The notion of graph polynomials definable in Monadic Second Order Logic, MSOL, was introduced in [Mak04]. It was shown that the Tutte polynomial and its generalization, as well as the matching polynomial, the cover polynomial and the various interlace polynomials fall into this category. In this paper we present a framework of graph polynomials based on counting functions of generalized colorin...

We know that counting perfect matchings is polynomial time when we restrict ourselves to the class of planar graphs. Generally speaking, the decision and search versions of a problem turn out to be “easier” than the counting question. For example, the problem of determining if a perfect matching exists, and finding one when it does, is polynomial time in general graphs, while the question of co...

We present a novel extension to the permutation group enumeration technique which is well known to have polynomial time algorithms. This extended technique allows each perfect matching in a bipartite graph on 2n nodes to be expressed as a unique directed path in a directed acyclic graph of size O(n). Thus it transforms the perfect matching counting problem into a directed path counting problem ...

We present a novel extension to the permutation group enumeration technique which is well known to have polynomial time algorithms. This extended technique allows each perfect matching in a bipartite graph of size O(n) to be expressed as a unique directed path in a directed acyclic graph of size O(n). Thus it transforms the perfect matching counting problem into a directed path counting problem...

The major goal of this study was to investigate the broad application of graph polynomials to the analysis of Henry’s law constants (solubility) of nonane isomers. In this context, Henry’s law constants of nonane isomers were modelled using characteristic and counting polynomials. The characteristic and counting polynomials on the distance matrix (CDi), on the maximal fragments matrix (CMx), on...

Recent work by Birnbaum & Lozinskii [1999] demonstrated that a clever yet simple extension of the well-known DavisPutnam procedure for solving instances of propositional satisfiability yields an efficient scheme for counting the number of satisfying assignments (models). We present a new extension, based on recursively identifying connected constraint-graph components, that substantially improv...

We characterize the class of committee scoring rules that satisfy the fixed-majority criterion. In some sense, the committee scoring rules in this class are multiwinner analogues of the singlewinner Plurality rule, which is uniquely characterized as the only single-winner scoring rule that satisfies the simple majority criterion. We define top-k-counting committee scoring rules and show that th...

We propose an algorithm based on Barvinok’s counting algorithm for P→max{c′x|Ax ≤ b;x ∈ Z}. It runs in time polynomial in the input size of P when n is fixed, and under a condition on c, provides the optimal value of P. We also relate Barvinok’s counting formula and Gomory relaxations.

We study the computational complexity of the counting version of the POSSIBLE-WINNER problem for elections. In the POSSIBLE-WINNER problem we are given a profile of voters, each with a partial preference order, and ask if there are linear extensions of the votes such that a designated candidate wins. We also analyze a special case of POSSIBLE-WINNER, the MANIPULATION problem. We provide polynom...

Holant problems are a general framework to study the algorithmic complexity of counting problems. Both counting constraint satisfaction problems and graph homomorphisms are special cases. All previous results of Holant problems are over the Boolean domain. In this paper, we give the first dichotomy theorem for Holant problems for domain size > 2. We discover unexpected tractable families of cou...