Title: 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., $\rm
PP^{\mbox{Legitimate Deck}} = PP$) 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 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
$\rm C_{=}P$ remains unchanged even if one allows a polynomial number of target
values for the number of accepting paths of the machine.