On Modular Counting with Polynomials

2006 ◽  
Author(s):  
K.A. Hansen
Keyword(s):  
Modular Counting

scholarly journals Languages Defined with Modular Counting Quantifiers

2001 ◽  
Vol 166 (2) ◽  
pp. 112-132 ◽  
Author(s):  
Howard Straubing
Keyword(s):  
Modular Counting

Algorithms for Modular Counting of Roots of Multivariate Polynomials

Algorithmica ◽  
2007 ◽  
Vol 50 (4) ◽  
pp. 479-496 ◽  
Author(s):  
Parikshit Gopalan ◽  
Venkatesan Guruswami ◽  
Richard J. Lipton
Keyword(s):  
Multivariate Polynomials ◽  
Modular Counting

scholarly journals Counting Homomorphisms to Trees Modulo a Prime

2021 ◽  
Vol 13 (3) ◽  
pp. 1-33
Author(s):  
Andreas Göbel ◽  
J. A. Gregor Lagodzinski ◽  
Karen Seidel
Keyword(s):  
Statistical Physics ◽  
Independent Sets ◽  
Partition Functions ◽  
Interim Result ◽  
Graph Homomorphism ◽  
Graph Theoretic ◽  
Modular Counting ◽  
Graph Homomorphisms ◽  
Counting Functions ◽  
The Common

Many important graph-theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article, we study the complexity of  # p H OMS T O H , the problem of counting graph homomorphisms from an input graph to a graph H modulo a prime number  p . Dyer and Greenhill proved a dichotomy stating that the tractability of non-modular counting graph homomorphisms depends on the structure of the target graph. Many intractable cases in non-modular counting become tractable in modular counting due to the common phenomenon of cancellation. In subsequent studies on counting modulo 2, however, the influence of the structure of  H on the tractability was shown to persist, which yields similar dichotomies. Our main result states that for every tree  H and every prime  p the problem # p H OMS T O H is either polynomial time computable or # p P-complete. This relates to the conjecture of Faben and Jerrum stating that this dichotomy holds for every graph H when counting modulo 2. In contrast to previous results on modular counting, the tractable cases of # p H OMS T O H are essentially the same for all values of the modulo when H is a tree. To prove this result, we study the structural properties of a homomorphism. As an important interim result, our study yields a dichotomy for the problem of counting weighted independent sets in a bipartite graph modulo some prime  p . These results are the first suggesting that such dichotomies hold not only for the modulo 2 case but also for the modular counting functions of all primes  p .

Complex polynomials and circuit lower bounds for modular counting

1994 ◽  
Vol 4 (4) ◽  
pp. 325-338 ◽  
Author(s):  
David A. Mix Barrington ◽  
Howard Straubing
Keyword(s):  
Lower Bounds ◽  
Complex Polynomials ◽  
Circuit Lower Bounds ◽  
Modular Counting
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