Sanjeev Mahajan

Derandomizing Approximation Algorithms Based on Semidefinite Programming

Remarkable breakthroughs have been made recently in obtaining approximate solutions to some fundamental NP-hard problems, namely Max-Cut, Max k-Cut, Max-Sat, Max-Dicut, Max-bisection, k-vertex coloring, maximum independent set, etc. All these breakthroughs involve polynomial time randomized algorithms based upon semidefinite programming, a technique pioneered by Goemans and Williamson. In this paper, we give techniques to derandomize the above class of randomized algorithms, thus obtaining polynomial time deterministic algorithms with the same approximation ratios for the above problems. At the heart of our technique is the use of spherical symmetry to convert a nested sequence of n integrations, which cannot be approximated sufficiently well in polynomial time, to a nested sequence of just a constant number of integrations, which can be approximated sufficiently well in polynomial time.

Using amplification to compute majority with small majority gates

In this paper, we consider the formula complexity of the majority function over a basis consisting only of small (bounded-size) majority gates. Using Valiants amplification technique, we show that there is a formula of sizeO(n4.29) when only the gateM3 (the majority gate on three inputs) is used. Then, based on a result of Boppana, we show that not only is our result optimal with respect to the amplification technique, there is no smaller formula over the basis of all monotone 3-input functions (again with respect to amplification). Finally, we show that no better bounds are possible even with respect to more general input distributions. In particular, we show that it is not possible to use amplification to “bootstrap”, that is, use smaller majority functions in the initial distribution to find an optimal formula for a larger majority function.

Approximate Hypergraph Coloring

A coloring of a hypergraph is a mapping of vertices to colors such that no hyperedge is monochromatic. We are interested in the problem of coloring 2-colorable hypergraphs. For the special case of graphs (hypergraphs of dimension 2) this can easily be done in linear time. The problem for general hypergraphs is much more difficult since a result of Lovasz implies that the problem is NP-hard even if all hyperedges have size three.

Vertex Partitioning Problems On Partial k-Trees

We describe a general approach to obtain polynomial-time algorithms over partial k-trees for graph problems in which the vertex set is to be partitioned in some way. We encode these problems with formulae of the Extended Monadic Second-order (or EMS) logic. Such a formula can be translated into a polynomial-time algorithm automatically. We focus on the problem of partitioning a partial k-tree into induced subgraphs isomorphic to a fixed pattern graph; a distinct algorithm is derived for each pattern graph and each value of k. We use a “pumping lemma” to show that (for some pattern graphs) this problem cannot be encoded in the “ordinary” Monadic Second-order logic—from which a linear-time algorithm over partial k-trees would be obtained. Hence, an EMS formula is in some sense the strongest possible. As a further application of our general approach, we derive a polynomial-time algorithm to determine the maximum number of co-dominating sets into which the vertices of a partial k-tree can be partitioned. (A co-dominating set of a graph is a dominating set of its complement graph).

The Cost of Derandomization: Computability or Competitiveness

Recently, much work has been done in game theory towards understanding the bounded rationality of players in infinite games. This requires the strategies of realistic players to be restricted to have bounded resources of reasoning. (See [H. Simon, Decision and Organization, North--Holland, Amsterdam, 1972, pp. 161--176] for an extensive discussion; also see [X. Deng and C. H. Papadimitriou, Math. Oper. Res., 19 (1994), pp. 257--266], [C. Futia, J. Math. Econom., 4 (1977), pp. 289--299], [V. Knoblauch, Games Econom. Behav., 7 (1994), pp. 381--389], [E. Kalai and W. Stanford, Econometria, 56 (1988), pp. 397--410], [A. Neyman, Econom. Lett., 19 (1985), pp. 227--229], and [C. H. Papadimitriou, Game Theory Econom. Behav., 4 (1992), pp. 122--131].) In this paper, we discuss infinite two-person games, focusing on the case where our player follows a computable strategy and the adversary may use any strategy, which formulates the notion of computer against extremely formidable nature. In this context, we say that an infinite game is semicomputably determinate if either the adversary has a winning strategy or our player has a computable winning strategy. We show that, whereas all open games are semicomputably determinate, there is a semicomputably indeterminate closed game. Since we want to prove an indeterminacy result for closed games and since the adversarys strategy set is uncountable and our players strategy set is countable, our proof for the indeterminacy result requires a new diagonalization technique, which might be useful in other similar cases. Our study of semicomputable games was inspired by online computing problems. In this direction, we discuss several possible applications to derandomization in online computing, with the restriction that the strategies of our player should be computable. We also study the power of randomization for the classical case where our player is allowed to play according to unrestricted strategies. An indeterminate game is obtained for which both players have a simple randomized winning strategy against all of the deterministic strategies of the opponent.