Ch. Sobhan Babu

Nash equilibria in fisher market

Much work has been done on the computation of market equilibria. However due to strategic play by buyers, it is not clear whether these are actually observed in the market. Motivated by the observation that a buyer may derive a better payoff by feigning a different utility function and thereby manipulating the Fisher market equilibrium, we formulate the Fisher market game in which buyers strategize by posing different utility functions. We show that existence of a conflict-free allocation is a necessary condition for the Nash equilibria (NE) and also sufficient for the symmetric NE in this game. There are many NE with very different payoffs, and the Fisher equilibrium payoff is captured at a symmetric NE. We provide a complete polyhedral characterization of all the NE for the two-buyer market game. Surprisingly, all the NE of this game turn out to be symmetric and the corresponding payoffs constitute a piecewise linear concave curve. We also study the correlated equilibria of this game and show that third-party mediation does not help to achieve a better payoff than NE payoffs.

Degree conditions for forests in graphs

If H is any forest of order n with m edges, then any graph G of order >=n with d(u)+d(v)>=2m-1 for any two non-adjacent vertices u,v contains H.

A simplex-like algorithm for fisher markets

We propose a new convex optimization formulation for the Fisher market problem with linear utilities. Like the Eisenberg-Gale formulation, the set of feasible points is a polyhedral convex set while the cost function is non-linear; however, unlike that, the optimum is always attained at a vertex of this polytope. The convex cost function depends only on the initial endowments of the buyers. This formulation yields an easy simplex-like pivoting algorithm which is provably strongly polynomial for many special cases.

Oriented Forests in Directed Graphs

Abstract If H is any oriented forest with m edges, then any directed graph of order ⩾ | H | with minimum out-degree as well as minimum in-degree ⩾m contains H.

Which sort orders are interesting

Sort orders play an important role in query evaluation. Algorithms that rely on sorting are widely used to implement joins, grouping, duplicate elimination and other set operations. The notion of interesting orders has allowed query optimizers to consider plans that could be locally sub-optimal, but produce ordered output beneficial for other operators, and thus be part of a globally optimal plan. However, the number of interesting orders for most operators is factorial in the number of attributes involved. Optimizer implementations use heuristics to prune the number of interesting orders, but the quality of the heuristics is unclear. Increasingly complex decision support queries and increasing use of query-covering indices, which provide multiple alternative sort orders for relations, motivate us to better address the problem of choosing interesting orders. We show that even a simplified version of the problem is NP-hard and provide a 1/2-benefit approximation algorithm for a special case of the problem. We then present principled heuristics for the general case of choosing interesting orders. We have implemented the proposed techniques in a Volcano-style cost-based optimizer, and our performance study shows significant improvements in estimated cost. We also executed our plans on a widely used commercial database system, and on PostgreSQL, and found that actual execution times for our plans were significantly better than for plans generated by those systems in several cases.