We study the complexity of approximately solving the weighted counting constraint satisfaction problem #CSP(F). In the conservative case, where F contains all unary functions, there is a classification known for the case in which the domain of functions in F is Boolean. In this paper, we give a classification for the more general problem where functions in F have an arbitrary finite domain. We define the notions of weak log-modularity and weak log-supermodularity. We show that if F is weakly log-modular, then #CSP(F)is in FP. Otherwise, it is at least as difficult to approximate as #BIS, the problem of counting independent sets in bipartite graphs. #BIS is complete with respect to approximation-preserving reductions for a logically-defined complexity class #RHPi1, and is believed to be intractable. We further sub-divide the #BIS-hard case. If F is weakly log-supermodular, then we show that #CSP(F) is as easy as a (Boolean) log-supermodular weighted #CSP. Otherwise, we show that it is NP-hard to approximate. Finally, we give a full trichotomy for the arity-2 case, where #CSP(F) is in FP, or is #BIS-equivalent, or is equivalent in difficulty to #SAT, the problem of approximately counting the satisfying assignments of a Boolean formula in conjunctive normal form. We also discuss the algorithmic aspects of our classification.
A dichotomy theorem for conservative general-valued CSPs
研究了有值约束满足问题（VCSP）的复杂性.一个问题，从VCSP的特点是一个\emph{constraint language}，一个固定的成本函数在有限域。问题的一个实例由语言中的成本函数之和指定，目标是最小化该和。我们考虑的情况下，所谓的\emph{conservative}语言，也就是说，语言包含所有的一元成本函数，从而允许任意限制域的变量。在这种情况下，我们证明了一个类似Schaefer的二分法定理：如果语言中的所有代价函数都满足一定的条件（由\emph{STP and MJN multimorphisms}的互补组合指定），那么任何实例都可以在多项式时间内通过Kolmogorov和Zivny的算法（arXiv：1008.3104v1）求解，否则语言是NP-难的。这推广了Takhanov（STACS'10）和Kolmogorov和Zivny（arXiv：1008.1555v1）最近的结果，Takhanov（STACS'10）认为$\{0,\infty\}$值语言包含额外的所有有限值一元代价函数，Kolmogorov和Zivny（arXiv：1008.1555v1）认为\emph{finite-valued}保守语言。
The expressibility of functions on the Boolean domain, with applications to Counting CSPs
An important tool in the study of the complexity of Constraint Satisfaction Problems (CSPs) is the notion of a relational clone, which is the set of all relations expressible using primitive positive formulas over a particular set of base relations. Post's lattice gives a complete classification of all Boolean relational clones, and this has been used to classify the computational difficulty of CSPs. Motivated by a desire to understand the computational complexity of (weighted) counting CSPs, we develop an analogous notion of functional clones and study the landscape of these clones. One of th…
Linear Space Streaming Lower Bounds for Approximating CSPs
Flatiron Institute（平顿研究所）；Georgetown University（乔治城大学）；Harvard University（哈佛大学）；Google Research（谷歌研究院）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
研究在流模型下近似约束满足问题的可行性，证明改进近似因子需线性空间，针对Max k-LIN mod q问题提出最优近似下界。
Algebraic Properties of Valued Constraint Satisfaction Problem
本文提出了一个代数框架的最优化问题表示为价值约束满足问题。我们的结果推广了Bulatov等人[SICOMP 2005]提供的决策版本（CSP）的代数框架。我们引入了加权代数和簇的概念，并使用Cohen等人[SICOMP 2013]的Galois连接将VCSP语言与加权代数联系起来。我们表明，VCSP的难度只取决于相关的加权代数生成的加权品种。通过对CSP结果的分析，我们展示了核和刚性核的减少，这使我们能够专注于幂等加权品种。此外，我们提出了一个类似的代数CSP二分法猜想;证明了硬度方向，并验证了它与二元集上的VCSP [Cohen et al. 2006]，有限值VCSP [Thapper and Zivny 2013]和保守VCSP [Kolmogorov and Zivny 2013]的已知结果一致。
