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

TL;DR

This paper develops approximation algorithms for binary quadratic programming problems with low cp-rank matrices, providing polynomial-time factorizations and approximation schemes for maximization and minimization under certain conditions.

Contribution

It introduces polynomial-time cp-rank factorizations and approximation schemes for a class of binary quadratic problems with low cp-rank matrices, extending to various objective functions.

Findings

Polynomial-time cp-rank factorizations for completely positive matrices.

A (1/4 - ε)-approximation for nonnegative submodular maximization.

A PTAS for linear and certain nonlinear objective functions.

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 $m=1$.