Approximating sparse quadratic programs

Given a matrix A is an element of R-nxn, we consider the problem of maximizing x(T) Ax subject to the constraint x is an element of {-1, 1}(n). This problem, called MAXQP by Charikar and Wirth [FOCS'04], generalizes MAXCUT and has natural applications in data clustering and in the study of disordered magnetic phases of matter. Charikar and Wirth showed that the problem admits an Omega(1/lg n) approximation via semidefinite programming, and Alon, Makarychev, Makarychev, and Naor [STOC'05] showed that the same approach yields an Omega(1) approximation when.. corresponds to a graph of bounded chromatic number. Both these results rely on solving the semidefinite relaxation of MAXQP, whose currently best running time is (O) over tilde ((n1.5) center dot min{N, (n1.5)}), where.. is the number of nonzero entries in.. and (O) over tilde ignores polylogarithmic factors. In this sequel, we abandon the semidefinite approach and design purely combinatorial approximation algorithms for special cases of MAXQP where.. is sparse (i.e., has O(n) nonzero entries). Our algorithms are superior to the semidefinite approach in terms of running time, yet are still competitive in terms of their approximation guarantees. More specifically, we show that: MAXQP admits a (1/2 Delta)- approximation in O(n lg n) time, where Delta = O(1) is the maximum degree of the corresponding graph. UNIT MAXQP, where A is an element of {-1, 0, 1}(nxn), admits a (1/2d)-approximation in O(n) time when the corresponding graph is d-degenerate, and a (1/3 delta)- approximation in O (n(1.5)) time when the corresponding graph has delta n edges for delta = O(1). MAXQP admits a (1 - epsilon)- approximation in O(n) time when the corresponding graph and each of its minors have bounded local treewidth. UNIT MAXQP admits a (1 - epsilon)- approximation in O(n(2)) time for H-minor free graphs.