Abstract

We prove a complexity dichotomy theorem for all non-negatively weighted counting Constraint Satisfaction Problems (#CSP). This caps a long series of important results on counting problems, including unweighted and weighted graph homomorphisms and the celebrated dichotomy theorem for unweighted #CSP. Our dichotomy theorem gives a succinct criterion for tractability. If a set F of constraint functions satisfies the criterion, then the #CSP problem defined by F is solvable in polynomial time; if it does not satisfy the criterion, then the problem is #P-hard. We furthermore show that the question of whether F satisfies the criterion is decidable in NP. Surprisingly, our tractability criterion is simpler than the previous tractability criteria for the more restricted classes of problems, although when specialized to those cases, they are logically equivalent. Our proof mainly uses Linear Algebra and represents a departure from Universal Algebra, the dominant methodology in recent years.

Full Text

Submitted Version (Free)

Published Version

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.