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Approximation Schemes for Binary Quadratic Programming Problems with Low cp-Rank Decompositions
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 , and a
system of constrains (), ,
we seek to find a vector that maximizes (minimizes) a given
function . 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 and the cp-ranks of the matrices 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
-approximation algorithm, for any ; when the
function 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