On Quadratic Programming with a Ratio Objective

Abstract

Quadratic Programming (QP) is the well-studied problem of maximizing over { − 1,1} values the quadratic form ∑  _i ≠ j a _ij x _i x _j . QP captures many known combinatorial optimization problems, and assuming the Unique Games conjecture, Semidefinite Programming (SDP) 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 problem we study is

$$\begin{aligned} \textsf{QP-Ratio} &: \mbox{\ \ } \max_{\{-1,0,1\}^n} \frac{\sum_{i \not = j} a_{ij} x_i x_j}{\sum x_i^2} \end{aligned}$$

This is a natural relative of several well studied problems (in fact Trevisan introduced a normalized variant as a stepping stone towards a spectral algorithm for Max Cut Gain). Quadratic ratio problems are good testbeds 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 these problems.

We consider an SDP relaxation obtained by adding constraints to the natural eigenvalue (or SDP) relaxation for this problem. Using this, we obtain an $\tilde{O}(n^{1/3})$ approximation algorithm for QP-ratio. We also give a $\tilde{O}(n^{1/4})$ 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 QP-Ratio is hard to approximate to within any constant factor: one is based on the assumption that random instances of Max k-AND are hard to approximate, and the other makes a connection to a ratio version of Unique Games. We also give a natural distribution on instances of QP-Ratio for which an n ^ε approximation (for ε roughly 1/10) seems out of reach of current techniques.

The full version of the paper [BCMV11] can be accessed at http://arxiv.org/abs/1101.1710