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DM18 Abstracts 1 DM18 Abstracts 1 IP1 called game bias), in each round Waiter offers to Client Analytic Methods in Graph Theory q+1 previously unoffered edges of G. Client chooses one of the edges offered, the rest go to Waiter. Waiter wins The theory of graph limits provides analytic tools to study the game if by the time every edge of G has been claimed, large graphs. Such tools have found applications in various Client’s graph possesses a given graph theoretic property areas of computer science and mathematics; they are also P, Client wins otherwise. We will present several recent closely linked to the flag algebra method, which changed results about Waiter-Client games played on complete and the landscape of extremal combinatorics. We will present random graphs and discuss the role of the so called prob- an introduction to this rapidly developing area of graph abilistic intuition in their analysis. Based on joint works theory and survey some of the recent results obtained in with M. Bednarska-Bzdega, D. Hefetz, T. Luczak, W. E. the area. Tan, N. Trumer. Daniel Kral Michael Krivelevich University of Warwick Tel Aviv University [email protected] [email protected] IP2 IP5 Interpolation Polynomials, Operator Method, and New Developments in Hypergraph Ramsey Theory Theory of Enumeration The Ramsey number rk(s, n) is the minimum integer N, Goncarov Polynomials are the basis of solutions of the clas- such that for any red/blue coloring of the k-tuples of { } sical Goncarov Interpolation Problem, which have been 1, 2, ..., N ,thereares integers such that every k-tuple studied extensively by analysts due to their significance in among them is red, or there are n integers such that every the interpolation theory of smooth and analytic functions. k-tuple among them is blue. In this talk, I will discuss new These Polynomials also play an important role in combi- lower bounds for rk(s, n) which nearly settles a question natorics due to their close relations to parking functions. of Erdos and Hajnal from 1972. I will also discuss a more This is not just a coincidence. In this talk we will present general function introduced by Erdos and Hajnal, and sev- the interpolation problems with delta-operators, develop eral interesting open problems in the area. This is joint the algebraic and analytic theory of delta-Goncarov poly- work with Dhruv Mubayi. nomials, and apply these results to problems in binomial Andrew Suk enumeration and order statistics. University of California, San Diego Catherine Yan [email protected] Texas A&M University [email protected] IP6 Random Graph Processes IP3 When dealing with random objects, it is often useful to A Simply Exponential Upper Bound on the Maxi- reveal the randomness gradually, rather than all at once; mum Number of Stable Matchings that is, to turn a static random object into a random pro- cess. In this talk we will describe some classical proofs of Stable matching is a classical combinatorial problem that this type, and a few more recent applications, for example has been the subject of intense theoretical and empirical to Ramsey numbers, and to determining sharp thresholds study since its introduction in 1962 in a seminal paper by in G(n,p) and in random sets of integers. Various parts of Gale and Shapley. In this talk, we describe a new upper the talk are based on joint work with Paul Balister, B´ela bound on f(n), the maximum number of stable matchings Bollob´as, Asaf Ferber, Gonzalo Fiz Pontiveros, Simon Grif- that a stable matching instance with n men and nwomen fiths, Oliver Riordan, Wojciech Samotij, and Paul Smith. can have. It has been a long-standing open problem to understand the asymptotic behavior of f(n) as n goes to infinity, first posed by Donald Knuth in the 1970s. Until n Robert Morris now the best lower bound was approximately 2.28 ,and IMPA, Rio de Janeiro n log n−O(n) the best upper bound was 2 .Inthiswork,we [email protected] show that for all n, f(n)isatmostcn for some universal constant c. This matches the lower bound up to the base of the exponent. Joint work with Shayan Oveis Gharan IP7 and Robbie Weber. Deciphering Cellular Networks: From Normal Functioning to Disease Anna R. Karlin University of Washington Each cell in our body accomplishes its functions via a com- [email protected] plex network of molecular interactions. Analyses of these networks are thus key to understanding cellular function- ing (and, in the case of disease, malfunctioning). I will IP4 overview what has been discovered about the basic struc- Waiter-Client Games ture and organization of cellular networks, and present frameworks and algorithms that leverage these properties Waiter-Client games (also called Picker-Chooser games) is in order to gain a better understanding of diseases such as a type of positional games that gained popularity recently. cancer. When played on the edge set of a graph G (typically a com- plete graph Kn, or a random graph drawn from G(n,p)), Mona Singh the game goes as follows. For a positive integer q (the so Princeton University 2 DM18 Abstracts [email protected] is joint work with Arnaud Casteigts and Luis Goddyn. Kevin C. Halasz IP8 Simon Fraser University Algorithms for the Asymmetric Traveling Salesman [email protected] Problem Arnaud Casteigts The traveling salesman problem is one of the most funda- Universit´edeBordeaux mental optimization problems. Given n cities and pairwise [email protected] distances, it is the problem of finding a tour of minimum distance that visits each city once. In spite of significant Luis Goddyn research efforts, current techniques seem insufficient for set- Simon Fraser University tling the approximability of the traveling salesman prob- [email protected] lem. The gap in our understanding is especially large in the general asymmetric setting where the distance from city i to j is *not* assumed to equal the distance from j CP1 to i. In this talk, we will give an overview of old and new approaches for settling this question. We shall, in par- Algebraic Analysis of Spiking Neural Networks for ticular, talk about our new approach that gives the first Graph Partitioning constant-factor approximation algorithm for the asymmet- ric traveling salesman problem. This is based on joint work Spiking neural networks (SNNs) are weighted graphs of with Jakub Tarnawski and L´aszl´oV´egh. logistical units which are used for event-based computa- tion. Understanding the discrete spiking dynamics of these Ola Svensson systems is integral to designing algorithms for neuromor- EPFL phic hardware. Recently, we have shown how systems of ola.svensson@epfl.ch deterministic leaky-integrate and fire neurons can be in- corporated into label propagation for community detec- tion using a fully connected SNN with static edge weights SP1 [K.E. Hamilton and T.S. Humble, “Spiking spin-glass mod- 2018 Dnes Knig Prize Lecture: Pseudorandom els for label propagation and community detection” arXiv: Graphs and the Green-Tao Theorem 1801.03571]. However, deploying spiking label propagation on neuromorphic hardware requires sparse representations. The celebrated Green-Tao theorem states that there are In this talk we explore how sparsity affects our spiking la- arbitrarily long arithmetic progressions in the primes. I bel propagation method. We focus on how sparse can a will explain some of the main ideas of the proof from a SNN be in order to be used for label propagation and how graph theoretic perspective, with a focus on the role of the sparsity of the underlying graph affects the resolution pseudorandomness in the proof. (Based on joint work with limit of spike-based label propagation. DavidConlonandJacobFox) Kathleen Hamilton Yufei Zhao Oak Ridge National Laboratory Massachusetts Institute of Technology Complex Systems Division [email protected] [email protected] Catherine Schuman, Travis Humble SP2 Oak Ridge National Laboratory Hot Topics Session [email protected], [email protected] Abstract Not Available At Time Of Publication. CP1 Henry Cohn Large Degree Asymptotics and the Reconstruction Microsoft Research New England Threshold of Asymmetric Ising Model on Regular [email protected] D-Ary Trees Determining the reconstruction threshold of a broadcast CP1 models on on d-ary regular tree, as the interdisciplinary Robust Maximal Independent Sets subject, has attracted more and more attention from prob- abilists, statistical physicists, biologists, etc. It is known A maximal independent set (MIS) in a simple connected that the Kesten-Stigum reconstruction bound is tight for graph G is robust if it remains maximal in every span- roughly symmetric binary channels. However, rigorous re- ning connected subgraph of G. We refer to such a set construction thresholds have only been established in a as an RMIS. The notion of robustness was introduced by small number of models. By means of a refined analysis of Casteigts et al. as a means of studying highly dynamic net- moment recursion on a weighted version of the magnetiza- works. They characterized the class of graphs in which ev- tion, concentration investigation, and large degree asymp- ery MIS is robust, calling this class RMIS∀,andraisedthe ∃ totics, we establish the exact reconstruction threshold of question of whether the corresponding graph class RMIS the asymmetric Ising model on regular d-ary trees, when (graphs which contain some RMIS) admits a mathemati- the Kesten-Stigum bound is not tight for the asymmetric cally natural characterization. Towards a resolution of this ∃ channel. Furthermore, we figure out the critical asymme- question, we show that every graph G ∈ RMIS has at try threshold to keep the tightness of Kesten-Stigum recon- n2 most 4 edges, and that this bound is tight if and only if n struction bound, and develop an algorithm of determining is even and G is a balanced complete bipartite graph.
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