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arXiv 2007.01252cs.DS
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稀疏二次规划的逼近

Approximating Sparse Quadratic Programs

Danny Hermelin, Leon Kellerhals, Rolf Niedermeier, Rami Pugatch

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摘要：

给定一个矩阵$A \in \mathbb{R}^{n\times n}$，我们考虑在约束$x \in \{-1,1\}^n$下最大化$x^TAx$的问题.这个问题，称为MaxQP的Charikar和Wirth [FOCS'04]，概括了MaxCut，并在数据聚类和无序磁相的物质的研究中有自然的应用。Charikar和Wirth表明，这个问题允许通过半定规划的$Ω(1/\lg n)$近似，而Alon，Makarychev，Makarychev和Naor [STOC'05]表明，当$A$对应于有界色数的图时，同样的方法会产生$Ω(1)$近似。这两个结果都依赖于求解MaxQP的半定松弛，MaxQP的当前最佳运行时间是$\tilde{O}(n^{1.5}\cdot \min\{N,n^{1.5}\})$，其中$N$是$A$中非零条目的数量，而$\tilde{O}$忽略多对数因子。在这个续集中，我们放弃了半定方法，并为MaxQP的特殊情况设计了纯粹的组合近似算法，其中$A$是稀疏的（即，具有$O(n)$非零条目）。我们的算法是优于半定的方法在运行时间方面，但仍然有竞争力的近似保证。更具体地说，我们表明：- MaxQP承认$(1/2Δ)$ -近似在$O(n \lg n)$时间，其中$Δ$是相应的图的最大程度。- UnitMaxQP，其中$A \in \{-1,0,1\}^{n\times n}$，当对应的图是$d$ -退化的时，允许在$O(n)$时间上的$(1/2d)$ -近似，当对应的图具有$δn$边时，允许在$O(n^{1.5})$时间上的$(1/3δ)$ -近似. - MaxQP允许在$O(n)$时间内的$(1-\varepsilon)$ -近似，当相应的图和它的每个子图有界局部树宽时。- 当相应的图是$H$ -子图自由时，UnitMaxQP允许在$O(n^2)$时间上的$(1-\varepsilon)$ -近似.

英文摘要：

Given a matrix $A \in \mathbb{R}^{n\times n}$, we consider the problem of maximizing $x^TAx$ 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 $Ω(1/\lg n)$ approximation via semidefinite programming, and Alon, Makarychev, Makarychev, and Naor [STOC'05] showed that the same approach yields an $Ω(1)$ 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: - MaxQP admits a $(1/2Δ)$-approximation in $O(n \lg n)$ time, where $Δ$ is the maximum degree of the corresponding graph. - UnitMaxQP, 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 $(1/3δ)$-approximation in $O(n^{1.5})$ time when the corresponding graph has $δn$ edges. - MaxQP admits a $(1-\varepsilon)$-approximation in $O(n)$ time when the corresponding graph and each of its minors have bounded local treewidth. - UnitMaxQP admits a $(1-\varepsilon)$-approximation in $O(n^2)$ time when the corresponding graph is $H$-minor free.

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