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

Edith Hemaspaandra,Holger Spakowski,Lane A Hemaspaandra,Osamu Watanabe

doi:10.1007/s00037-020-00197-5

Abstract

We show that the counting class LWPP 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., $$\rm PP^{Legitimate Deck} = PP$$ ) 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 Kobler, Schoning & Toran 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 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 $$\#{\rm P}$$ -based analogue 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 $$\#{\rm P}$$ -based class, we show that the $$\#{\rm P}$$ -based “exact counting” class $${\rm C}_{=}{\rm P}$$ remains unchanged even if one allows a polynomial number of target values for the number of accepting paths of the machine.

Highlights

Nothing is more natural than wanting to better understand an object by knowing what it can and cannot do
To summarize: In this paper, we prove that LWPP and WPP are robust enough that they remain unchanged when their single target gap is allowed to be expanded to a polynomial-sized list; we apply this new robustness of LWPP to show that the PP-lowness of the Legitimate Deck Problem follows from a weaker hypothesis than was previously known; we show that our polynomial robustness of LWPP is optimal with respect to relativizable proofs; and we prove a number of related results on limitations and extensions
We proved that LWPP and WPP are robust enough that they remain unchanged when their single target gap is allowed to be expanded to polynomial-sized lists

Summary

Introduction

Nothing is more natural than wanting to better understand an object by knowing what it can and cannot do. This paper shows that the counting classes LWPP and WPP, defined in 1994 in the seminal work of Fenner, Fortnow, and Kurtz [8] on gap-based counting classes, are quite robust Even though their definitions are in terms of having the gap function (the difference between the number of accepting and rejecting paths of a machine) hit a single target value, we prove (in Section 3) that one can allow a list of up to polynomially many target values without altering the descriptive richness of the class, i.e., without changing the class. To summarize: In this paper, we prove that LWPP and WPP are robust enough that they remain unchanged when their single target gap is allowed to be expanded to a polynomial-sized list; we apply this new robustness of LWPP to show that the PP-lowness of the Legitimate Deck Problem follows from a weaker hypothesis than was previously known; we show that our polynomial robustness of LWPP is optimal with respect to relativizable proofs; and we prove a number of related results on limitations and extensions

Preliminaries

Main Result

Applying the Main Result to Graph Reconstruction

Conclusions and Open Questions