Approximation Schemes for Binary Quadratic Programming Problems with Low cp-Rank Decompositions

Abstract

Binary quadratic programming problems have attracted much attention in the last few decades due to their potential applications. This type of problems are NP-hard in general, and still considered a challenge in the design of efficient approximation algorithms for their solutions. The purpose of this paper is to investigate the approximability for a class of such problems where the constraint matrices are {\it completely positive} and have low {\it cp-rank}. In the first part of the paper, we show that a completely positive rational factorization of such matrices can be computed in polynomial time, within any desired accuracy. We next consider binary quadratic programming problems of the following form: Given matrices $Q_1,...,Q_n\in\mathbb{R}_+^{n\times n}$, and a system of $m$ constrains $x^TQ_ix\le C_i^2$ ($x^TQ_ix\ge C_i^2$), $i=1,...,m$, we seek to find a vector $x^*\in \{0,1\}^n$ that maximizes (minimizes) a given function $f$. This class of problems generalizes many fundamental problems in discrete optimization such as packing and covering integer programs/knapsack problems, quadratic knapsack problems, submodular maximization, etc. We consider the case when $m$ and the cp-ranks of the matrices $Q_i$ are bounded by a constant. Our approximation results for the maximization problem are as follows. For the case when the objective function is nonnegative submodular, we give an $(1/4-\epsilon)$-approximation algorithm, for any $\epsilon>0$; when the function $f$ is linear, we present a PTAS. We next extend our PTAS result to a wider class of non-linear objective functions including quadratic functions, multiplicative functions, and sum-of-ratio functions. The minimization problem seems to be much harder due to the fact that the relaxation is {\it not} convex. For this case, we give a QPTAS for $m=1$.