Parameterised Holant Problems

2024-09-20

arXiv.org

AI Summary

This paper establishes the first complete trichotomy for the parameterized Holant problem p-Holant(𝑆), classifying its computational complexity: either (i) solvable in FPT near-linear time, (ii) solvable in FPT matrix-multiplication time—but not faster—or (iii) #W[1]-complete. Crucially, it reveals that *all* FPT tractable instances of p-Holant reside precisely within the FPT matrix-multiplication time class—a sharp departure from the uncolored variant p-UnColHolant, thereby establishing a fine-grained complexity dichotomy. Methodologically, the proof integrates parameterized analysis, holographic transformations, algebraic techniques, and fine-grained hardness assumptions including ETH and the Triangle Detection Conjecture. The framework unifies and characterizes the parameterized complexity of fundamental counting problems—including edge-colored 𝑘-matchings, graph factors, and Eulerian orientations—providing the first structured, decidable classification paradigm for Holant theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

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. $f(k)cdot ilde{mathcal{O}}(|x|)$); (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 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.

Problem

• Classify complexity of parameterised holant problems for signature families.
• Establish trichotomy: FPT-near-linear, FPT-matrix-multiplication, or #W[1]-complete.
• Compare complexity between coloured and uncoloured parameterised holant problems.

Innovation

• Parameterised Holant framework for complexity analysis
• Complexity trichotomy based on signature sets
• FPT-near-linear time solvable or #W[1]-complete