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40 DM12 Abstracts IP0 [email protected] Dnes Knig Prize Lecture: Talk Title TBD Abstract not available at time of publication. IP4 Adding and Counting Zeev Dvir Princeton University In mathematics, the stuff of partitions seems like mere [email protected] child’s play. The speaker will explain how the sim- ple task of adding and counting has fascinated many of the world’s leading mathematicians: Euler, Ramanujan, IP1 Hardy, Rademacher, Dyson, to name a few. And as is typ- Cell Complexes in Combinatorics ical in number theory, many of the most fundamental (and simple to state) questions have remained open. In 2010, Cell complexes of various kinds have been invented in the speaker, with the support of the American Institute for topology to help analyze manifolds and other spaces. By Mathematics and the National Science Foundation, assem- introducing a combinatorial structure they make algo- bled an international team of researchers to attack some rithms for computing topological invariants possible. Sim- of these problems. Come hear Professor Ono speak about plicial complexes are well-known examples. In the other their findings: new theories which solve some of the famous direction, several structures studied in combinatorics nat- old questions. urally suggest associated cell complexes. Can the link to topology provided by these cell complexes be of use for Ken Ono dealing with purely combinatorial questions, or are they Emory University just idle curiosities? The answer is definitely ”yes” in the Department of Mathematics and Computer Science simplicial case, as testified by several successes of what has [email protected] come to be called ”topological combinatorics”. But what about more general cell complexes? In the talk this ques- tion will be discussed. Examples, old and new, arising in IP5 graph, group and number theory will be reviewed. Coloring 3-colorable Graphs; Graph Theory Fi- nally Strikes Back! Anders Bj¨orner Royal Institute of Technology (KTH) We consider the problem of coloring a 3-colorable graph in [email protected] polynomial time using as few colors as possible. Starting with Wigderson in 1982 (O(n1/2) colors), Blum in 1990 came with the first polynomial improvements (O(n3/8) IP2 colors). His improvement is based on graph theoretical On Sidorenko’s Conjecture approach. Karger, Motwani, Sudan in 1994 is the first to use semi-definite programming (SDP) to give improve- The Erdos-Simonovits-Sidorenko conjecture is well-known ment, and then Karger and Blum in 1997 combines Blum’s in combinatorics but it has equivalent formulations in anal- method with the SDP improvement to show that O(n0.2142 ) ysis and probability theory. The shortest formulation is an colors suffices. Since then, the only improvements in semi- integral inequality related to Mayer integrals in statistical definite programming have been made (Arora, Chlamtac, mechanics and Feynman integrals in quantum field theory. and Charikar in 2006 (O(n0.2111 ) colors), and Chlamtac We present new progress in the area. Part of the talk is in 2007 (O(n0.2072 ) colors)). We present the first im- based on joint results with J.L. Xiang Li. In particular we provement on the graph theory side since Blum in 1990. present a type of calculus (based on logarithmic functions) With a purely graph theoretical approach, we get down to which can be used to prove inequalities between subgraph O(n4/11) colors (over Blum’s O(n3/8) colors). Combining densities. it with SDP, we get down to O(n0.2038 ) colors. Joint work Bal´azs Szegedy with Mikkel Thorup (ATT Research) University of Toronto Ken-ichi Kawarabayashi Department of Mathematics National Institute of Informatics, Japan [email protected] k [email protected] IP3 IP6 Forcing Large Transitive Subtournamets Talk Title TBA - Singh The Erdos Hajnal Conjecture states roughly that a graph Abstract not available at time of publication. with some induced subgraph excluded has a large clique or a large stable set. A similar statement can be formu- Mona Singh lated for tournaments (a tournament is an orientation of a Princeton University complete graph), replacing cliques and stable sets by tran- [email protected] sitive subtournaments; and the two conjectures turn out to be equivalent. This talk will survey a number of recent results related to the latter conjecture. In particular, we IP7 will discuss a new infinite class of tournaments excluding The Hub Labeling Algorithm which forces large transitive subtournaments; to the best of our knowledge this is the first such class not obtained This is a survey of Hub Labeling results for general and by the so-called substitution operation. road networks. Given a weighted graph, a distance oracle takes as an input a pair of vertices and returns the dis- Maria Chudnovsky tance between them. The labeling approach to distance Columbia DM12 Abstracts 41 oracle design is to precompute a label for every vertex so [email protected] that distances can be computed from the corresponding labels. This approach has been introduced by [Gavoille et Muhammad Javaid al. ’01], who also introduced the Hub Labeling algorithm FAST-National University of Computer and Emerging (HL). HL has been further studied by [Cohen et al. ’02]. Sciences We study HL in the context of graphs with small high- Lahore Campus, Lahore, Pakistan way dimension (e.g., road networks). We show that under [email protected] this assumption HL labels are small and the queries are sublinear. We also give an approximation algorithm for computing small HL labels that uses the fact that short- CP1 est path set systems have small VC-dimension. Although Distingushing with Nordhaus Gaddum graphs polynomial-time, precomputation given by theory is too slow for continental-size road networks. However, heuris- Albertson and Collins introduced the distinguishing num- tics guided by the theory are fast, and compute very small ber in 1996 and Collins and Trenk introduced the distin- labels. This leads to the fastest currently known practi- guishing chromatic number of a graph which requires that cal distance oracles for road networks. The simplicity of the coloring be proper as well as distinguishing in 2006. HL queries allows their implementation inside of a rela- We revisit the Nordhaus-Gaddum inequalities which give tional database (e.g., in SQL), and query efficiency assures bounds on the sum and the product of χ(G)andχ(G)for real-time response. Furthermore, including HL data in the any graph G, and give analogues of these inequalities for database allows efficient implementation of more sophis- the distinguishing chromatic number. We provide a new ticated location-based queries. This approach brings the characterization of those graphs that achieve equality for power of location-based services to SQL programmers the upper bound in the sum Nordhaus-Gaddum inequality, which leads to a polynomial-time recognition algorithm for Andrew Goldberg this class and efficient computation of their chromatic num- Microsoft Research Silicon Valley bers. [email protected] Karen Collins Wesleyan University IP8 [email protected] Algorithms, Graph Theory, and the Solution of Laplacian Linear Equations Ann N. Trenk Wellesley College We survey several fascinating concepts and algorithms in [email protected] graph theory that arise in the design of fast algorithms for solving linear equations in the Laplacian matrices of graphs. We will begin by explaning why linear equations CP1 in these matrices are so interesting. The problem of solving On Super (a,d)-Edge Antimagic Total Labeling of linear equations in these matrices motivates a new notion Subdivided Caterpillar of what it means for one graph to approximate another. This leads to a problem of graph sparsification–the ap- Let G =(V,E) be a graph with v = |V (G)| vertices and proximation of a graph by a sparser graph. Our algorithms e = |E(G)| edges. An(a, d)-edge antimagic total labeling for solving Laplacian linear equations will exploit surpris- is a bijection “f” from V (G)UE(G)to the set of consec- ingly strong approximations of graphs by sparse graphs, utive integers 1, 2, .., v + e, such that the weights of the and even by trees. We will survey the roles that spectral edge W = {w(xy):xy in E(G)} form an arithmetic pro- graph theory, random matrix theory, graph sparsification, gression with the initial term “a” and common difference low-stretch spanning trees and local clustering algorithms “d” where w(xy)=f(x)+f(y)+f(xy). “W” is called play in the design of fast algorithms for solving Laplacian the set of edge-weights of the graph G. Additionally, if linear equations. f(V )=1, 2, ..., |V (G)| then f is called super (a, d)-edge antimagic total labeling and G is called super(a, d)-edge Daniel Spielman antimagic total. In this paper we formulate, super (a, d)- Yale University edge antimagic total labeling on subdivided caterpillar for [email protected] different integral values of d. Muhammad Javaid CP1 FAST-National University of Computer and Emerging On Antimagic Vertex Labeling Of Hypergraphs Sciences Lahore Campus, Lahore, Pakistan Let H =(V,E) be a graph with vertex set V (H)andedge [email protected] set E(H). Moreover suppose that v = |V (H)|,e = |E(H)| and N denotes the set of non negative integers. An an- timagic (magic) vertex labeling is a bijection “f” from Akhlaq Bhatti V (H) to the set of consecutive integers 1, 2,...,v if the FAST-NUCES LAHORE, PAKISTAN induced edge labeling “g” from E(H) to the set of non neg- [email protected] ative integers N defined by g(e)=? f(v) for all v in E(G)is injective (constant) function. A hypergraph H is called an- CP1 timagic (magic) iff there exist an antimagic (magic) vertex labeling of H.
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