[Paper Review] The complexity of counting solutions to Generalised Satisfiability Problems modulo k

John Faben|ArXiv.org|Sep 10, 2008

Advanced Graph Theory Research10 references13 citations

TL;DR

This paper establishes a dichotomy theorem for the complexity of counting solutions to Generalised Satisfiability Problems modulo an integer $k$, showing that the problem is solvable in polynomial time if all constraints are affine or, for $k=2$, if the constraint set is C-closed; otherwise, it is $\#_k\text{P}$-complete. The result extends Creignou and Hermann's $\#\text{P}$ dichotomy to modular counting, with only one exception at $k=2$ for non-affine, non-C-closed sets.

ABSTRACT

Generalised Satisfiability Problems (or Boolean Constraint Satisfaction Problems), introduced by Schaefer in 1978, are a general class of problem which allow the systematic study of the complexity of satisfiability problems with different types of constraints. In 1979, Valiant introduced the complexity class parity P, the problem of counting the number of solutions to NP problems modulo two. Others have since considered the question of counting modulo other integers. We give a dichotomy theorem for the complexity of counting the number of solutions to Generalised Satisfiability Problems modulo integers. This follows from an earlier result of Creignou and Hermann which gave a counting dichotomy for these types of problem, and the dichotomy itself is almost identical. Specifically, counting the number of solutions to a Generalised Satisfiability Problem can be done in polynomial time if all the relations are affine. Otherwise, except for one special case with k = 2, it is #_kP-complete.

Motivation & Objective

• To determine the computational complexity of counting the number of solutions to Generalised Satisfiability Problems modulo an integer $k$.
• To extend Creignou and Hermann's $\#\text{P}$ dichotomy theorem to modular counting settings.
• To identify the precise conditions under which counting modulo $k$ becomes tractable or remains hard.
• To resolve whether the $\#_k\text{P}$-completeness threshold differs from the classical $\#\text{P}$-completeness for modular counting.

Proposed method

• Define the complexity class $\#_k\text{P}$ as the class of function problems that compute the number of accepting paths of a non-deterministic Turing machine modulo $k$, analogous to $\#\text{P}$.
• Use parsimonious reductions modulo $k$ to preserve solution counts modulo $k$, enabling reductions between $\#_k\text{P}$-complete problems.
• Leverage the concept of 'faithful implementation' of basic functions (e.g., OR, XOR, F, T) using constraint sets to propagate hardness.
• Prove hardness via reductions from known $\#_k\text{P}$-complete problems, using closure properties like C-closure and affine closure.
• Establish that if a constraint set can implement OR or XOR functions, then $\#_k\text{SAT}(\mathcal{F})$ is $\#_k\text{P}$-hard for all $k$.
• Use induction and lifting lemmas (e.g., Lemma 4.8 and 4.10) to extend hardness from $\#_{2^{k-1}}\text{P}$ to $\#_{2^k}\text{P}$, and from odd primes to composite moduli.

Experimental results

Research questions

• RQ1 Under what conditions is counting the number of solutions to a Generalised Satisfiability Problem modulo $k$ tractable?
• RQ2 Does the $\#_k\text{P}$-completeness threshold for modular counting differ from the classical $\#\text{P}$-completeness for any $k$?
• RQ3 Is there a non-trivial case where counting modulo $k$ is easy even when counting modulo 2 is hard?
• RQ4 Can the $\#_k\text{P}$-hardness of a constraint set be lifted from smaller moduli to larger ones, such as from $2^{k-1}$ to $2^k$?
• RQ5 How do closure properties like C-closure and affine closure affect the complexity of modular counting?

Key findings

• Counting solutions modulo $k$ is in polynomial time if all constraints in the set $\mathcal{F}$ are affine, regardless of $k$.
• For $k=2$, counting modulo 2 is tractable if $\mathcal{F}$ is C-closed, even if not affine.
• If $\mathcal{F}$ is not affine and $k=2$, then $\#_2\text{SAT}(\mathcal{F})$ is $\#_2\text{P}$-complete.
• For $k \geq 3$, $\#_k\text{SAT}(\mathcal{F})$ is $\#_k\text{P}$-complete unless $\mathcal{F}$ is affine or $k=2$ and $\mathcal{F}$ is C-closed.
• The only exception to the dichotomy is when $k=2$ and $\mathcal{F}$ is not C-closed but still not $\#_2\text{P}$-complete — but this case is ruled out by the proof structure.
• Hardness is preserved under modular lifting: if $\#_{2^{k-1}}\text{SAT}(\mathcal{F} \cup \{\text{F}, \text{T}\})$ is $\#_{2^{k-1}}\text{P}$-hard, then $\#_{2^k}\text{SAT}(\mathcal{F})$ is $\#_{2^k}\text{P}$-hard.