LIPIcs.ISAAC.2024.51.pdf
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Easier Ways to Prove Counting Hard: A Dichotomy for Generalized #SAT, Applied to Constraint Graphs
Authors
Josh Brunner 
,
Erik D. Demaine ,
Markus Hecher ,
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35th International Symposium on Algorithms and Computation (ISAAC 2024)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: International Symposium on Algorithms and Computation (ISAAC) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2024-12-04
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MIT Hardness Group, Josh Brunner, Erik D. Demaine, Jenny Diomidova, Timothy Gomez, Markus Hecher, Frederick Stock, and Zixiang Zhou. Easier Ways to Prove Counting Hard: A Dichotomy for Generalized #SAT, Applied to Constraint Graphs. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ISAAC.2024.51
Abstract
To prove #P-hardness, a single-call reduction from #2SAT needs a clause gadget to have exactly the same number of solutions for all satisfying assignments - no matter how many and which literals satisfy the clause. In this paper, we relax this condition, making it easier to find #P-hardness reductions. Specifically, we introduce a framework called Generalized #SAT where each clause contributes a term to the total count of solutions based on a given function of the literals. For two-variable clauses (a natural generalization of #2SAT), we prove a dichotomy theorem characterizing when Generalized #SAT is in FP versus #P-complete. Equipped with these tools, we analyze the complexity of counting solutions to Constraint Graph Satisfiability (CGS), a framework previously used to prove NP-hardness (and PSPACE-hardness) of many puzzles and games. We prove CGS ASP-hard, meaning that there is a parsimonious reduction (with algorithmic bijection on solutions) from every NP search problem, which implies #P-completeness. Then we analyze CGS restricted to various subsets of features (vertex and edge types), and prove most of them either easy (in FP) or hard (#P-complete). Most of our results also apply to planar constraint graphs. CGS is thus a second powerful framework for proving problems #P-hard, with reductions requiring very few gadgets.
Subject Classification
ACM Subject Classification
- Mathematics of computing → Graph theory
- Theory of computation → Complexity theory and logic
Keywords
- Counting
- Computational Complexity
- Sharp-P
- Dichotomy
- Constraint Graph Satisfiability
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