Arkadev Chattopadhyay

Discrepancy and the Power of Bottom Fan-in in Depth-three Circuits

We present efficient approximation algorithms for finding Nash equilibria in anonymous games, that is, games in which the players utilities, though different, do not differentiate between other players. Our results pertain to such games with many players but few strategies. We show that any such game has an approximate pure Nash equilibrium, computable in polynomial time, with approximation O(s2lambda), where s is the number of strategies and lambda is the Lipschitz constant of the utilities. Finally, we show that there is a PTAS for finding an isin-approximate Nash equilibrium when the number of strategies is two.

The story of set disjointness

The satisfiability problem has emerged as the queen of the complexity zoo. She is the quintessential NP-complete hard-to-find but easy-to-recognize search problem in computer science. There are hundreds if not thousands of problems that are now known to be equivalent to SAT, and our rich theory of complexity classes is centered around its queen. In the world of communication complexity, the set disjointness problem has similarly emerged as the quintessential hard-to-find but easy-to-recognize problem. There is an impressive collection of problems in many diverse areas whose hardness boils down to the hardness of the set disjointness problem in some model of communication complexity. Moreover, we will argue that proving lower bounds for the set disjointness function in a particular communication

Lower bounds for circuits with MOD_m gates

Let CCo(n)[m] be the class of circuits that have size o(n) and in which all gates are MOD[m] gates. We show that CC [m] circuits cannot compute MODq in sub-linear size when m, q > 1 are co-prime integers. No non-trivial lower bounds were known before on the size of CC [m] circuits of constant depth for computing MODq. On the other hand, our results show circuits of type MAJ o CCo(n)[m] need exponential size to compute MODq . Using Bourgains recent breakthrough result on estimates of exponential sums, we extend our bound to the case where small fan-in AND gates are allowed at the bottom of such circuits i.e. circuits of type MAJ o CC[m] o AND epsiv log n, where epsiv > 0 is a sufficiently small constant. CC [m] circuits of constant depth need superlinear number of wires to compute both the AND and MODq functions. To prove this, we show that any circuit computing such functions has a certain connectivity property that is similar to that of superconcentration. We show a superlinear lower bound on the number of edges of such graphs extending results on superconcentrators

Lower bounds for circuits with few modular and symmetric gates

We consider constant depth circuits augmented with few modular, or more generally, arbitrary symmetric gates. We prove that circuits augmented with o(log2n) symmetric gates must have size n

The Hardness of Being Private

^{\Omega({\rm log}\ {\it n})}

Graph Isomorphism is Not AC0-Reducible to Group Isomorphism

to compute a certain (complicated) function in ACC0. This function is also hard on the average for circuits of size n

Graph Isomorphism is not AC 0 reducible to Group Isomorphism

^{\epsilon log {\it n}}

Lower Bounds on Interactive Compressibility by Constant-Depth Circuits

augmented with o(log n) symmetric gates, and as a consequence we can get a pseudorandom generator for circuits of size m containing

Properly 2-Colouring Linear Hypergraphs

o(\sqrt{{\rm log} \ m})

Lower Bounds for Elimination via Weak Regularity

symmetric gates. For a composite integer m having r distinct prime factors, we prove that circuits augmented with sMODm gates must have size