Approximate Graph Coloring by Semidefinite Programming

We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on $n$ vertices with min O(Delta^{1/3} log^{1/2} Delta log n), O(n^{1/4} log^{1/2} n) colors where Delta is the maximum degree of any vertex. Besides giving the best known approximation ratio in terms of n, this marks the first non-trivial approximation result as a function of the maximum degree Delta. This result can be generalized to k-colorable graphs to obtain a coloring using min O(Delta^{1-2/k} log^{1/2} Delta log n), O(n^{1-3/(k+1)} log^{1/2} n) colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovasz theta-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the theta-function

Bounded Independence Fools Degree-2 Threshold Functions

Let x be a random vector coming from any k-wise independent distribution over {-1,1}^n. For an n-variate degree-2 polynomial p, we prove that E[sgn(p(x))] is determined up to an additive epsilon for k = poly(1/epsilon). This answers an open question of Diakonikolas et al. (FOCS 2009). Using standard constructions of k-wise independent distributions, we obtain a broad class of explicit generators that epsilon-fool the class of degree-2 threshold functions with seed length log(n)*poly(1/epsilon). Our approach is quite robust: it easily extends to yield that the intersection of any constant number of degree-2 threshold functions is epsilon-fooled by poly(1/epsilon)-wise independence. Our results also hold if the entries of x are k-wise independent standard normals, implying for example that bounded independence derandomizes the Goemans-Williamson hyperplane rounding scheme. To achieve our results, we introduce a technique we dub multivariate FT-mollification, a generalization of the univariate form introduced by Kane et al. (SODA 2010) in the context of streaming algorithms. Along the way we prove a generalized hypercontractive inequality for quadratic forms which takes the operator norm of the associated matrix into account. These techniques may be of independent interest.Comment: Using v1 numbering: removed Lemma G.5 from the Appendix (it was wrong). Net effect is that Theorem G.6 reduces the m^6 dependence of Theorem 8.1 to m^4, not m^

An algorithm for orienting graphs based on cause-effect pairs and its applications to orienting protein networks.

Acknowledgments: I would like to thank my thesis advisor, Prof. Roded Sharan, for the initial idea and the excellent guidance throughout the research. I would like to thank Prof. Uri Zwick and Prof. Vineet Bafna for substantial contribution to this work and for co-authoring the paper, upon which this thesis is based. I also thank Andreas Beyer and Silpa Suthram for providing the kinase-substrate data, Oved Ourfali for his help with Integer Programming implementation, and Rani Hod for his help with some theoretical issues. Abstract In recent years we have seen a vast increase in the amount of protein-protein interaction data. Study of the resulting biological networks can provide us a better understanding of the processes taking place within a cell. In this work we consider a graph orientation problem arising in the study of biological networks. Given an undirected graph and a list of ordered source-target pairs, the goal is to orient the graph so that a maximum number of pairs will admit a directed path from the source to the target. We show that the problem is NP-hard and hard to approximate to within a constant ratio. We then study restrictions of the problem to various graph classes, and provide an O(log n) approximation algorithm for the general case. We show that this algorithm achieves very tight approximation ratios in practice and is able to infer edge directions with high accuracy on both simulated and real network data

Applications of Derandomization Theory in Coding

Randomized techniques play a fundamental role in theoretical computer science and discrete mathematics, in particular for the design of efficient algorithms and construction of combinatorial objects. The basic goal in derandomization theory is to eliminate or reduce the need for randomness in such randomized constructions. In this thesis, we explore some applications of the fundamental notions in derandomization theory to problems outside the core of theoretical computer science, and in particular, certain problems related to coding theory. First, we consider the wiretap channel problem which involves a communication system in which an intruder can eavesdrop a limited portion of the transmissions, and construct efficient and information-theoretically optimal communication protocols for this model. Then we consider the combinatorial group testing problem. In this classical problem, one aims to determine a set of defective items within a large population by asking a number of queries, where each query reveals whether a defective item is present within a specified group of items. We use randomness condensers to explicitly construct optimal, or nearly optimal, group testing schemes for a setting where the query outcomes can be highly unreliable, as well as the threshold model where a query returns positive if the number of defectives pass a certain threshold. Finally, we design ensembles of error-correcting codes that achieve the information-theoretic capacity of a large class of communication channels, and then use the obtained ensembles for construction of explicit capacity achieving codes. [This is a shortened version of the actual abstract in the thesis.]Comment: EPFL Phd Thesi

Essays on optimization and incentive contracts

Includes bibliographical references (p. 167-176).Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2007.(cont.) In the second part of the thesis, we focus on the design and analysis of simple, possibly non-coordinating contracts in a single-supplier, multi-retailer supply chain where retailers make both pricing and inventory decisions. Specifically, we introduce a buy-back menu contract to improve supply chain efficiency, and compare two systems, one in which the retailers compete against each other, and another in which the retailers coordinate their decisions to maximize total expected retailer profit. In a linear additive demand setting, we show that for either retailer configuration, the proposed buy-back menu guarantees the supplier, and hence the supply chain, at least 50% of the optimal global supply chain profit. In particular, in a coordinated retailers system, the contract guarantees the supply chain at least 75% of the optimal global supply chain profit. We also analyze the impact of retail price caps on supply chain performance in this setting.In this thesis, we study important facets of two problems in methodological and applied operations research. In the first part of the thesis, motivated by optimization problems that arise in the context of Internet advertising, we explore the performance of the greedy algorithm in solving submodular set function maximization problems over various constraint structures. Most classic results about the greedy algorithm assume the existence of an optimal polynomial-time incremental oracle that identifies in any iteration, an element of maximum incremental value to the solution at hand. In the presence of only an approximate incremental oracle, we generalize the performance bounds of the greedy algorithm in maximizing nondecreasing submodular functions over special classes of matroids and independence systems. Subsequently, we unify and improve on various results in the literature for problems that are specific instances of maximizing nondecreasing submodular functions in the presence of an approximate incremental oracle. We also propose a randomized algorithm that improves upon the previous best-known 2-approximation result for the problem of maximizing a submodular function over a partition matroid.by Pranava Raja Goundan.Ph.D

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library