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Parity Systems and the Delta-Matroid Intersection Problem
Parity Systems and the Delta-Matroid Intersection Problem Andr´eBouchet ∗ and Bill Jackson † Submitted: February 16, 1998; Accepted: September 3, 1999. Abstract We consider the problem of determining when two delta-matroids on the same ground-set have a common base. Our approach is to adapt the theory of matchings in 2-polymatroids developed by Lov´asz to a new abstract system, which we call a parity system. Examples of parity systems may be obtained by combining either, two delta- matroids, or two orthogonal 2-polymatroids, on the same ground-sets. We show that many of the results of Lov´aszconcerning ‘double flowers’ and ‘projections’ carry over to parity systems. 1 Introduction: the delta-matroid intersec- tion problem A delta-matroid is a pair (V, ) with a finite set V and a nonempty collection of subsets of V , called theBfeasible sets or bases, satisfying the following axiom:B ∗D´epartement d’informatique, Universit´edu Maine, 72017 Le Mans Cedex, France. [email protected] †Department of Mathematical and Computing Sciences, Goldsmiths’ College, London SE14 6NW, England. [email protected] 1 the electronic journal of combinatorics 7 (2000), #R14 2 1.1 For B1 and B2 in and v1 in B1∆B2, there is v2 in B1∆B2 such that B B1∆ v1, v2 belongs to . { } B Here P ∆Q = (P Q) (Q P ) is the symmetric difference of two subsets P and Q of V . If X\ is a∪ subset\ of V and if we set ∆X = B∆X : B , then we note that (V, ∆X) is a new delta-matroid.B The{ transformation∈ B} (V, ) (V, ∆X) is calledB a twisting. -
Prahladh Harsha
Prahladh Harsha Toyota Technological Institute, Chicago phone : +1-773-834-2549 University Press Building fax : +1-773-834-9881 1427 East 60th Street, Second Floor Chicago, IL 60637 email : [email protected] http://ttic.uchicago.edu/∼prahladh Research Interests Computational Complexity, Probabilistically Checkable Proofs (PCPs), Information Theory, Prop- erty Testing, Proof Complexity, Communication Complexity. Education • Doctor of Philosophy (PhD) Computer Science, Massachusetts Institute of Technology, 2004 Research Advisor : Professor Madhu Sudan PhD Thesis: Robust PCPs of Proximity and Shorter PCPs • Master of Science (SM) Computer Science, Massachusetts Institute of Technology, 2000 • Bachelor of Technology (BTech) Computer Science and Engineering, Indian Institute of Technology, Madras, 1998 Work Experience • Toyota Technological Institute, Chicago September 2004 – Present Research Assistant Professor • Technion, Israel Institute of Technology, Haifa February 2007 – May 2007 Visiting Scientist • Microsoft Research, Silicon Valley January 2005 – September 2005 Postdoctoral Researcher Honours and Awards • Summer Research Fellow 1997, Jawaharlal Nehru Center for Advanced Scientific Research, Ban- galore. • Rajiv Gandhi Science Talent Research Fellow 1997, Jawaharlal Nehru Center for Advanced Scien- tific Research, Bangalore. • Award Winner in the Indian National Mathematical Olympiad (INMO) 1993, National Board of Higher Mathematics (NBHM). • National Board of Higher Mathematics (NBHM) Nurture Program award 1995-1998. The Nurture program 1995-1998, coordinated by Prof. Alladi Sitaram, Indian Statistical Institute, Bangalore involves various topics in higher mathematics. 1 • Ranked 7th in the All India Joint Entrance Examination (JEE) for admission into the Indian Institutes of Technology (among the 100,000 candidates who appeared for the examination). • Papers invited to special issues – “Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding” (with Eli Ben- Sasson, Oded Goldreich, Madhu Sudan, and Salil Vadhan). -
Oasics-SOSA-2019-14.Pdf (0.4
Simple Greedy 2-Approximation Algorithm for the Maximum Genus of a Graph Michal Kotrbčík Department of Computer Science, Comenius University, 842 48 Bratislava, Slovakia [email protected] Martin Škoviera1 Department of Computer Science, Comenius University, 842 48 Bratislava, Slovakia [email protected] Abstract The maximum genus γM (G) of a graph G is the largest genus of an orientable surface into which G has a cellular embedding. Combinatorially, it coincides with the maximum number of disjoint pairs of adjacent edges of G whose removal results in a connected spanning subgraph of G. In this paper we describe a greedy 2-approximation algorithm for maximum genus by proving that removing pairs of adjacent edges from G arbitrarily while retaining connectedness leads to at least γM (G)/2 pairs of edges removed. As a consequence of our approach we also obtain a 2-approximate counterpart of Xuong’s combinatorial characterisation of maximum genus. 2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms → Graph algorithms analysis, Mathematics of computing → Graph algorithms, Mathematics of computing → Graphs and surfaces Keywords and phrases maximum genus, embedding, graph, greedy algorithm Digital Object Identifier 10.4230/OASIcs.SOSA.2019.14 Acknowledgements The authors would like to thank Rastislav Královič and Jana Višňovská for reading preliminary versions of this paper and making useful suggestions. 1 Introduction One of the paradigms in topological graph theory is the study of all surface embeddings of a given graph. The maximum genus γM (G) parameter of a graph G is then the maximum integer g such that G has a cellular embedding in the orientable surface of genus g.A result of Duke [12] implies that a graph G has a cellular embedding in the orientable surface of genus g if and only if γ(G) ≤ g ≤ γM (G) where γ(G) denotes the (minimum) genus of G. -
A Tight Extremal Bound on the Lovász Cactus Number in Planar Graphs
A Tight Extremal Bound on the Lovász Cactus Number in Planar Graphs Parinya Chalermsook Aalto University, Espoo, Finland parinya.chalermsook@aalto.fi Andreas Schmid Max Planck Institute for Informatics, Saarbrücken, Germany [email protected] Sumedha Uniyal Aalto University, Espoo, Finland sumedha.uniyal@aalto.fi Abstract A cactus graph is a graph in which any two cycles are edge-disjoint. We present a constructive proof of the fact that any plane graph G contains a cactus subgraph C where C contains at least 1 a 6 fraction of the triangular faces of G. We also show that this ratio cannot be improved by showing a tight lower bound. Together with an algorithm for linear matroid parity, our bound implies two approximation algorithms for computing “dense planar structures” inside any graph: (i) 1 A 6 approximation algorithm for, given any graph G, finding a planar subgraph with a maximum 1 number of triangular faces; this improves upon the previous 11 -approximation; (ii) An alternate (and 4 arguably more illustrative) proof of the 9 approximation algorithm for finding a planar subgraph with a maximum number of edges. Our bound is obtained by analyzing a natural local search strategy and heavily exploiting the exchange arguments. Therefore, this suggests the power of local search in handling problems of this kind. 2012 ACM Subject Classification Mathematics of computing → Graph theory Keywords and phrases Graph Drawing, Matroid Matching, Maximum Planar Subgraph, Local Search Algorithms Digital Object Identifier 10.4230/LIPIcs.STACS.2019.19 Related Version Full Version: https://arxiv.org/abs/1804.03485. Funding Parinya Chalermsook: Part of this work was done while PC and AS were visiting the Simons Institute for the Theory of Computing. -
Professor Shang-Hua Teng
Professor Shang-Hua Teng Department of Computer Science, University of Southern California 2054 N. New Hampshire Ave 941 Bloom Walk, SAL 300, Los Angeles, CA 90089 Los Angeles, CA 90027 (213) 440-4498, [email protected] (617) 440-4281 EMPLOYMENT: Seeley G. Mudd Professor and Chair (Computer Science) USC (Fall 2009 – ) Professor (Computer Science) Boston University (2002 – 2009) Research Affiliate Professor (Mathematics) MIT (1999 – Present) Visiting Research Scientist Microsoft Research Asia and New England (2004 – Present) Senior Research Scientist Akamai (1997 – Present) Visiting Professor (Computer Science) Tsinghua University (2004 – 2008) Professor (Computer Science)[Associate Professor before 1999] UIUC (2000 – 2002) Research Scientist IBM Almaden Research Center (1997 – 1999) Assistant Professor (Computer Science) Univ. of Minnesota (1994 – 1997) Research Scientist Intel Corporation (Summer 1994, 1995) Instructor (Mathematics) MIT (1992 – 1994) Computer Scientist NASA Ames Research Center (Summer 1993) Research Scientist Xerox Palo Alto Research Center (1991 – 1992) EDUCATION: Ph.D. (Computer Science) Carnegie Mellon University (1991) Thesis: “A Unified Geometric Approach to Graph Partitioning.” Advisor: Gary Lee Miller M.S. (Computer Science) University of Southern California (1988) Thesis: “Security, Verifiablity, and Universality in Distributed Computing”. Advised by Leonard Adleman and Ming-Deh Huang B.S. (Computer Science) & B.A. (Electrical Engineering) Shanghai Jiao-Tong University, China (1985) Thesis: “Efficient Methods for Implementing Logic Programs.” Advisor: Yixing Hou HONORS/AWARDS: 2011 ACM STOC Best Paper 2009 ACM Fellow, 2009 2009 Fulkerson Prize (American Mathematical Society/Mathematical Programming Society) 2008 G¨odel Prize (ACM-EATCS). 2003 The NSF’s 2003 Budget Request to Congress listed the Smoothed Analysis of Spielman-Teng as one of three most significant accomplishments funded by its computer science division. -
Algebraic Algorithms in Combinatorial Optimization
Algebraic Algorithms in Combinatorial Optimization CHEUNG, Ho Yee A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Philosophy in Computer Science and Engineering The Chinese University of Hong Kong September 2011 Thesis/Assessment Committee Professor Shengyu Zhang (Chair) Professor Lap Chi Lau (Thesis Supervisor) Professor Andrej Bogdanov (Committee Member) Professor Satoru Iwata (External Examiner) Abstract In this thesis we extend the recent algebraic approach to design fast algorithms for two problems in combinatorial optimization. First we study the linear matroid parity problem, a common generalization of graph matching and linear matroid intersection, that has applications in various areas. We show that Harvey's algo- rithm for linear matroid intersection can be easily generalized to linear matroid parity. This gives an algorithm that is faster and simpler than previous known al- gorithms. For some graph problems that can be reduced to linear matroid parity, we again show that Harvey's algorithm for graph matching can be generalized to these problems to give faster algorithms. While linear matroid parity and some of its applications are challenging generalizations, our results show that the al- gebraic algorithmic framework can be adapted nicely to give faster and simpler algorithms in more general settings. Then we study the all pairs edge connectivity problem for directed graphs, where we would like to compute minimum s-t cut value between all pairs of vertices. Using a combinatorial approach it is not known how to solve this problem faster than computing the minimum s-t cut value for each pair of vertices separately. -
Derandomizing the Ahlswede-Winter Matrix-Valued Chernoff Bound Using Pessimistic Estimators, and Applications
THEORY OF COMPUTING, Volume 4 (2008), pp. 53–76 http://theoryofcomputing.org Derandomizing the Ahlswede-Winter matrix-valued Chernoff bound using pessimistic estimators, and applications Avi Wigderson∗ David Xiao† Received: November 28, 2007; published: May 15, 2008. Abstract: Ahlswede and Winter [IEEE Trans. Inf. Th. 2002] introduced a Chernoff bound for matrix-valued random variables, which is a non-trivial generalization of the usual Chernoff bound for real-valued random variables. We present an efficient derandomization of their bound using the method of pessimistic estimators (see Raghavan [JCSS 1988]). As a consequence, we derandomize an efficient construction by Alon and Roichman [RSA 1994] of an expanding Cayley graph of logarithmic degree on any (possibly non-abelian) group. This gives an optimal solution to the homomorphism testing problem of Shpilka and Wigderson [STOC 2004]. We also apply these pessimistic estimators to the problem of solving semidefinite covering problems, thus giving a deterministic algorithm for the quantum hypergraph cover problem of Ahslwede and Winter. ∗Partially supported by NSF grant CCR-0324906 †Supported by an NDSEG Graduate Fellowship and a NSF Graduate Fellowship ACM Classification: G.3, G.2.2, F.2.1, F.1.2 AMS Classification: 68W20, 68R10, 60F10, 20D60, 81P68, 15A18 Key words and phrases: Chernoff bounds, matrix-valued random variables, derandomization, pes- simistic estimators Authors retain copyright to their work and grant Theory of Computing unlimited rights to publish the work electronically and in hard copy. Use of the work is permitted as long as the author(s) and the journal are properly acknowledged. For the detailed copyright statement, see http://theoryofcomputing.org/copyright.html. -
Matroid Matching: the Power of Local Search
Matroid Matching: the Power of Local Search [Extended Abstract] Jon Lee Maxim Sviridenko Jan Vondrák IBM Research IBM Research IBM Research Yorktown Heights, NY Yorktown Heights, NY San Jose, CA [email protected] [email protected] [email protected] ABSTRACT can be easily transformed into an NP-completeness proof We consider the classical matroid matching problem. Un- for a concrete class of matroids (see [46]). An important weighted matroid matching for linear matroids was solved result of Lov´asz is that (unweighted) matroid matching can by Lov´asz, and the problem is known to be intractable for be solved in polynomial time for linear matroids (see [35]). general matroids. We present a PTAS for unweighted ma- There have been several attempts to generalize Lov´asz' re- troid matching for general matroids. In contrast, we show sult to the weighted case. Polynomial-time algorithms are that natural LP relaxations have an Ω(n) integrality gap known for some special cases (see [49]), but for general linear and moreover, Ω(n) rounds of the Sherali-Adams hierarchy matroids there is only a pseudopolynomial-time randomized are necessary to bring the gap down to a constant. exact algorithm (see [8, 40]). More generally, for any fixed k ≥ 2 and > 0, we obtain a In this paper, we revisit the matroid matching problem (k=2 + )-approximation for matroid matching in k-uniform for general matroids. Our main result is that while LP- hypergraphs, also known as the matroid k-parity problem. based approaches including the Sherali-Adams hierarchy fail As a consequence, we obtain a (k=2 + )-approximation for to provide any meaningful approximation, a simple local- the problem of finding the maximum-cardinality set in the search algorithm gives a PTAS (in the unweighted case). -
Algorithms for Incremental Planar Graph Drawing and Two-Page Book Embeddings
Algorithms for Incremental Planar Graph Drawing and Two-page Book Embeddings Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨at der Universit¨at zu Koln¨ vorgelegt von Martin Gronemann aus Unna Koln¨ 2015 Berichterstatter (Gutachter): Prof. Dr. Michael Junger¨ Prof. Dr. Markus Chimani Prof. Dr. Bettina Speckmann Tag der m¨undlichenPr¨ufung: 23. Juni 2015 Zusammenfassung Diese Arbeit besch¨aftigt sich mit zwei Problemen bei denen es um Knoten- ordnungen in planaren Graphen geht. Hierbei werden als erstes Ordnungen betrachtet, die als Grundlage fur¨ inkrementelle Zeichenalgorithmen dienen. Solche Algorithmen erweitern in der Regel eine vorhandene Zeichnung durch schrittweises Hinzufugen¨ von Knoten in der durch die Ordnung gegebene Rei- henfolge. Zu diesem Zweck kommen im Gebiet des Graphenzeichnens verschie- dene Ordnungstypen zum Einsatz. Einigen dieser Ordnungen fehlen allerdings gewunschte¨ oder sogar fur¨ einige Algorithmen notwendige Eigenschaften. Diese Eigenschaften werden genauer untersucht und dabei ein neuer Typ von Ord- nung entwickelt, die sogenannte bitonische st-Ordnung, eine Ordnung, welche Eigenschaften kanonischer Ordnungen mit der Flexibilit¨at herk¨ommlicher st- Ordnungen kombiniert. Die zus¨atzliche Eigenschaft bitonisch zu sein erm¨oglicht es, eine st-Ordnung wie eine kanonische Ordnung zu verwenden. Es wird gezeigt, dass fur¨ jeden zwei-zusammenh¨angenden planaren Graphen eine bitonische st-Ordnung in linearer Zeit berechnet werden kann. Im Ge- gensatz zu kanonischen Ordnungen, k¨onnen st-Ordnungen naturgem¨aß auch fur¨ gerichtete Graphen verwendet werden. Diese F¨ahigkeit ist fur¨ das Zeichnen von aufw¨artsplanaren Graphen von besonderem Interesse, da eine bitonische st-Ordnung unter Umst¨anden es erlauben wurde,¨ vorhandene ungerichtete Zei- chenverfahren fur¨ den gerichteten Fall anzupassen. -
Inductive Tools for Connected Delta-Matroids and Multimatroids
European Journal of Combinatorics 63 (2017) 59–69 Contents lists available at ScienceDirect European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc Inductive tools for connected delta-matroids and multimatroids Carolyn Chun a, Deborah Chun b, Steven D. Noble c a Mathematics Department, United States Naval Academy, Chauvenet Hall, 572C Holloway Road, Annapolis, MD 21402-5002, United States b Department of Mathematics, West Virginia University Institute of Technology, Montgomery, WV, United States c Department of Economics, Mathematics and Statistics, Birkbeck, University of London, Malet St, London, WC1E 7HX, United Kingdom article info a b s t r a c t Article history: We prove a splitter theorem for tight multimatroids, generalizing Received 25 January 2016 the corresponding result for matroids, obtained independently by Accepted 21 February 2017 Brylawski and Seymour. Further corollaries give splitter theorems for delta-matroids and ribbon graphs. ' 2017 Elsevier Ltd. All rights reserved. 1. Introduction A matroid M D .E; B/ is a finite ground set E together with a non-empty collection of subsets of the ground set, B, that are called bases, satisfying the following conditions, which are stated in a slightly different way from what is most common in order to emphasize the connection with other combinatorial structures discussed in this paper. 1. If B1 and B2 are bases and x 2 B1 4 B2, then there exists y 2 B1 4 B2 such that B1 4 fx; yg is a basis. 2. All bases are equicardinal. Matroid theory is often thought of as a generalization of graph theory, as a matroid .M; B/ may be constructed from a graph G by taking E to be set of edges of G and B to be the edge sets of maximal spanning forests of G. -
Mathematics People
Mathematics People information, sometimes even one incorrect bit can de- Chudnovsky and Spielman stroy the integrity of the data stream; adding verification Selected as MacArthur Fellows ‘codes’ to the data helps to test its accuracy. Spielman and collaborators developed several families of codes Maria Chudnovsky of Columbia University and Dan- that optimize speed and reliability, some approaching the iel Spielman of Yale University have been named Mac- theoretical limit of performance. One, an enhanced version Arthur Fellows for 2012. Each Fellow receives a grant of of low-density parity checking, is now used to broadcast US$500,000 in support, with no strings attached, over the and decode high-definition television transmissions. In a next five years. separate line of research, Spielman resolved an enduring Chudnovsky was honored for her work on the classifica- mystery of computer science: why a venerable algorithm tions and properties of graphs. The prize citation reads, for optimization (e.g., to compute the fastest route to the “Maria Chudnovsky is a mathematician who investigates airport, picking up three friends along the way) usually the fundamental principles of graph theory. In mathemat- works better in practice than theory would predict. He and ics, a graph is an abstraction that represents a set of similar a collaborator proved that small amounts of randomness things and the connections between them—e.g., cities and convert worst-case conditions into problems that can be the roads connecting them, networks of friendship among solved much faster. This finding holds enormous practi- people, websites and their links to other sites. When used cal implications for a myriad of calculations in science, to solve real-world problems, like efficient scheduling for engineering, social science, business, and statistics that an airline or package delivery service, graphs are usually depend on the simplex algorithm or its derivatives. -
The Gödel Prize 2020 - Call for Nominatonn
The Gödel Prize 2020 - Call for Nominatonn Deadline: February 15, 2020 The Gödel Prize for outntanding papern in the area of theoretial iomputer niienie in nponnored jointly by the European Annoiiaton for Theoretial Computer Siienie (EATCS) and the Annoiiaton for Computng Maihinery, Speiial Innterent Group on Algorithmn and Computaton Theory (AC M SInGACT) The award in prenented annually, with the prenentaton taaing plaie alternately at the Innternatonal Colloquium on Automata, Languagen, and Programming (InCALP) and the AC M Symponium on Theory of Computng (STOC) The 28th Gödel Prize will be awarded at the 47th Innternatonal Colloquium on Automata, Languagen, and Programming to be held during 8-12 July, 2020 in Beijing The Prize in named in honour of Kurt Gödel in reiogniton of hin major iontributonn to mathematial logii and of hin interent, diniovered in a leter he wrote to John von Neumann nhortly before von Neumann’n death, in what han beiome the famoun “P vernun NP” quenton The Prize iniluden an award of USD 5,000 Award Committee: The 2020 Award Commitee ionnintn of Samnon Abramnay (Univernity of Oxford), Anuj Dawar (Chair, Univernity of Cambridge), Joan Feigenbaum (Yale Univernity), Robert Krauthgamer (Weizmann Innnttute), Daniel Spielman (Yale Univernity) and David Zuiaerman (Univernity of Texan, Auntn) Eligibility: The 2020 Prize rulen are given below and they nupernede any diferent interpretaton of the generii rule to be found on webniten of both SInGACT and EATCS Any renearih paper or nerien of papern by a ningle author or by