Restrictive acceptance suffices for equivalence problems

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 [17,12]) and the interchange equivalence problem for 2-dags are in EP. We also show that for boolean negation [20] the equivalence problem is in EPNP, thus tightening the existing NPNP upper bound. We show that FewP [2], 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 EPNP, no proof of membership in UPNP is known. Thus, EP is indeed a tool that gives evidence against NP-completeness in natural cases where UP cannot currently be applied.