Recognizing When Heuristics Can Approximate Minimum Vertex Covers Is Complete for Parallel Access to NP

For both the edge deletion heuristic and the maximum-degree greedy heuristic, we study the problem of recognizing those graphs for which that heuristic can approximate the size of a minimum vertex cover within a constant factor of r, where r is a fixed rational number. Our main results are that these problems are complete for the class of problems solvable via parallel access to NP. To achieve these main results, we also show that the restriction of the vertex cover problem to those graphs for which either of these heuristics can find an optimal solution remains NP-hard.Comment: 16 pages, 2 figure

LWPP and WPP are not uniformly gap-definable

AbstractResolving an issue open since Fenner, Fortnow, and Kurtz raised it in [S. Fenner, L. Fortnow, S. Kurtz, Gap-definable counting classes, J. Comput. System Sci. 48 (1) (1994) 116–148], we prove that LWPP is not uniformly gap-definable and that WPP is not uniformly gap-definable. We do so in the context of a broader investigation, via the polynomial degree bound technique, of the lowness, Turing hardness, and inclusion relationships of counting and other central complexity classes

The Robustness of LWPP and WPP, with an Application to Graph Reconstruction

We show that the counting class LWPP [FFK94] remains unchanged even if one allows a polynomial number of gap values rather than one. On the other hand, we show that it is impossible to improve this from polynomially many gap values to a superpolynomial number of gap values by relativizable proof techniques. The first of these results implies that the Legitimate Deck Problem (from the study of graph reconstruction) is in LWPP (and thus low for PP, i.e., \rm PP^{\mbox{Legitimate Deck}} = PP) if the weakened version of the Reconstruction Conjecture holds in which the number of nonisomorphic preimages is assumed merely to be polynomially bounded. This strengthens the 1992 result of K\"{o}bler, Sch\"{o}ning, and Tor\'{a}n [KST92] that the Legitimate Deck Problem is in LWPP if the Reconstruction Conjecture holds, and provides strengthened evidence that the Legitimate Deck Problem is not NP-hard. We additionally show on the one hand that our main LWPP robustness result also holds for WPP, and also holds even when one allows both the rejection- and acceptance- gap-value targets to simultaneously be polynomial-sized lists; yet on the other hand, we show that for the #P-based analog of LWPP the behavior much differs in that, in some relativized worlds, even two target values already yield a richer class than one value does. Despite that nonrobustness result for a #P-based class, we show that the #P-based "exact counting" class $\rm C_{=}P$ remains unchanged even if one allows a polynomial number of target values for the number of accepting paths of the machine