On Satisfiability Problems with a Linear Structure

TL;DR

This paper extends polynomial-time solvability of satisfiability problems to broader classes of bipartite graphs with linear structures, introduces methods to find minimal such structures, and proves complexity limits for related problems.

Contribution

It generalizes the class of graphs for which satisfiability is polynomially solvable and provides algorithms to find minimal k-interval bigraphs compatible with given orders.

Findings

Polynomial-time solvability for k-interval bigraphs.

Algorithm to find minimal k-interval bigraphs given variable and clause orders.

NP-hardness of recognizing 1-interval bigraphs.

Abstract

It was recently shown \cite{STV} that satisfiability is polynomially solvable when the incidence graph is an interval bipartite graph (an interval graph turned into a bipartite graph by omitting all edges within each partite set). Here we relax this condition in several directions: First, we show that it holds for $k$-interval bigraphs, bipartite graphs which can be converted to interval bipartite graphs by adding to each node of one side at most $k$ edges; the same result holds for the counting and the weighted maximization version of satisfiability. Second, given two linear orders, one for the variables and one for the clauses, we show how to find, in polynomial time, the smallest $k$ such that there is a $k$-interval bigraph compatible with these two orders. On the negative side we prove that, barring complexity collapses, no such extensions are possible for CSPs more general than satisfiability. We also show NP-hardness of recognizing 1-interval bigraphs.