32 research outputs found
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
remains unchanged even if one allows a polynomial number of target
values for the number of accepting paths of the machine