Chinmay Karande

A note on the problem of reporting maximal cliques

Reporting the maximal cliques of a graph is a fundamental problem arising in many areas. This note bridges the gap between three papers addressing this problem: the original paper of Bron-Kerbosh [C. Bron, J. Kerbosch, Algorithm 457: Finding all cliques of an undirected graph, Communication of ACM 16 (9) (1973) 575-577], and two papers recently published in TCS, namely that of Tomita et al. [Tomita, A. Tanaka, H. Takahashi, The worst-case time complexity for generating all maximal cliques and computational experiments, Theoretical Computer Science 363 (1) (2006) 28-42], and that of Koch [I. Koch, Fundamental study: Enumerating all connected maximal common subgraphs in two graphs, Theoretical Computer Science 250 (1-2) (2001) 1-30]. In particular, we show that the strategy of Tomita et al. is a simple modification of the Bron-Kerbosch algorithm, based on an (un-exploited) observation raised in Kochs paper.

Approximability of Combinatorial Problems with Multi-agent Submodular Cost Functions

Abstract— Applications in complex systems such as the Internet have spawned recent interest in studying situations involving multiple agents with their individual cost or utility functions. In this paper, we introduce an algorithmic framework for studying combinatorial problems in the presence of multiple agents with submodular cost functions. We study several fundamental covering problems (Vertex Cover, Shortest Path, Perfect Matching, and Spanning Tree) in this setting and establish tight upper and lower bounds for the approximability of these problems

Approximability of combinatorial problems with multi-agent submodular cost functions

Abstract— Applications in complex systems such as the Internet have spawned recent interest in studying situations involving multiple agents with their individual cost or utility functions. In this paper, we introduce an algorithmic framework for studying combinatorial problems in the presence of multiple agents with submodular cost functions. We study several fundamental covering problems (Vertex Cover, Shortest Path, Perfect Matching, and Spanning Tree) in this setting and establish tight upper and lower bounds for the approximability of these problems

Market equilibrium with transaction costs

Identical products being sold at different prices in different locations is a common phenomenon. To model such scenarios, we supplement the classical Fisher market model by introducing transaction costs. For every buyer i and good j, there is a transaction cost of cij ; if the price of good j is pj, then the cost to the buyer i per unit of j is pj + cij. The same good can thus be sold at different (effective) prices to different buyers. We provide a combinatorial algorithm that computes e-approximate equilibrium prices and allocations in O(1/e (n + logm)mnlog(B/e)) operations - where m is the number goods, n is the number of buyers and B is the sum of the budgets of all the buyers.

Computing market equilibrium: beyond weak gross substitutes

The property ofWeak Gross Substitutibility (WGS) of goods in a market has been found to be conducive to efficient algorithms for finding equilibria. In this paper, we give a natural definition of a δ-approximate WGS property, and show that the auction algorithm of [GK04][GKV04] can be extended to give an (Ɛ+δ)-approximate equilibrium for markets with this property.