Holographic reduction for some counting problems
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ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Guest column: complexity dichotomies of counting problems
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ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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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.