Approximating Sparse Quadratic Programs

Given a matrix $A \in \mathbb{R}^{n\times n}$, we consider the problem of maximizing $x^TAx$ subject to constraint $x \{-1,1\}^n$. This problem, called MaxQP by Charikar and Wirth [FOCS'04], generalizes MaxCut has natural applications in data clustering study disordered magnetic phases matter. showed that admits an $\Omega(1/\lg n)$ approximation via semidefinite programming, Alon, Makarychev, Naor [STOC'05] same approach yields $\Omega(1)$ when $A$ corresponds graph bounded chromatic number. Both these results rely on solving relaxation MaxQP, whose currently best running time is $\tilde{O}(n^{1.5}\cdot \min\{N,n^{1.5}\})$, where $N$ number nonzero entries $\tilde{O}$ ignores polylogarithmic factors. In this sequel, abandon design purely combinatorial algorithms for special cases sparse (i.e., $O(n)$ entries). Our are superior terms time, yet still competitive their guarantees. More specifically, show that: - $(1/2\Delta)$-approximation $O(n \lg $\Delta$ maximum degree corresponding graph. UnitMaxQP, \{-1,0,1\}^{n\times $(1/2d)$-approximation $d$-degenerate, $(1/3\delta)$-approximation $O(n^{1.5})$ $\delta n$ edges. $(1-\varepsilon)$-approximation each its minors have local treewidth. UnitMaxQP $O(n^2)$ $H$-minor free.