复杂加权计数无循环约束满足问题的复杂性分类

Tomoyuki Yamakami
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引用次数: 0

摘要

我们研究了计数约束满足问题(#CSPs)的计算复杂度,当相应的约束超图是非循环的,这些约束将复数分配给布尔输入。这些问题被称为非循环 #CSPs 或简明的 #ACSPs。我们希望确定当任意一元约束可以自由使用时,所有此类 #ACSP 的计算复杂度。根据我们是否进一步允许或不允许自由使用特定约束 XOR(二元不等式),我们将根据问题所使用的约束类型对 #ACSP 进行两种复杂度分类。当 XOR 免费提供时,我们首先得到一个完整的二分法分类。相反,当 XOR 不能免费使用时,我们会得到一个三分法分类。为了处理这些分类中约束条件的非循环性质,我们开发了一种新的技术工具,称为非循环-T-可构造性或 AT-可构造性,并利用它来分析每个 #ACSP 的复杂度上限。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Complexity Classification of Complex-Weighted Counting Acyclic Constraint Satisfaction Problems
We study the computational complexity of counting constraint satisfaction problems (#CSPs) whose constraints assign complex numbers to Boolean inputs when the corresponding constraint hypergraphs are acyclic. These problems are called acyclic #CSPs or succinctly, #ACSPs. We wish to determine the computational complexity of all such #ACSPs when arbitrary unary constraints are freely available. Depending on whether we further allow or disallow the free use of the specific constraint XOR (binary disequality), we present two complexity classifications of the #ACSPs according to the types of constraints used for the problems. When XOR is freely available, we first obtain a complete dichotomy classification. On the contrary, when XOR is not available for free, we then obtain a trichotomy classification. To deal with an acyclic nature of constraints in those classifications, we develop a new technical tool called acyclic-T-constructibility or AT-constructibility, and we exploit it to analyze a complexity upper bound of each #ACSPs.
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