Achieving new upper bounds for the hypergraph duality problem through logic

Abstract

The hypergraph duality problem Dual is defined as follows: given two simple hypergraphs G and H, decide whether H consists precisely of all minimal transversals of G (in which case we say that G is the dual of H). This problem is equivalent to decide whether two given non-redundant monotone DNFs are dual. It is known that DUAL, the complementary problem to Dual, is in GC (log2 n, PTIME), where GC(f(n), C) denotes the complexity class of all problems that after a nondeterministic guess of O(f(n)) bits can be decided (checked) within complexity class C. It was conjectured that DUAL is in GC(log2 n, LOGSPACE). In this paper we prove this conjecture and actually place the DUAL problem into the complexity class GC(log2 n, TC0) which is a subclass of GC(log2 n, LOGSPACE). We here refer to the logtime-uniform version of TC0, which corresponds to FO(COUNT), i.e., first order logic augmented by counting quantifiers. We achieve the latter bound in two steps. First, based on existing problem decomposition methods, we develop a new nondeterministic algorithm for DUAL that requires to guess O(log2 n) bits. We then proceed by a logical analysis of this algorithm, allowing us to formulate its deterministic part in FO(COUNT).