Summary: Given a matrix \(A \in \mathbb{R}^{n \times n}\), we consider the problem of maximizing \(x^T Ax\) subject to the constraint \(x \in \{-1, 1\}^n\). This problem, called MaxQP by Charikar and Wirth [FOCS’04], generalizes MaxCut and has natural applications in data clustering and in the study of disordered magnetic phases of matter. Charikar and Wirth showed that the problem admits an \(\Omega(1 / \lg n)\) approximation via semidefinite programming, and Alon, Makarychev, Makarychev, and Naor [STOC’05] showed that the same approach yields an \(\Omega(1)\) approximation when \(A\) corresponds to a graph of bounded chromatic number. Both these results rely on solving the semidefinite relaxation of MaxQP, whose currently best running time is \(\tilde{O}(n^{1.5} \cdot \min \{N, n^{1.5}\})\), where \(N\) is the number of nonzero entries in \(A\) and \(\tilde{O}\) ignores polylogarithmic factors.
In this sequel, we abandon the semidefinite approach and design purely combinatorial approximation algorithms for special cases of MaxQP where \(A\) is sparse (i.e., has \(O(n)\) nonzero entries). Our algorithms are superior to the semidefinite approach in terms of running time, yet are still competitive in terms of their approximation guarantees. More specifically, we show that:

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MaxQP admits a \((1/2\Delta)\)-approximation in \(O(n \lg n)\) time, where \(\Delta = O(1)\) is the maximum degree of the corresponding graph.

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Unit MaxQP, where \(A \in \{- 1, 0, 1\}^{n \times n}\), admits a \((1/2d)\)-approximation in \(O(n)\) time when the corresponding graph is \(d\)-degenerate, and a \((1/3\delta)\)-approximation in \(O(n^{1.5})\) time when the corresponding graph has \(\delta n\) edges for \(\delta = O(1)\).

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MaxQP admits a \((1 - \varepsilon)\)-approximation in \(O(n)\) time when the corresponding graph and each of its minors have bounded local treewidth.

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Unit MaxQP admits a \((1 - \varepsilon)\)-approximation in \(O(n^2)\) time for \(H\)-minor free graphs.