eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-03-03
49:1
49:18
10.4230/LIPIcs.STACS.2023.49
article
Gap Preserving Reductions Between Reconfiguration Problems
Ohsaka, Naoto
1
https://orcid.org/0000-0001-9584-4764
CyberAgent, Inc., Tokyo, Japan
Combinatorial reconfiguration is a growing research field studying problems on the transformability between a pair of solutions for a search problem. For example, in SAT Reconfiguration, for a Boolean formula Ο and two satisfying truth assignments Ο_π and Ο_π for Ο, we are asked to determine whether there is a sequence of satisfying truth assignments for Ο starting from Ο_π and ending with Ο_π, each resulting from the previous one by flipping a single variable assignment. We consider the approximability of optimization variants of reconfiguration problems; e.g., Maxmin SAT Reconfiguration requires to maximize the minimum fraction of satisfied clauses of Ο during transformation from Ο_π to Ο_π. Solving such optimization variants approximately, we may be able to obtain a reasonable reconfiguration sequence comprising almost-satisfying truth assignments.
In this study, we prove a series of gap-preserving reductions to give evidence that a host of reconfiguration problems are PSPACE-hard to approximate, under some plausible assumption. Our starting point is a new working hypothesis called the Reconfiguration Inapproximability Hypothesis (RIH), which asserts that a gap version of Maxmin CSP Reconfiguration is PSPACE-hard. This hypothesis may be thought of as a reconfiguration analogue of the PCP theorem. Our main result is PSPACE-hardness of approximating Maxmin 3-SAT Reconfiguration of bounded occurrence under RIH. The crux of its proof is a gap-preserving reduction from Maxmin Binary CSP Reconfiguration to itself of bounded degree. Because a simple application of the degree reduction technique using expander graphs due to Papadimitriou and Yannakakis (J. Comput. Syst. Sci., 1991) does not preserve the perfect completeness, we modify the alphabet as if each vertex could take a pair of values simultaneously. To accomplish the soundness requirement, we further apply an explicit family of near-Ramanujan graphs and the expander mixing lemma. As an application of the main result, we demonstrate that under RIH, optimization variants of popular reconfiguration problems are PSPACE-hard to approximate, including Nondeterministic Constraint Logic due to Hearn and Demaine (Theor. Comput. Sci., 2005), Independent Set Reconfiguration, Clique Reconfiguration, and Vertex Cover Reconfiguration.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol254-stacs2023/LIPIcs.STACS.2023.49/LIPIcs.STACS.2023.49.pdf
combinatorial reconfiguration
hardness of approximation
gap reduction