Ravi Kannan

Adaptive Sampling for k-Means Clustering

We show that adaptively sampled O (k ) centers give a constant factor bi-criteria approximation for the k -means problem, with a constant probability. Moreover, these O (k ) centers contain a subset of k centers which give a constant factor approximation, and can be found using LP-based techniques of Jain and Vazirani [JV01] and Charikar et al. [CGTS02]. Both these algorithms run in effectively O (nkd ) time and extend the O (logk )-approximation achieved by the k -means++ algorithm of Arthur and Vassilvitskii [AV07].

Market Equilibria in Polynomial Time for Fixed Number of Goods or Agents

We consider markets in the classical Arrow-Debreu model. There are n agents and m goods. Each buyer has a concave utility function (of the bundle of goods he/she buys) and an initial bundle. At an ldquoequilibriumrdquo set of prices for goods, if each individual buyer separately ex-changes the initial bundle for an optimal bundle at the set prices, the market clears, i.e., all goods are exactly consumed. Classical theorems guarantee the existence of equilibria, but computing them has been the subject of much recent research. In the related area of Multi-Agent Games,much attention has been paid to the complexity as well as algorithms. While most general problems are hard, polynomial time algorithms have been developed for restricted classes of games, when one assumes the number of strategies is constant.For the Market Equilibrium problem, several important special cases of utility functions have been tackled. Here we begin a program for this problem similar to that for multi-agent games, where general utilities are considered. We begin by showing that if the utilities are separable piece-wise linear concave (PLC) functions, and the number of goods(or alternatively the number of buyers) is constant, then we can compute an exact equilibrium in polynomial time.Our technique for the constant number of goods is to de-compose the space of price vectors into cells using certain hyperplanes, so that in each cell, each buyerpsilas threshold marginal utility is known. Still, one needs to solve a linear optimization problem in each cell. We then show the main result - that for general (non-separable) PLC utilities, an exact equilibrium can be found in polynomial time provided the number of goods is constant. The starting point of the algorithm is a ldquocell-decompositionrdquo of the space of price vectors using polynomial surfaces (instead of hyperplanes).We use results from computational algebraic geometry to bound the number of such cells. For solving the problem inside each cell, we introduce and use a novel LP-duality based method. We note that if the number of buyers and agents both can vary, the problem is PPAD hard even for the very special case of PLC utilities - namely Leontief utilities.

A simple randomised algorithm for convex optimisation

We consider maximising a concave function over a convex set by a simple randomised algorithm. The strength of the algorithm is that it requires only approximate function evaluations for the concave function and a weak membership oracle for the convex set. Under smoothness conditions on the function and the feasible set, we show that our algorithm computes a near-optimal point in a number of operations which is bounded by a polynomial function of all relevant input parameters and the reciprocal of the desired precision, with high probability. As an application to which the features of our algorithm are particularly useful we study two-stage stochastic programming problems. These problems have the property that evaluation of the objective function is #P-hard under appropriate assumptions on the models. Therefore, as a tool within our randomised algorithm, we devise a fully polynomial randomised approximation scheme for these function evaluations, under appropriate assumptions on the models. Moreover, we deal with smoothing the feasible set, which in two-stage stochastic programming is a polyhedron.

Discovering Global Patterns in Linguistic Networks through Spectral Analysis: A Case Study of the Consonant Inventories

Recent research has shown that language and the socio-cognitive phenomena associated with it can be aptly modeled and visualized through networks of linguistic entities. However, most of the existing works on linguistic networks focus only on the local properties of the networks. This study is an attempt to analyze the structure of languages via a purely structural technique, namely spectral analysis, which is ideally suited for discovering the global correlations in a network. Application of this technique to PhoNet, the co-occurrence network of consonants, not only reveals several natural linguistic principles governing the structure of the consonant inventories, but is also able to quantify their relative importance. We believe that this powerful technique can be successfully applied, in general, to study the structure of natural languages.

Finding dense subgraphs in G ( n ,1/2)

Finding the largest clique in random graphs is a well known hard problem. It is known that a random graph G(n, 1/2) almost surely has a clique of size about 2logn. A simple greedy algorithm finds a clique of size logn, and it is a long-standing open problem to find a clique of size (1+e)logn in randomized polynomial time. In this paper, we study the generalization of finding the largest subgraph of any given edge density. We show that a simple modification of the greedy algorithm finds a subset of 2logn vertices with induced edge density at least 0.951. We also show that almost surely there is no subset of 2.784logn vertices whose induced edge density is at least 0.951.

Optimization of a convex program with a polynomial perturbation

We consider the problem of minimizing a convex function plus a polynomial p over a convex body K. We give an algorithm that outputs a solution x whose value is within @erangeK(p) of the optimum value, where rangeK(p)=supx@?Kp(x)-infx@?Kp(x). When p depends only on a constant number of variables, the algorithm runs in time polynomial in 1/@e, the degree of p, the time to round K and the time to solve the convex program that results by setting p=0.

Markets with Production: A Polynomial Time Algorithm and a Reduction to Pure Exchange

The classic Arrow-Debreu market model captures both production and consumption, two equally important blocks of an economy, however most of the work in theoretical computer science has so far concentrated on markets without production, i.e., the exchange economy. In this paper we show two new results on markets with production. Our first result gives a polynomial time algorithm for Arrow-Debreu markets under piecewise linear concave (PLC) utilities and polyhedral production sets provided the number of goods is constant. This is the first polynomial time result for the most general case of Arrow-Debreu markets. Our second result gives a novel reduction from an Arrow-Debreu market M (with production firms) to an equivalent exchange market M such that the equilibria of M are in one-to-one correspondence with the equilibria of M. Unlike the previous reduction by Rader where M is artificially constructed, our reduction gives an explicit market M and we also get: (i) when M has concave utilities and convex production sets (standard assumption in Arrow-Debreu markets), then M has concave utilities, (ii) when M has PLC utilities and polyhedral production sets, then M has PLC utilities, and (iii) when M has nested CES-Leontief utilities and nested CES-Leontief production, then M has nested CES-Leontief utilities.

Decoupling and Partial Independence

The initial motivation of this note was the question: How many samples are needed to approximate the inertia matrix (variance-covariance matrix) of a density on R n ? It first arose in a joint paper with L. Lovasz and M. Simonovits on an algorithm for computing volumes of convex sets. Rudelson proved a very interesting result (answering the question) based on a classical theorem from Functional Analysis (see Square Form Theorem below) due to Lust-Piquard, which is proved using the beautiful technique of Decoupling. This note gives a self-contained proof of the theorem and its application to this problem as well as a different question dealing with extending the basic result of Random Matrix Theory to partially random matrices (see Theorem 3) below.