We study the complexity of the parameterised counting constraint satisfaction problem: given a set of constraints over a set of variables and a positive integer $k$, how many ways are there to assign $k$ variables to 1 (and the others to 0) such that all constraints are satisfied. Existing work has so far exclusively focused on restricted settings such as finding and counting homomorphisms between relational structures due to Grohe (JACM 2007) and Dalmau and Jonsson (TCS 2004), or the case of finite constraint languages due to Creignou and Vollmer (SAT 2012), and Bulatov and Marx (SICOMP 2014). In this work, we tackle a more general setting of Valued Parameterised Counting Constraint Satisfaction Problems (VCSPs) with infinite constraint languages. In this setting we are able to model significantly more general problems such as (weighted) parameterised factor problems on hypergraphs and counting weight-$k$ solutions of systems of linear equations, not captured by existing complexity classifications. We express parameterised VCSPs as parameterised Holant problems on uniform hypergraphs, and we establish complete and explicit complexity dichotomy theorems. For resolving the $\mathrm{P}$ vs. $\#\mathrm{P}$ question, we mainly rely on hypergraph gadgets, the existence of which we prove using properties of degree sequences necessary for realisability in uniform hypergraphs. For the $\mathrm{FPT}$ vs. $\#\mathrm{W}[1]$ question, we build upon the recently established combinatorial toolkit for parameterised holants on the special case of graphs by Aivasiliotis et al. (ICALP 2025) and also rely on an extension of the framework of the homomorphism basis due to Curticapean, Dell and Marx (STOC 17) to uniform hypergraphs. As a technical highlight, we also employ Curticapean's "CFI Filters'' (SODA 2024) to establish polynomial-time algorithms for isolating vectors in the homomorphism basis.

翻译：我们研究参数化计数约束满足问题的复杂性：给定一组变量上的约束集和一个正整数$k$，有多少种方式将$k$个变量赋值为1（其余变量赋值为0），使得所有约束得到满足。现有工作迄今仅关注受限设置，例如Grohe（JACM 2007）、Dalmau和Jonsson（TCS 2004）的关系结构同态查找与计数，或Creignou和Vollmer（SAT 2012）以及Bulatov和Marx（SICOMP 2014）的有限约束语言情形。在本工作中，我们处理具有无限约束语言的值参数化计数约束满足问题（VCSPs）这一更一般的设置。在此设置下，我们能够建模显著更一般的问题，例如超图上的（加权）参数化因子问题以及线性方程组的权为$k$的解计数问题，这些问题未被现有复杂性分类所涵盖。我们将参数化VCSPs表示为一致超图上的参数化Holant问题，并建立完整且显式的复杂性二分定理。为解决$\mathrm{P}$与$\#\mathrm{P}$问题，我们主要依赖超图工具，并通过一致超图中可实现性所需的度序列性质证明了这些工具的存在性。针对$\mathrm{FPT}$与$\#\mathrm{W}[1]$问题，我们基于Aivasiliotis等人（ICALP 2025）近期为图的特例建立的参数化Holant组合工具箱，并借助Curticapean、Dell和Marx（STOC 17）的同态基框架到一致超图的扩展。作为技术亮点，我们还采用Curticapean的“CFI滤波器”（SODA 2024）来建立用于在同态基中隔离向量的多项式时间算法。