The Robustness of LWPP and WPP, with an Application to Graph Reconstruction
We show that the counting class LWPP [S. Fenner et al., 1994] 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 [J. Köbler et al., 1992] 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.
structural complexity theory
robustness of counting classes
the legitimate deck problem
PP-lowness
the Reconstruction Conjecture
Theory of computation~Complexity classes
51:1-51:14
Regular Paper
https://arxiv.org/abs/1711.01250v2
Edith
Hemaspaandra
Edith Hemaspaandra
Rochester Institute of Technology, Rochester, NY, USA
Lane A.
Hemaspaandra
Lane A. Hemaspaandra
University of Rochester, Rochester, NY, USA
Holger
Spakowski
Holger Spakowski
University of Cape Town, Rondebosch, South Africa
Osamu
Watanabe
Osamu Watanabe
Tokyo Institute of Technology, Tokyo, Japan
10.4230/LIPIcs.MFCS.2018.51
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Edith Hemaspaandra, Lane A. Hemaspaandra, Holger Spakowski, and Osamu Watanabe
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