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Series: ACO Student Seminar

In the first part of the talk, I am going to give an introduction and overview of linear and semidefinite programming hierarchies. I will mostly review known integrality gaps for such programs and try to give an intuition of why we currently lack strong techniques for designing rounding algorithms. In the second part of the talk I will focus on the stronger SDP Lasserre hierarchy. In contrast with the previous LP and SDP hierarchies, very few examples of integrality gap instances are known to date. I will present a recent technique for designing such instances and discuss open problems in the area.

Series: ACO Student Seminar

We will survey some old, some new, and some open problems in the area of efficient sampling. We will focus on sampling combinatorial structures (such as perfect matchings and independent sets) on regular lattices. These problems arise in statistical physics, where sampling objects on lattices can be used to determine many thermodynamic properties of simple physical systems. For example, perfect matchings on the Cartesian lattice, more commonly referred to as domino tilings of the chessboard, correspond to systems of diatomic molecules. But most importantly they are just cool problems with some beautiful solutions and a surprising number of unsolved challenges!

Series: ACO Student Seminar

Scarf's lemma is one of the fundamental results in combinatorics, originally introduced to study the core of an N-person game. Over the last four decades, the usefulness of Scarf's lemma has been demonstrated in several important combinatorial problems seeking stable solutions. However, the complexity of the computational version of Scarf's lemma (Scarf) remained open. In this talk, I will prove that Scarf is complete for the complexity class PPAD. This shows that Scarf is as hard as the computational versions of Brouwer's fixed point theorem and Sperner's lemma. Hence, there is no polynomial-time algorithm for Scarf unless PPAD \subseteq P. I will also talk about fractional stable paths problem, finding fractional kernels in digraphs, finding fractional stable matching in hypergraphic preference systems and finding core in an N-person balanced game with non-transferable utilities. I will show the connection between these problems through Scarf's lemma and address the complexity of these problems.

Series: ACO Student Seminar

In this talk, I will introduce the class of logconcave and s-concave functions, illustrate their properties, and explain their connections to convex geometry. I will present a simple and unified approach for proving probabilistic inequalities for logconcave and s-concave densities on the real line. Lastly I will use these techniques to prove two important theorems in convex geometry: Grunbaum's theorem, every halfspace cut through the centroid of a convex body contains at least a 1/e volume fraction of the body, and the Milman-Pajor inequality, a convex body in R^n is sandwiched between its inertial ellipsoid and a factor n scaling of it. Joint work with Santosh Vempala.

Series: ACO Student Seminar

In this talk I will give an introduction of the Markov Chain Monte Carlo Method, which uses markov chains to sample interesting combinatorial objects such as proper colorings, independent sets and perfect matchings of a graph. I will introduce methods such as Couplings and Canonical Paths which have been widely used to analyze how many steps Markov Chains needs to go (mixing time) in order to get a sufficiently random combinatorial object. I will also give a brief survey of some recent results in the sampling of colorings.

Series: ACO Student Seminar

Shrouded in mystery and kept hidden for decades, Richard Lipton's vault of open problems will be revealed...

Series: ACO Student Seminar

In this article, we disprove the uniform shortest path routing conjecture for vertex-transitive graphs by constructing an infinite family of counterexamples.

Series: ACO Student Seminar

Two independent proofs of the polyhedrality of the split closure of Mixed Integer Linear Program have been previously presented. Unfortunately neither of these proofs is constructive. In this paper, we present a constructive version of this proof. We also show that split cuts dominate a family of inequalities introduced by Koppe and Weismantel.

Series: ACO Student Seminar

Nash bargaining was first modeled in John Nash's seminal 1950 paper. In his paper, he used a covex program to give the Nash bargaining solution, which satifies many nature properties. Recently, V.Vazirani defined a class of Nash bargaining problem as Uniform Nash Bargaining(UNB) and also defined a subclass called Submodular Nash Bargaining (SNB). In this talk, we will consider some game theoretic issues of UNB: (1) price of bargaining; (2) fully competitiveness; (3) min-max and max-min fairness and we show that each of these properties characterizes the subclass SNB.

Series: ACO Student Seminar

We consider the following Maximum Budgeted Allocation(MBA) problem: Given a set of m indivisible items and n agents; each agent i is willing to pay b_ij amount of money on item j, and in addition he species the maximum amount (budget of B_i) he is willing to pay in total over all the items he receives. Goal is to allocate items to agents so as to maximize the total payment received from all the agents. The problem naturally arises as auctioneer revenue maximization in first price budget-constrained Auctions (For e.g. auctioning of TV/Radio ads by Google). Our main results are: 1) We give a 3/4-approximation algorithm for MBA improving upon the previous best of 0.632 [Anelman-Mansour, 04]. Our factor matches the integrality gap of the LP used by the previous results. 2) We prove it is NP-hard to approximate MBA to any factor better than 15/16, previously only NP-hardness was known. Our result also implies NP-hardness of approximating maximum submodular welfare with demand oracle to a factor better than 15/16, improving upon the best known hardness of 275/276 [Feige-Vondrak, 07]. Our hardness techniques can be modified to prove that it is NP-hard to approximate the Generalized Assignment Problem (GAP) to any factor better than 10/11. This improves upon the 422/423 hardness of [Chekuri-Kumar, 04]. We use iterative rounding on a natural LP relaxation of MBA to obtain the 3/4-approximation. Recently iterative rounding has achieved considerable success in designing approximation algorithms. However, these successes have been limited to minimization problems, and as per our knowledge, this work is the first iterative rounding based approximation algorithm for a natural maximization problem. We also give a (3/4 - \epsilon)-factor algorithm based on the primal-dual schema which runs in O(nm) time, for any constant \epsilon > 0. In this talk, I will present the iterative rounding based algorithm, show the hardness reductions, and put forward some directions which can help in solving the natural open question of closing the approximation gap. Joint work with Deeparnab Chakrabarty.