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LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students. -
Prizes and Awards Session
PRIZES AND AWARDS SESSION Wednesday, July 12, 2021 9:00 AM EDT 2021 SIAM Annual Meeting July 19 – 23, 2021 Held in Virtual Format 1 Table of Contents AWM-SIAM Sonia Kovalevsky Lecture ................................................................................................... 3 George B. Dantzig Prize ............................................................................................................................. 5 George Pólya Prize for Mathematical Exposition .................................................................................... 7 George Pólya Prize in Applied Combinatorics ......................................................................................... 8 I.E. Block Community Lecture .................................................................................................................. 9 John von Neumann Prize ......................................................................................................................... 11 Lagrange Prize in Continuous Optimization .......................................................................................... 13 Ralph E. Kleinman Prize .......................................................................................................................... 15 SIAM Prize for Distinguished Service to the Profession ....................................................................... 17 SIAM Student Paper Prizes .................................................................................................................... -
Kombinatorische Optimierung – Blatt 1
Prof. Dr. Volker Kaibel M.Sc. Benjamin Peters Wintersemester 2016/2017 Kombinatorische Optimierung { Blatt 1 www.math.uni-magdeburg.de/institute/imo/teaching/wise16/kombopt/ Pr¨asentation in der Ubung¨ am 20.10.2016 Aufgabe 1 Betrachte das Hamilton-Weg-Problem: Gegeben sei ein Digraph D = (V; A) sowie ver- schiedene Knoten s; t ∈ V . Ein Hamilton-Weg von s nach t ist ein s-t-Weg, der jeden Knoten in V genau einmal besucht. Das Problem besteht darin, zu entscheiden, ob ein Hamilton-Weg von s nach t existiert. Wir nehmen nun an, wir h¨atten einen polynomiellen Algorithmus zur L¨osung des Kurzeste-¨ Wege Problems fur¨ beliebige Bogenl¨angen. Konstruiere damit einen polynomiellen Algo- rithmus fur¨ das Hamilton-Weg-Problem. Aufgabe 2 Der folgende Graph abstrahiert ein Straßennetz. Dabei geben die Kantengewichte die (von einander unabh¨angigen) Wahrscheinlichkeiten an, bei Benutzung der Straßen zu verunfallen. Bestimme den sichersten Weg von s nach t durch Aufstellen und L¨osen eines geeignetes Kurzeste-Wege-Problems.¨ 2% 2 5% 5% 3% 4 1% 5 2% 3% 6 t 6% s 4% 2% 1% 2% 7 2% 1% 3 Aufgabe 3 Lesen Sie den Artikel The Year Combinatorics Blossomed\ (erschienen 2015 im Beijing " Intelligencer, Springer) von William Cook, Martin Gr¨otschel und Alexander Schrijver. S. 1/7 {:::a äi, stq: (: W illianr Cook Lh t ivers it1' o-l' |'\'at ttlo t), dt1 t t Ll il Martin Grötschel ZtLse lnstitrttt; orttl'l-U Ilt:rlin, ()trrnnt1, Alexander Scfu ijver C\\/ I nul Ltn ivtrsi !)' o-l' Anrstt rtlan, |ietherlattds The Year Combinatorics Blossomed One summer in the mid 1980s, Jack Edmonds stopped Much has been written about linear programming, by the Research Institute for Discrete Mathematics including several hundred texts bearing the title. -
Matroid Partitioning Algorithm Described in the Paper Here with Ray’S Interest in Doing Ev- Erything Possible by Using Network flow Methods
Chapter 7 Matroid Partition Jack Edmonds Introduction by Jack Edmonds This article, “Matroid Partition”, which first appeared in the book edited by George Dantzig and Pete Veinott, is important to me for many reasons: First for per- sonal memories of my mentors, Alan J. Goldman, George Dantzig, and Al Tucker. Second, for memories of close friends, as well as mentors, Al Lehman, Ray Fulker- son, and Alan Hoffman. Third, for memories of Pete Veinott, who, many years after he invited and published the present paper, became a closest friend. And, finally, for memories of how my mixed-blessing obsession with good characterizations and good algorithms developed. Alan Goldman was my boss at the National Bureau of Standards in Washington, D.C., now the National Institutes of Science and Technology, in the suburbs. He meticulously vetted all of my math including this paper, and I would not have been a math researcher at all if he had not encouraged it when I was a university drop-out trying to support a baby and stay-at-home teenage wife. His mentor at Princeton, Al Tucker, through him of course, invited me with my child and wife to be one of the three junior participants in a 1963 Summer of Combinatorics at the Rand Corporation in California, across the road from Muscle Beach. The Bureau chiefs would not approve this so I quit my job at the Bureau so that I could attend. At the end of the summer Alan hired me back with a big raise. Dantzig was and still is the only historically towering person I have known. -
The Traveling Preacher Problem, Report No
Discrete Optimization 5 (2008) 290–292 www.elsevier.com/locate/disopt The travelling preacher, projection, and a lower bound for the stability number of a graph$ Kathie Camerona,∗, Jack Edmondsb a Wilfrid Laurier University, Waterloo, Ontario, Canada N2L 3C5 b EP Institute, Kitchener, Ontario, Canada N2M 2M6 Received 9 November 2005; accepted 27 August 2007 Available online 29 October 2007 In loving memory of George Dantzig Abstract The coflow min–max equality is given a travelling preacher interpretation, and is applied to give a lower bound on the maximum size of a set of vertices, no two of which are joined by an edge. c 2007 Elsevier B.V. All rights reserved. Keywords: Network flow; Circulation; Projection of a polyhedron; Gallai’s conjecture; Stable set; Total dual integrality; Travelling salesman cost allocation game 1. Coflow and the travelling preacher An interpretation and an application prompt us to recall, and hopefully promote, an older combinatorial min–max equality called the Coflow Theorem. Let G be a digraph. For each edge e of G, let de be a non-negative integer. The capacity d(C) of a dicircuit C means the sum of the de’s of the edges in C. An instance of the Coflow Theorem (1982) [2,3] says: Theorem 1. The maximum cardinality of a subset S of the vertices of G such that each dicircuit C of G contains at most d(C) members of S equals the minimum of the sum of the capacities of any subset H of dicircuits of G plus the number of vertices of G which are not in a dicircuit of H. -
What's the Matter with Tie-Breaking? Improving Efficiency in School Choice
American Economic Review 2008, 98:3, 669–689 http://www.aeaweb.org/articles.php?doi=10.1257/aer.98.3.669 What’s the Matter with Tie-Breaking? Improving Efficiency in School Choice By Aytek Erdil and Haluk Ergin* In several school choice districts in the United States, the student proposing deferred acceptance algorithm is applied after indifferences in priority orders are broken in some exogenous way. Although such a tie-breaking procedure preserves stability, it adversely affects the welfare of the students since it intro- duces artificial stability constraints. Our main finding is a polynomial-time algorithm for the computation of a student-optimal stable matching when pri- orities are weak. The idea behind our construction relies on a new notion which we call a stable improvement cycle. We also investigate the strategic properties of the student-optimal stable mechanism. (JEL C78, D82, I2) Until about a decade ago, children in the United States were assigned to public schools by the district they live in, without taking into account the preferences of their families. Such systems overlooked reallocations of seats which could Pareto improve welfare. Motivated by such con- cerns, several US cities, including New York City, Boston, Cambridge, Charlotte, Columbus, Denver, Minneapolis, Seattle, and St. Petersburg-Tampa, started centralized school choice pro- grams. Typically in these programs, each family submits a preference list of schools, including those outside their district, and then a centralized mechanism assigns students to schools based on the preferences. The mechanisms initially adopted by school choice programs were ad hoc, and did not perform well in terms of efficiency, incentives, and/or stability. -
Submodular Functions, Optimization, and Applications to Machine Learning — Spring Quarter, Lecture 11 —
Submodular Functions, Optimization, and Applications to Machine Learning | Spring Quarter, Lecture 11 | http://www.ee.washington.edu/people/faculty/bilmes/classes/ee596b_spring_2016/ Prof. Jeff Bilmes University of Washington, Seattle Department of Electrical Engineering http://melodi.ee.washington.edu/~bilmes May 9th, 2016 f (A) + f (B) f (A B) + f (A B) Clockwise from top left:v Lásló Lovász Jack Edmonds ∪ ∩ Satoru Fujishige George Nemhauser ≥ Laurence Wolsey = f (A ) + 2f (C) + f (B ) = f (A ) +f (C) + f (B ) = f (A B) András Frank r r r r Lloyd Shapley ∩ H. Narayanan Robert Bixby William Cunningham William Tutte Richard Rado Alexander Schrijver Garrett Birkho Hassler Whitney Richard Dedekind Prof. Jeff Bilmes EE596b/Spring 2016/Submodularity - Lecture 11 - May 9th, 2016 F1/59 (pg.1/60) Logistics Review Cumulative Outstanding Reading Read chapters 2 and 3 from Fujishige's book. Read chapter 1 from Fujishige's book. Prof. Jeff Bilmes EE596b/Spring 2016/Submodularity - Lecture 11 - May 9th, 2016 F2/59 (pg.2/60) Logistics Review Announcements, Assignments, and Reminders Homework 4, soon available at our assignment dropbox (https://canvas.uw.edu/courses/1039754/assignments) Homework 3, available at our assignment dropbox (https://canvas.uw.edu/courses/1039754/assignments), due (electronically) Monday (5/2) at 11:55pm. Homework 2, available at our assignment dropbox (https://canvas.uw.edu/courses/1039754/assignments), due (electronically) Monday (4/18) at 11:55pm. Homework 1, available at our assignment dropbox (https://canvas.uw.edu/courses/1039754/assignments), due (electronically) Friday (4/8) at 11:55pm. Weekly Office Hours: Mondays, 3:30-4:30, or by skype or google hangout (set up meeting via our our discussion board (https: //canvas.uw.edu/courses/1039754/discussion_topics)). -
Ecole Polytechnique Departement D'economie
ECOLE POLYTECHNIQUE CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE Judge : Don't Vote! Michel BALINSKI Rida LARAKI Novembre 2010 Cahier n° 2010-27 DEPARTEMENT D'ECONOMIE Route de Saclay 91128 PALAISEAU CEDEX (33) 1 69333033 http://www.economie.polytechnique.edu/ / mailto:[email protected] Judge : Don't Vote! 1 Michel Balinski 2 Rida Laraki Novembre 2010 Cahier n° 2010-27 Résumé: Cet article explique pourquoi (1) le modèle traditionnel de choix social n’est pas réaliste, (2) il ne peut en aucun cas proposer une méthode acceptable pour classer et élire, et (3) qu’un modèle plus réaliste implique inévitablement une seule méthode pour classer et élire ---le jugement majoritaire--- qui satisfait le mieux qu’il se peut les critères traditionnels de ce qui constitue une bonne méthode. Abstract: This article explains why (1) the traditional model of the theory of social choice misrepresents reality, (2) it cannot lead to acceptable methods of ranking and electing in any case, and (3) a more realistic model leads inevitably to one method of ranking and electing---majority judgment---that best meets the traditional criteria of what constitutes a good method. Mots clés : Paradoxe d’Arrow, paradoxe de Condorcet, patinage artistique, choix social, jugement majoritaire, manipulation stratégique, vote Key Words : Arrow’s paradox, Condorcet’s paradox, majority judgment, skating, social choice, strategic manipulation, voting Classification JEL: D71, C72 Classification AMS: 91A, 91C, 90B 1 Economics Department of the Ecole Polytechnique and CNRS, France. 2 Economics Department of the Ecole Polytechnique and CNRS, France. Judge : Don’t Vote! Michel Balinski and Rida Laraki The final test of a theory is its capacity to solve the problems which originated it. -
Basic Polyhedral Theory 3
BASIC POLYHEDRAL THEORY VOLKER KAIBEL A polyhedron is the intersection of finitely many affine halfspaces, where an affine halfspace is a set ≤ n H (a, β)= x Ê : a, x β { ∈ ≤ } n n n Ê Ê for some a Ê and β (here, a, x = j=1 ajxj denotes the standard scalar product on ). Thus, every∈ polyhedron is∈ the set ≤ n P (A, b)= x Ê : Ax b { ∈ ≤ } m×n of feasible solutions to a system Ax b of linear inequalities for some matrix A Ê and some m ≤ n ′ ′ ∈ Ê vector b Ê . Clearly, all sets x : Ax b, A x = b are polyhedra as well, as the system A′x = b∈′ of linear equations is equivalent{ ∈ the system≤ A′x }b′, A′x b′ of linear inequalities. A bounded polyhedron is called a polytope (where bounded≤ means− that≤ there − is a bound which no coordinate of any point in the polyhedron exceeds in absolute value). Polyhedra are of great importance for Operations Research, because they are not only the sets of feasible solutions to Linear Programs (LP), for which we have beautiful duality results and both practically and theoretically efficient algorithms, but even the solution of (Mixed) Integer Linear Pro- gramming (MILP) problems can be reduced to linear optimization problems over polyhedra. This relationship to a large extent forms the backbone of the extremely successful story of (Mixed) Integer Linear Programming and Combinatorial Optimization over the last few decades. In Section 1, we review those parts of the general theory of polyhedra that are most important with respect to optimization questions, while in Section 2 we treat concepts that are particularly relevant for Integer Programming. -
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Experimental Democracy – Collective Intelligence for a Diverse and Complex World Felix Gerlsbeck Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2013 © 2013 Felix Gerlsbeck All rights reserved Abstract Experimental Democracy – Collective Intelligence for a Diverse and Complex World Felix Gerlsbeck My dissertation is motivated by the following observation: while we care very much about the outcomes of the democratic process, there is widespread uncertainty about ex ante how to produce them—and quite often there is also disagreement and uncertainty about what they are in the first place. Consequently, unless we have a definite idea what “better decision-making” might be, it is not obvious which institutional reforms or changes in democratic structures would actually promote it. Democracy is a wide concept, and not all institutional constellations and rules and regulations that can be called democratic function equally well. In this dissertation therefore I offer a specific model of democracy— “Experimental Democracy”—that unites the view that the quality of decisions matter, with taking into account the circumstances of uncertainty and disagreement that define political problems. On this account, a desirable political mechanism is one that realizes an experimental method of policy-making directed at solving problems, such that we can expect it to make progress over time, even though we cannot rule out that it will get things wrong—possibly even frequently. I also show how democracy may best realize such an experimental method, and which particular institutional features of democracy could serve this purpose. -
Paths, Trees, and Flowers
Paths, Trees, and Flowers “Jack Edmonds has been one of the creators of the ered as arcs, each joining a pair of the four nodes which field of combinatorial optimization and polyhedral correspond to the two islands and the two shores. combinatorics. His 1965 paper ‘Paths, Trees, and Flowers’ [1] was one of the first papers to suggest the possibility of establishing a mathematical theory of efficient combinatorial algorithms . ” [from the award citation of the 1985 John von Neumann Theory Prize, awarded annually since 1975 by the Institute for Operations Research and the Management Sciences]. In 1962, the Chinese mathematician Mei-Ko Kwan proposed a method that could be used to minimize the lengths of routes walked by mail carriers. Much earlier, in 1736, the eminent Swiss mathematician Leonhard Euler, who had elevated calculus to unprecedented heights, had investigated whether there existed a walk across all seven bridges that connected two islands in the river Pregel with the rest of the city of Ko¨nigsberg Fig. 2. The graph of the bridges of Ko¨nigsberg. on the adjacent shores, a walk that would cross each bridge exactly once. The commonality between the two graph-theoretical problem formulations reaches deeper. Given any graph, for one of its nodes, say i0,findanarca1 that joins node i0 and another node, say i1. One may try to repeat this process and find a second arc a2 which joins node i1 to a node i2 and so on. This process would yield a sequence of nodes—some may be revisited—successive pairs of which are joined by arcs (Fig. -
Election by Majority Judgement: Experimental Evidence Michel Balinski, Rida Laraki
Election by Majority Judgement: Experimental Evidence Michel Balinski, Rida Laraki To cite this version: Michel Balinski, Rida Laraki. Election by Majority Judgement: Experimental Evidence. 2007. hal- 00243076 HAL Id: hal-00243076 https://hal.archives-ouvertes.fr/hal-00243076 Preprint submitted on 8 Feb 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ECOLE POLYTECHNIQUE CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE Election by Majority Judgement: Experimental Evidence Michel Balinski Rida Laraki 17 décembre 2007 Cahier n° 2007-28 LABORATOIRE D'ECONOMETRIE 1rue Descartes F-75005 Paris (33) 1 55558215 http://ceco.polytechnique.fr/ mailto:[email protected] Election by Majority Judgement: Experimental Evidence Michel Balinski1 Rida Laraki2 17 décembre 2007 Cahier n° 2007-28 Résumé: Le jugement majoritaire est une méthode d'élection. Cette méthode est l'aboutissement d'une nouvelle théorie du choix social où les électeurs jugent les candidats au lieu de les ranger. La théorie est développée dans d'autres publications ([2, 4]). Cet article décrit et analyse des expériences électorales conduites pendant les deux dernières élections présidentielles françaises dans plusieurs buts: (1) démontrer que le jugement majoritaire est une méthode pratique, (2) la décrire et établir ses principales propriétés, (3) démontrer qu'elle échappe aux paradoxes classiques, et (4) illustrer comment dans la pratique tous les mécanismes de vote connus violent certains critères importants.