NDLI: Holographic Algorithms with Matchgates Capture Precisely Tractable Planar

Content Provider	IEEE Xplore Digital Library
Author	Jin-Yi Cai Pinyan Lu Mingji Xia
Copyright Year	2010
Abstract	Valiant introduced match gate 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, match gate computations and holographic algorithms based on them provide a universal methodology to a broad class of counting problems studied in 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 ractable 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.
Starting Page	427
Ending Page	436
File Size	432520
Page Count	10
File Format	PDF
ISBN	9781424485253
ISSN	02725428
DOI	10.1109/FOCS.2010.48
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2010-10-23
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Tensile stress Polynomials Complexity theory Software algorithms Bipartite graph Interpolation
Content Type	Text
Resource Type	Article

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