On Quadratic Programming with a Ratio Objective

Quadratic Programming (QP) is the well-studied problem of maximizing over {−1, 1} values the quadratic form i=j aijxixj. QP captures many known combinatorial optimization problems and semidefinite programming techniques have given optimal approximation algorithms for many of these problems (assuming the Unique Games Conjecture). We extend this body of work by initiating the study of Quadratic Programming problems where the variables take values in the domain {−1, 0, 1}. The specific problem we study is QP-Ratio : max {−1,0,1}n xT Ax xT x . This objective function is a fairly natural relative of several well studied problems. Yet, this is a challenging problem and a good testbed for both algorithms and complexity because the techniques used for quadratic problems for the {−1, 1} and {0, 1} domains do not seem to carry over to the {−1, 0, 1} domain. We give approximation algorithms and evidence for the hardness of approximating the QP-Ratio problem. We consider an SDP relaxat...

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Aditya Bhaskara, Moses Charikar, Rajsekar Manokaran, Aravindan Vijayaraghavan
• Keywords: Approximation Algorithms, CORR 2011, Education, Problems, Quadratic Programming