The complexity of complex weighted Boolean #CSP

We prove a complexity dichotomy theorem for the most general form of Boolean #CSP where every constraint function takes values in the field of complex numbers C. We first give a non-trivial tractable class of Boolean #CSP which was inspired by holographic reductions. The tractability crucially depends on algebraic cancelations which are absent for non-negative numbers. We then completely characterize all the tractable Boolean #CSP with complex-valued constraints and show that we have found all the tractable ones, and every remaining problem is #P-hard. We also improve our result by proving the same dichotomy theorem holds for Boolean #CSP with maximum degree 3 (every variable appears at most three times). The concept of Congruity and Semi-congruity provides a key insight and plays a decisive role in both the tractability and hardness proofs. We also introduce local holographic reductions as a technique in hardness proofs.

[1]  Mingji Xia,et al.  The Complexity of Weighted Boolean #CSP Modulo k , 2011, STACS.

[2]  Martin E. Dyer,et al.  The complexity of counting graph homomorphisms (extended abstract) , 2000, SODA '00.

[3]  Jin-Yi Cai,et al.  Holographic Algorithms by Fibonacci Gates and Holographic Reductions for Hardness , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[4]  Jin-Yi Cai,et al.  From Holant to #CSP and Back: Dichotomy for Holantc Problems , 2010, ISAAC.

[5]  Martin E. Dyer,et al.  On the complexity of #CSP , 2010, STOC '10.

[6]  Jin-Yi Cai,et al.  A complete dichotomy rises from the capture of vanishing signatures: extended abstract , 2012, STOC '13.

[7]  Jin-Yi Cai,et al.  Computational Complexity of Holant Problems , 2011, SIAM J. Comput..

[8]  Jin-Yi Cai,et al.  Graph Homomorphisms with Complex Values: A Dichotomy Theorem , 2009, SIAM J. Comput..

[9]  Nadia Creignou,et al.  Complexity of Generalized Satisfiability Counting Problems , 1996, Inf. Comput..

[10]  Martin E. Dyer,et al.  The Complexity of Weighted Boolean #CSP with Mixed Signs , 2009, Theor. Comput. Sci..

[11]  Pinyan Lu,et al.  A Dichotomy for Real Weighted Holant Problems , 2012, Computational Complexity Conference.

[12]  Leslie Ann Goldberg,et al.  A Complexity Dichotomy for Partition Functions with Mixed Signs , 2008, SIAM J. Comput..

[13]  Martin E. Dyer,et al.  The Complexity of Weighted Boolean #CSP , 2009, SIAM J. Comput..

[14]  Jin-Yi Cai,et al.  Non-negatively Weighted #CSP: An Effective Complexity Dichotomy , 2010, 2011 IEEE 26th Annual Conference on Computational Complexity.

[15]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[16]  Sanjeev Khanna,et al.  Complexity classifications of Boolean constraint satisfaction problems , 2001, SIAM monographs on discrete mathematics and applications.

[17]  Jin-Yi Cai,et al.  Holant problems and counting CSP , 2009, STOC '09.

[18]  Salil P. Vadhan,et al.  The Complexity of Counting in Sparse, Regular, and Planar Graphs , 2002, SIAM J. Comput..

[19]  Ker-I Ko,et al.  Complexity Theory of Real Functions , 1991, Progress in Theoretical Computer Science.

[20]  Jin-Yi Cai,et al.  Holographic algorithms: from art to science , 2007, STOC '07.

[21]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[22]  Andrei A. Bulatov.  The Complexity of the Counting Constraint Satisfaction Problem , 2008, ICALP.

[23]  Leslie G. Valiant,et al.  Holographic Algorithms (Extended Abstract) , 2004, FOCS.

[24]  Andrei A. Bulatov,et al.  The complexity of partition functions , 2005, Theor. Comput. Sci..

[25]  Leslie G. Valiant,et al.  Holographic Algorithms (Extended Abstract) , 2004, FOCS.

[26]  Catherine S. Greenhill,et al.  The complexity of counting graph homomorphisms , 2000 .

[27]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[28]  Andrei A. Bulatov,et al.  A dichotomy theorem for constraint satisfaction problems on a 3-element set , 2006, JACM.

[29]  Martin E. Dyer,et al.  On counting homomorphisms to directed acyclic graphs , 2006, JACM.

[30]  Jin-Yi Cai,et al.  From Holant to #CSP and Back: Dichotomy for Holantc Problems , 2012, Algorithmica.