Georgia Institute of Technology College of Sciences

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.

We discuss the inverse problem of determining elastic parameters in the interior of an anisotropic elastic media from dynamic measurements made at the surface. This problem has applications in medical imaging and seismology. The boundary data is modeled by the Dirichlet-to-Neumann map, which gives the correspondence between surface displacements and surface tractions. We first show that, without a priori information on the anisotropy type, uniqueness can hold only up to change of coordinates fixing the boundary. In particular, we study orbits of elasticity tensors under diffeomorphisms. Then, we obtain partial uniqueness for special classes of transversely isotropic media. This is joint work with L. Rachele (RPI).

It is easy to ask for the number T(g,n) of (rooted) graphs with n edges on a surface of genus g. Bender et al gave an asymptotic expansion for fixed g and large n. The contant t_g remained missing for over 20 years, although it satisfied a complicated nonlinear recursion relation. The relation was vastly simplified last year. But a further simplification was made possible last week, thus arriving to Painleve I. I will review many trivialities and lies about this famous non-linear differential equation, from a post modern point of view.

Given a set of linear equations Mx=b, we say that a set of integers S is (M,b)-free if it contains no solution to this system of equations. Motivated by questions related to testing linear-invariant Boolean functions, as well as recent investigations in additive number theory, the following conjecture was raised (implicitly) by Green and by Bhattacharyya, Chen, Sudan and Xie: we say that a set of integers S \subseteq [n], is \epsilon-far from being (M,b)-free if one needs to remove at least \epsilon n elements from S in order to make it (M,b)-free. The conjecture was that for any system of homogeneous linear equations Mx=0 and for any \epsilon > 0 there is a *constant* time algorithm that can distinguish with high probability between sets of integers that are (M,0)-free from sets that are \epsilon-far from being (M,0)-free. Or in other words, that for any M there is an efficient testing algorithm for the property of being (M,0)-free. In this paper we confirm the above conjecture by showing that such a testing algorithm exists even for non-homogeneous linear equations. As opposed to most results on testing Boolean functions, which rely on algebraic and analytic arguments, our proof relies on results from extremal hypergraph theory, such as the recent removal lemmas of Gowers, R\"odl et al. and Austin and Tao.