ is some fixed constant $\Delta^+$, so in particular
$|E|\leq \Delta^+|V|$. Assume we had an algorithm that for a given $\epsilon >
0$ finds an $(n/\Delta^+)^{1-\epsilon}$-approximate dd-step $\vey_\epsilon$; let $\vey$ refer to an exact dd-step, and let the associated dd-improvements be denoted as $c_{\vey_\epsilon}$ and~$c_{\vey}$,
respectively. Then we have that
$\frac{c_{\vey}}{c_{\vey_\epsilon}}\leq
(n/\Delta^+)^{1-\epsilon}=(|E|/\Delta^+)^{1-\epsilon}\leq |V|^{1-\epsilon}$. We
know that $c_{\vey}=-\vecc^T\vey=-\vecc^T\veg$ and
$c_{\vey_\epsilon}=-\vecc^T\vey_\epsilon=-\vecc^T(\alpha\veg_\epsilon)$ for some
$\alpha\in(0,1]$ and some circuits $\veg$ and $\veg_\epsilon$. By the above, we
may assume that $\alpha=1$, so $c_{\vey_\epsilon}=-\vecc^T(\veg_\epsilon)$.
Since
$\frac{-\vecc^T\veg}{-\vecc^T\veg_\epsilon}=\frac{c_{\vey}}{c_{\vey_\epsilon}}
\leq |V|^{1-\epsilon}$, $\veg_\epsilon$ corresponds to a cycle in $G''$ that
approximates the longest cycle within a factor of $|V|^{1-\epsilon}$ (since by
construction of the cost vector $\vecc$, a cycle has maximum cost if and only
if it has maximum length). This would imply a polynomial-time
$|V|^{1-\epsilon}$-approximation algorithm for general DLCP.

The polytope we used in this construction is a $0/1$-polytope with a TU-matrix,
and the LP at hand has a unique optimum and this fact is known apriori; see
above. This shows that the hardness of approximation holds even for LPs adhering
to all these restrictions. This proves the claim.
\end{proof}

As an immediate consequence, we obtain {\sf NP}\xspace-hardness of \textsc{dd-SP} from this inapproximability result.

\begin{reptheorem}{thm:deepest-descent-hardness}
\textsc{dd-SP} is \hbox{{\sf NP}-hard}\xspace, even for LPs over $0/1$-polytopes defined by a TU matrix and with a unique optimum.
 \end{reptheorem}

A direct proof of the {\sf NP}\xspace-hardness of \textsc{dd-SP} would be possible through
a reduction from Hamiltonian cycle instead of DWLCP, following a similar line
of arguments as in the proof of Theorem \ref{thm:deepest-descent-unique}. A
perturbation of the arc costs would not be necessary, and neither would be the
careful connection of $|V|$ and $|E|$ through the inapproximability of DWLCP for
graphs with fixed maximum out-degree. However, to obtain the final part of
Theorem \ref{thm:deepest-descent-hardness} -- that hardness persists even for
LPs with unique optima -- one would have to reduce from a variant of Hamiltonian
cycle where the underlying graph has a {\em unique} circulation with a maximal
number of arcs and one has the apriori information that there exists such a
circulation. (This property is what would guarantee the existence of a unique
optimum in (\ref{LP}), and apriori knowledge thereof.) To the best of the
authors' knowledge, hardness of this variant has not been studied yet in the
literature.

\section{Open Problems}\label{sec:open}
We conclude with two open problems related to our results. First, Theorem~\ref{thm:deepest-descent-approximation} shows how to $n$-approximate \textsc{dd-SP}. However, for the purposes of solving an LP using dd-steps, this is irrelevant, as the first step of the algorithm is to completely solve the LP itself. Is there a combinatorial $n$-approximation of \textsc{dd-SP}, i.e., an algorithm, which does not use the polynomial solvability of an LP as a black-box? Actually, this would yield a new algorithm for linear programming, so to make the question well-posed, we ask whether there is a combinatorial $n$-approximation of \textsc{dd-SP} for some non-trivial class of constraint matrices? Secondly, we have shown strong inapproximability of \textsc{dd-SP}. What are (natural) classes of LP instances for which \textsc{dd-SP} admits, e.g., $\log(n)$- or even $c$-approximation, for some constant $c \in \mathbb R_+$? Potential candidate classes include uni-- or bimodular LPs, and more generally, LPs with minors of bounded absolute value. Also, structurally restricted classes of LPs might be of interest. In particular, for $n$-fold LPs, which have a special block-structure, an approximation ratio for \textsc{dd-SP} polynomially depending only on the parameters (that is, block size) would be desirable, and would break below the barrier proved in this paper.

{\small\section*{Acknowledgments}
\vspace*{-0.2cm}
\noindent S. Borgwardt gratefully acknowledges support of this work through NSF award 2006183 {\em Circuit Walks in Optimization}, Algorithmic Foundations, CCF, Division of Computing and Communication Foundations, and through Simons Collaboration Grant 524210 {\em Polyhedral Theory in Data Analytics} before. C. Brand was supported by project 0016976 of OP VVV call 02\_18\_053. M. Kouteck\'y and A. Feldmann were partially supported by Charles University project UNCE/SCI/004 and by the project 19-27871X of GA \u{C}R.}

\bibliographystyle{elsarticle-num}

redpajama_set_name: RedPajamaArXiv


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Caroline m. Fremont was born in Toronto, Canada. Her education included commercial art and mechanical drafting. While still in high school a family move to Florida made it possible, after graduation, to enroll into the Ringling School of Art becoming an honour student. This was followed by a transfer with advanced standing to and graduation from The Ontario School of Art, in Interior Design.
It took retirement from an extended career in interior design then summer retreats from country life to cottage life to bring about a new interest, painting for the sake of painting. Serendipity played its part at The Haliburton School of the Arts in the finding of Toronto art educator and artist Steve Rose. It was through his mentorship, encouragement and direction that a hidden talent emerged. Annual studio opportunities with American artist Tammra Sigler help push techniques in mixed media and composition.
Caroline burst onto the Toronto art scene in 2012 as part of a four woman show at The Canadian Women's Art Association after being greatly encouraged by gallery showings in Naples, Florida and a first place award from their Von Liebig Art Center. She participates yearly in numerous gallery exhibitions and has been an annual contributor to The Toronto Gilda's Art Auction. Filled with artistic passion and now in her 70's she has art works that hang in Canadian, American and German collections.
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When Israelis say, "I worry about my grandchildren's future," it has a radically different dimension than similar concerns expressed in many other countries.
Europeans' current anxiety about the future derives mainly from darkening social and economic prospects. A number of Europeans and Americans are also apprehensive about climate change.
For Israelis, physical survival is a prime matter, often over and above their many other concerns.
Israeli society faces mortal risks from parts of the Muslim world where extreme anti-Semitic hatemongering is massive. Israel is threatened with a second Holocaust, for which the ideological basis is being laid today. The Islamic world has substantial components – such as Iran's leaders and Hamas – that promote the genocide of Israel and Jews.
In the future, significant threats may also come from others – for example, if and when atomic bombs or fissile material fall into the hands of terrorists.
Palestinian society is permeated with sympathy for the most criminal major Muslim movement, al Qaeda. Those who see a "peace agreement" as an interim stage toward the annihilation of Israel are unreliable partners and not exactly the kind of people to whom concessions can be made.
In view of possible, albeit presently unforeseeable, radical changes in the Middle East, true peace is not a totally impossible dream. Yet this can be the case only after many other problematic developments are dealt with.
Meanwhile, Israelis for generations to come will serve in the army and risk their lives. Once one's life is at stake, everything else becomes secondary.
As a result of their experiences, Israelis live in a reality that is unique among democratic nations and have worldviews that differ significantly from people who live in other societies.
Serving in the army means Israelis cannot live a life as fully defined by individualism as their counterparts do in Europe and the U.S. One can understand this from Prime Minister Netanyahu's words on Yom HaZikaron – Israel's Memorial Day – earlier this year.
"When you hear the siren tonight, we will turn into one family and the citizens of Israel will be united in our remembrance," Netanyahu told his countrymen.
Western Europeans rarely turn into one family, though it may happen occasionally to some extent in the face of a natural or terrorist catastrophe or upon winning a soccer championship.
In Israel, due to the ups and downs of the economy and the political situation generally, few people outside government services assume that their employment will be uninterrupted and lifelong. This reality has helped cultivate a more flexible mindset among Israelis than is the case among Europeans or Americans.
Contrary to the typical Westerner, many Israeli youngsters realize they owe much to society and that what Israeli society owes them has its limits. At the same time, Israel's unity is threatened in very different ways by major segments of two growing parts of the population – Israeli Arabs and haredim – as well by much smaller but far more vociferous groups of extreme leftists.
The threat of seeing one's country destroyed is far from theoretical in Israel. Given this type of reality, Israeli youngsters must continue informal learning throughout their active lifetimes. In other words, invest in one's brain as much as possible because that will be the main portable source of one's knowledge in crisis situations.
Israelis should learn as many skills as possible – preferably those that can be used abroad as well as in Israel. Further, it is necessary to learn to speak proper English, which will remain the lingua franca of this century. Spending a few years abroad in one's youth can be extremely useful for one's future, wherever that may be.
In an uncertain Israeli environment, the important skill of improvisation will frequently be required; further development of it will, therefore, be more than merely useful.
Murphy's Law is not necessarily valid: Not everything that can go wrong will go wrong, and if Israel continues to flourish in the remarkable way it has, the same skills will come in very handy in finding a place in Israeli society.
In Israel as elsewhere, there will be a small number of people who are extraordinarily talented. If they have reasonable emotional intelligence, they will enjoy unprecedented opportunities in a complex society.


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