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From Holant To #CSP And Back: Dichotomy For Holant^c Problems
University of Wisconsin - Madison. (Computer Sciences Department)
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0002-1600-5290
Microsoft Research Asia.
2010 (English)In: Algorithms and Computation - 21st International Symposium / [ed] Otfried Cheong and Kyung-Yong Chwa and Kunsoo Park, Springer , 2010, 253-265 p.Conference paper (Refereed)
Abstract [en]

We explore the intricate interdependent relationship among counting problems, considered from three frameworks for such problems: Holant Problems, counting CSP and weighted H-colorings. We consider these problems for general complex valued functions that take Boolean inputs. We show that results from one framework can be used to derive results in another, and this happens in both directions. Holographic reductions discover an underlying unity, which is only revealed when these counting problems are investigated in the complex domain . We prove three complexity dichotomy theorems, leading to a general theorem for Holantc problems. This is the natural class of Holant problems where one can assign constants 0 or 1. More specifically, given any signature grid on G=(VE) over a set  of symmetric functions, we completely classify the complexity to be in P or #P-hard, according to , of:E01vVfv(E(v))where fv01 (0, 1 are the unary constant 0, 1 functions). Not only is holographic reduction the main tool, but also the final dichotomy can be only naturally stated in the language of holographic transformations. The proof goes through another dichotomy theorem on Boolean complex weighted #CSP.

Place, publisher, year, edition, pages
Springer , 2010. 253-265 p.
, Lecture Notes in Computer Science, ISSN 0302-9743 ; 6506
National Category
Computer Science
URN: urn:nbn:se:kth:diva-42822DOI: 10.1007/978-3-642-17517-6ScopusID: 2-s2.0-78650905937ISBN: 978-3-642-17516-9OAI: diva2:447569
ISAAC 2010
EU, European Research Council, 6853
QC 20111013Available from: 2011-10-13 Created: 2011-10-12 Last updated: 2011-10-13Bibliographically approved

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