Approximating integer quadratic programs and MAXCUT in subdense graphs

Abstract

Let A be a real symmetric n × n-matrix with eigenvalues λ1, . . . , λn ordered after decreasing absolute value, and 6 an n × 1-vector. We present an algorithm finding approximate solutions to min x* (Ax + b) and max x* (Ax + 6) over x ε {-1, 1}n, with an absolute error of at most (c 1|λ1| + |λ[C2 log n]|)2n + O((αn + β)√n log n), where α and β are the largest absolute values of the entries in A and b, respectively, for any positive constants c1 and c2, in time polynomial in n. We demonstrate that the algorithm yields a PTAS for MAXCUT in regular graphs on n vertices of degree d of ω(√n log n), as long as they contain O(d4 log n) 4-cycles. The strongest previous result showed that Ω(n/ log n) average degree graphs admit a PTAS. We also show that smooth n-variate polynomial integer programs of constant degree k, always can be approximated in polynomial time leaving an absolute error of o(nk), answering in the affirmative a suspicion of Arora, Karger, and Karpinski in STOC 1995. © Springer-Verlag Berlin Heidelberg 2005.