Parameterised Holant Problems

TL;DR

This work develops a thorough parameterised complexity theory for Holant problems by introducing p-Holant and p-UnColHolant and classifying them via signature fingerprints into Lin, ω, and ∞ types. The authors prove a complete trichotomy for p-Holant(\mathcal{S}) and a dichotomy for p-UnColHolant(\mathcal{S}), with tractable cases achieving either near-linear or matrix-multiplication running times and intractable cases proving #W[1]-hardness under ETH; modular counting extensions are also provided. A key technical advance is expressing Holant values as finite linear combinations of graph-homomorphism counts and analyzing the surviving terms through treewidth, using Möbius inversion and the fingerprint χ(d,s). The framework yields almost-tight exponents for the faster FPT algorithms and applies to a broad class of problems, including colourful and uncoloured k-matchings and k-factors, illustrating a clear separation between tractable and intractable parameterised counting in Holant settings.

Abstract

We investigate the complexity of parameterised holant problems p-$\mathrm{Holant}(\mathcal{S})$ for families of signatures $\mathcal{S}$. The parameterised holant framework was introduced by Curticapean in 2015 as a counter-part to the classical theory of holographic reductions and algorithms and it constitutes an extensive family of coloured and weighted counting constraint satisfaction problems on graph-like structures, encoding as special cases various well-studied counting problems in parameterised and fine-grained complexity theory such as counting edge-colourful $k$-matchings, graph-factors, Eulerian orientations or, subgraphs with weighted degree constraints. We establish an exhaustive complexity trichotomy along the set of signatures $\mathcal{S}$: Depending on $\mathcal{S}$, p-$\mathrm{Holant}(\mathcal{S})$ is: (1) solvable in FPT-near-linear time (i.e. $f(k)\cdot \tilde{\mathcal{O}}(|x|)$); (2) solvable in "FPT-matrix-multiplication time" (i.e. $f(k)\cdot {\mathcal{O}}(n^ω)$) but not solvable in FPT-near-linear time unless the Triangle Conjecture fails; or (3) #W[1]-complete and no significant improvement over brute force is possible unless ETH fails. This classification reveals a significant and surprising gap in the complexity landscape of parameterised Holants: Not only is every instance either fixed-parameter tractable or #W[1]-complete, but additionally, every FPT instance is solvable in time $f(k)\cdot {\mathcal{O}}(n^ω)$. We also establish a complete classification for a natural uncoloured version of parameterised holant problem p-$\mathrm{UnColHolant}(\mathcal{S})$, which encodes as special cases the non-coloured analogues of the aforementioned examples. We show that the complexity of p-$\mathrm{UnColHolant}(\mathcal{S})$ is different: Depending on $\mathcal{S}$ all instances are either solvable in FPT-near-linear time, or #W[1]-complete.

Figures (1)

• Figure 1: Illustration of the construction of a fractured graph taken from PeyerimhoffRSSVW23. The left picture shows a vertex $v$ of a graph $H$ with incident edges $E_H(v)=\{e_1,\dots,e_6\}$. The right picture shows the splitting of $v$ in the construction of the fractured graph $H\raisebox{.2ex}{origin=c]{-15}{$\sharp$}}\vec{\rho}$ for a fracture $\vec{\rho}$ satisfying that the partition $\vec{\rho}(v)$ contains two blocks $B_1 =\{e_1,e_2,e_3\}$ and $B_2=\{e_4,e_5,e_6\}$.

Theorems & Definitions (90)

• definition 1: Signatures
• definition 2: The Parameterised Holant Problem
• definition 3: The Parameterised Uncoloured Holant Problem
• definition 4: Signature Fingerprints
• definition 5: Types of Signature Sets
• lemma 6
• definition 7
• theorem 8
• corollary 9
• theorem 10