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

Abstract

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., PP 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 Köbler, Schöning, and Torán [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 C = P remains unchanged even if one allows a polynomial number of target values for the number of accepting paths of the machine.

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

“…Note: Please see [12] for a complete version of this section, which includes the proof of Theorem 5.1 and additional results showing that LWPP with an exponential number of target gap values cannot equal LWPP without causing highly unlikely structural consequences such as PP NP = PP.…”

“…It is easy to see that 1-LWPP = LWPP, and that, of course, more flexibility as to targets never removes sets from the class, i.e., speaking loosely for the moment as to notation (and the log case will not be defined or used again in this paper, but it is clear from context here what we mean by it; the exponential case's definition can be found in the full version of the paper [12]),…”

“…So it is not the case that single targets and lists of targets inherently function identically as to descriptive richness for counting classes. Note: Please see [12] for a complete version of this section, which includes definitions and the proof of our relativized result (Theorem 6.…”