Complexity and Algorithms for Well-Structured k-SAT Instances

Abstract

This paper initiates the study of SAT instances of bounded diameter. The diameter of an ordered CNF formula is defined as the maximum difference between the index of the first and the last occurrence of a variable. We study the complexity of the satisfiability, the counting and the maximization problems for formulas of bounded diameter. We investigate the relation between the diameter of a formula, and the tree-width and the path-width of its corresponding incidence graph, and show that under highly parallel and efficient transformations, diameter and path-width are equal up to a constant factor.Our main technical contribution is that the computational complexity of SAT, Max-SAT, #SAT grows smoothly with the diameter (as a function of the number of variables). Our main focus is in providing space efficient and highly parallel algorithms, while the running time of our algorithms matches previously known results. Among others, we show NL-completeness of SAT and NC 2 algorithms for Max-SAT, #SAT when diameter is O(log n). Given the tree decomposition of a formula, we further improve on the space efficiency to decide SAT as asked by Alekhnovich and Razborov [1].

A note on width-parameterized SAT: An exact machine-model characterization

“…In particular, Theorem 1 and Corollary 4 are proved in Section 2. Lemma 3 directly follows by the foregoing results from [4], Lemma 11, and by padding. Here, by padding we mean that we extend a propositional formula by adding new clauses each of which contains a single, fresh variable.…”

“…For instance, it is crucial that the parameter is the number of passes over the nd-tape and not some other measure of "use" of nondeterminism. Furthermore, the same characterization wouldn't work for treewidth, since SAT pw [log n] is complete for NL, but we conjecture that the same problem for treewidth is complete for LOGCFL (one can show containment [4]), and it is thought that NL LOGCFL.…”

“…Making further progress seems difficult, partly because the precise complexity of SAT pw [w(n)] is unknown. In [4] it is observed that for k-CNFs with path decompositions of width w(n) the 2 O(w(n)) running time can be improved to nondeterministic space O(w(n)). This improvement is far from an exact characterization, since NSPACE(w(n)) is believed to contain sets not in NP, if w(n) ∈ ω(log n).…”

“…Our main result is the completeness part of Theorem 1. In [4] it is shown that 3-SAT pw [w(n)] ∈ NSPACE(w(n)), and that 3-SAT pw [w(n)] and 3-SAT diam [w(n)] are logspace reducible to each other. Thus, the results of the next section imply Lemma 3 and Corollary 4.…”

“…Lemma 3 is somewhat interesting since there is a family of 3-CNFs where every variable appears only a constant number of times, the formulas are of pathwidth O(log n) and every ordering of the clauses has diameter Ω( n log n ) [4]. This combinatorial exponential gap is being smoothed out through logspace transformations.…”