We investigate the complexity of parameterised holant problems $\textsc{p-Holant}(\mathcal{S})$ 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 $\mathcal{S}$, $\textsc{p-Holant}(\mathcal{S})$ 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. $f(k)\cdot {\mathcal{O}}(n^{\omega})$) 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 $f(k)\cdot {\mathcal{O}}(n^{\omega})$. We also establish a complete classification for a natural uncoloured version of parameterised holant problem $\textsc{p-UnColHolant}(\mathcal{S})$, which encodes as special cases the non-coloured analogues of the aforementioned examples. We show that the complexity of $\textsc{p-UnColHolant}(\mathcal{S})$ is different: Depending on $\mathcal{S}$ all instances are either solvable in FPT-near-linear time, or #W[1]-complete.

翻译：我们研究了签名族$\mathcal{S}$对应的参数化Holant问题$\textsc{p-Holant}(\mathcal{S})$的复杂度。参数化Holant框架由Curticapean于2015年提出，作为经典全息归约与算法理论的对偶框架，它构成了图状结构上带颜色与权重的计数约束满足问题的一个广泛家族，其特例编码了参数化与精细复杂度理论中诸多被深入研究的计数问题，例如计算边着色$k$-匹配、图因子、欧拉定向或带加权度约束的子图。我们沿签名集$\mathcal{S}$建立了完整的复杂度三分定理：根据$\mathcal{S}$的不同，$\textsc{p-Holant}(\mathcal{S})$属于以下三类之一：(1) 可在FPT近线性时间内求解（即$f(k)\cdot \tilde{\mathcal{O}}(|x|)$）；(2) 可在“FPT矩阵乘法时间”内求解（即$f(k)\cdot {\mathcal{O}}(n^{\omega})$），但除非三角形猜想不成立，否则无法在FPT近线性时间内求解；或(3) 是#W[1]-完全的，且除非指数时间假说不成立，否则无法显著改进暴力枚举法。该分类揭示了参数化Holant问题复杂度图景中一个显著且令人惊讶的间隙：不仅每个实例要么是固定参数可解的，要么是#W[1]-完全的，而且所有FPT实例均可在$f(k)\cdot {\mathcal{O}}(n^{\omega})$时间内求解。我们还对参数化Holant问题的自然非着色版本$\textsc{p-UnColHolant}(\mathcal{S})$建立了完整分类，该版本编码了上述例子的非着色类比。我们证明$\textsc{p-UnColHolant}(\mathcal{S})$的复杂度有所不同：根据$\mathcal{S}$的不同，所有实例要么可在FPT近线性时间内求解，要么是#W[1]-完全的。