Symmetric Parameterised Holants on Hypergraphs: Towards a Classification for Parameterised VCSPs

Abstract

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. While this problem, and its decision version, received significant attention during the last two decades, 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 parameterised (counting) valued constraint satisfaction problems (VCSPs) with infinite constraint languages: we allow our constraints to be chosen from an infinite set of permitted constraints and we allow our constraints to map an assignment of its variables not only to True or False, but to arbitrary values. In this setting we are able to model and classify significantly more general problems such as (weighted) parameterised factor problems on hypergraphs and counting weight-k solutions of systems of linear equations, none of which are captured by existing complexity classifications of parameterised constraint satisfaction problems.
On a formal level, we express parameterised VCSPs as parameterised holant problems on uniform hypergraphs, and we establish complete and explicit complexity dichotomy theorems for this family of problems both w.r.t. classical complexity theory (P vs. #P) and parameterised complexity (FPT vs. #W[1]). For resolving the P vs. #P question, we mainly rely on the use of hypergraph gadgets, the existence of which we prove using properties of degree sequences necessary for realisability in uniform hypergraphs. As a technical highlight, we also employ Curticapean’s "CFI Filters" (SODA 2024) - named after the Cai-Fürer-Immermann construction for bounding the expressiveness of the Weisfeiler-Leman heuristic - to establish polynomial-time algorithms for isolating vectors in the homomorphism basis of some of our holant problems. For the FPT vs. #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.