On Quadratic Programming with a Ratio Objective
Abstract
Quadratic Programming (QP) is the wellstudied problem of maximizing over {1,1} values the quadratic form \sum_{i \ne j} a_{ij} x_i x_j. QP captures many known combinatorial optimization problems, and assuming the unique games conjecture, semidefinite programming techniques give optimal approximation algorithms. 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 problems we study are QPRatio : \max_{\{1,0,1\}^n} \frac{\sum_{i \not = j} a_{ij} x_i x_j}{\sum x_i^2}, and Normalized QPRatio : \max_{\{1,0,1\}^n} \frac{\sum_{i \not = j} a_{ij} x_i x_j}{\sum d_i x_i^2}, where d_i = \sum_j a_{ij} We consider an SDP relaxation obtained by adding constraints to the natural eigenvalue (or SDP) relaxation for this problem. Using this, we obtain an $\tilde{O}(n^{1/3})$ algorithm for QPratio. We also obtain an $\tilde{O}(n^{1/4})$ approximation for bipartite graphs, and better algorithms for special cases. As with other problems with ratio objectives (e.g. uniform sparsest cut), it seems difficult to obtain inapproximability results based on P!=NP. We give two results that indicate that QPRatio is hard to approximate to within any constant factor. We also give a natural distribution on instances of QPRatio for which an n^\epsilon approximation (for \epsilon roughly 1/10) seems out of reach of current techniques.
 Publication:

arXiv eprints
 Pub Date:
 January 2011
 DOI:
 10.48550/arXiv.1101.1710
 arXiv:
 arXiv:1101.1710
 Bibcode:
 2011arXiv1101.1710B
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms