The Complexity of Boolean Constraint Isomorphism

TL;DR

This paper classifies the computational complexity of the Boolean constraint isomorphism problem, showing it is either in P, coNP-hard, or GI-hard, with clear criteria to distinguish these cases.

Contribution

It provides a complete complexity classification of Boolean constraint isomorphism, identifying when the problem is in P, coNP-hard, or GI-hard, and offers simple criteria for this determination.

Findings

Boolean constraint isomorphism is either in P, coNP-hard, or GI-hard.

The paper provides criteria to determine the complexity class of a given problem instance.

It extends the understanding of the complexity landscape beyond satisfiability problems.

Abstract

In 1978, Schaefer proved his famous dichotomy theorem for generalized satisfiability problems. He defined an infinite number of propositional satisfiability problems (nowadays usually called Boolean constraint satisfaction problems) and showed that all these satisfiability problems are either in P or NP-complete. In recent years, similar results have been obtained for quite a few other problems for Boolean constraints.Almost all of these problems are variations of the satisfiability problem. In this paper, we address a problem that is not a variation of satisfiability, namely, the isomorphism problem for Boolean constraints. Previous work by B\"ohler et al. showed that the isomorphism problem is either coNP-hard or reducible to the graph isomorphism problem (a problem that is in NP, but not known to be NP-hard), thus distinguishing a hard case and an easier case. However, they did not classify which cases are truly easy, i.e., in P. This paper accomplishes exactly that. It shows that Boolean constraint isomorphism is coNP-hard (and GI-hard), or equivalent to graph isomorphism, or in P, and it gives simple criteria to determine which case holds.