Restrictive Acceptance Suffices for Equivalence Problems

TL;DR

The paper introduces the class EP as a new, weaker evidence-based approach to suggest that certain NP problems are not NP-complete, broadening the tools available beyond UP.

Contribution

It defines the class EP, shows its relevance to NP-completeness evidence, and demonstrates its applicability to specific problems like OBDD equivalence and boolean negation.

Findings

EP contains certain problems like the negation and interchange equivalence problems.

EP tightens the upper bound for boolean negation equivalence to EP^{NP}.

FewP is contained in EP, providing new evidence against NP-completeness.

Abstract

One way of suggesting that an NP problem may not be NP-complete is to show that it is in the class UP. We suggest an analogous new approach---weaker in strength of evidence but more broadly applicable---to suggesting that concrete~NP problems are not NP-complete. In particular we introduce the class EP, the subclass of NP consisting of those languages accepted by NP machines that when they accept always have a number of accepting paths that is a power of two. Since if any NP-complete set is in EP then all NP sets are in EP, it follows---with whatever degree of strength one believes that EP differs from NP---that membership in EP can be viewed as evidence that a problem is not NP-complete. We show that the negation equivalence problem for OBDDs (ordered binary decision diagrams) and the interchange equivalence problem for 2-dags are in EP. We also show that for boolean negation the equivalence problem is in EP^{NP}, thus tightening the existing NP^{NP} upper bound. We show that FewP, bounded ambiguity polynomial time, is contained in EP, a result that is not known to follow from the previous SPP upper bound. For the three problems and classes just mentioned with regard to EP, no proof of membership/containment in UP is known, and for the problem just mentioned with regard to EP^{NP}, no proof of membership in UP^{NP} is known. Thus, EP is indeed a tool that gives evidence against NP-completeness in natural cases where UP cannot currently be applied.