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.