NDLI: Holographic Algorithms with Matchgates Capture Precisely Tractable Planar #CSP

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Lu, Pinyan Cai, Jin-Yi Xia, Mingji
Copyright Year	2017
Abstract	Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate computations and holographic algorithms based on them provide a universal methodology to a broad class of counting problems studied in the statistical physics community for decades. They capture precisely those problems which are #P-hard on general graphs but computable in polynomial time on planar graphs. More precisely, we prove complexity dichotomy theorems in the framework of counting CSP problems. The local constraint functions take Boolean inputs and can be arbitrary real-valued symmetric functions. We prove that every problem in this class belongs to precisely three categories: (1) those which are tractable (i.e., polynomial time computable) on general graphs, or (2) those which are #P-hard on general graphs but tractable on planar graphs, or (3) those which are #P-hard even on planar graphs. The classification criteria are explicit. Moreover, problems in category (2) are tractable on planar graphs precisely by holographic algorithms with matchgates.
Sponsorship	China National 973 Program. National Natural Science Foundation of China. National Science Foundation
Starting Page	853
Ending Page	889
Page Count	37
File Format	PDF
ISSN	00975397
DOI	10.1137/16M1073984
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	3
Volume Number	46
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-05-11
Access Restriction	Subscribed
Subject Keyword	Computational difficulty of problems holographic algorithms counting problems #P dichotomy theorems
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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