树宽与箱约束二次规划的复杂性

中文摘要 AI 辅助

我们考虑在单位超立方体上最小化稀疏二次函数的问题。在二元二次规划中，交互图的树宽是易处理性的核心参数：有界树宽可导致多项式时间可解性。受此事实启发，我们研究当二元域被单位超立方体替代时，树宽是否发挥类似作用。我们证明情况截然不同。若交互图是森林，我们给出一个基于动态规划和所得单变量值函数结构分析的强多项式时间算法，该值函数被证明是凹的且分段二次，具有线性多个片段。然而，当交互图树宽为二时，问题已变为强NP难。随后，我们识别出更广泛的、交互图可能具有无界树宽的多项式时间可解类。我们的方法利用了以下事实：存在一个最优解，其中每个平方项系数非正的变量取二元值，从而将问题分为组合部分和真正连续部分。我们表明，可通过控制这两部分及其交互的复杂性而非整个交互图的树宽来恢复易处理性。

英文摘要

We consider the problem of minimizing a sparse quadratic function over the unit hypercube. In binary quadratic programming, treewidth of the interaction graph is a central parameter for tractability: bounded treewidth yields polynomial-time solvability. Motivated by this fact, we investigate whether treewidth plays a similar role when the binary domain is replaced by the unit hypercube. We show that the situation is strikingly different. If the interaction graph is a forest, we give a strongly polynomial-time algorithm based on dynamic programming and a structural analysis of the resulting univariate value functions, which are shown to be concave and piecewise quadratic with linearly many pieces. However, the problem becomes strongly NP-hard already when the interaction graph has treewidth two. We then identify substantially more general polynomial-time solvable classes whose interaction graphs may have unbounded treewidth. Our approach exploits the fact that there exists an optimal solution in which every variable with a nonpositive coefficient for its square term is binary-valued, thereby separating the problem into a combinatorial part and a genuinely continuous part. We show that tractability can be recovered by controlling the complexity of these two parts and their interaction, rather than the treewidth of the entire interaction graph.

发表机构

• University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
• Lehigh University（理海大学）

机构由 AI 辅助整理，请以论文原文为准。