We show that the propositional model counting problem #SAT for CNFformulas with hypergraphs that allow a disjoint branches decomposition can be solved in polynomial time. We show that this class of hypergraphs is incomparable to hypergraphs of bounded incidence cliquewidth which were the biggest class of hypergraphs for which #SAT was known to be solvable in polynomial time so far. Furthermore,...

Motivated by the definition of the edge elimination polynomial of a graph we define the covered components polynomial counting spanning subgraphs with respect to their number of components, edges and covered components. We prove a recurrence relation, which shows that both graph polynomials are substitution instances of each other. We give some properties of the covered components polynomial an...

In this paper, a stability criterion based on counting the real roots of two specific polynomials is formulated. To establish this result, it is shown that a hyperbolicity condition and a strict positivity of a polynomial Wronskian are necessary and sufficient for the stability of any real polynomial. This result is extended to the stability study of some linear combinations of polynomials. Nec...

An explicit combinatorial expression is obtained for the Zhang-Zhang polynomial (also known as “Clar cover polynomial”) of a large class of pericondensed benzenoid systems, the multiple linear hexagonal chains Mn,m. By means of this result, various problems encountered in the Clar theory of Mn,m are also resolved: counting of Clar and Kekulé structures, determining the Clar number, and calculat...

We establish a connection between linear codes and hyperplane arrangements using the Thomas decomposition of polynomial systems and the resulting counting polynomial. This yields both a generalization and a refinement of the weight enumerator of a linear code. In particular, one can deal with infinitely many finite fields simultaneously by defining a weight enumerator for codes over infinite fi...

We examine two different ways of encoding a counting function, as a rational generating function and explicitly as a function (defined piecewise using the greatest integer function). We prove that, if the degree and number of input variables of the (quasi-polynomial) function are fixed, there is a polynomial time algorithm which converts between the two representations. Examples of such countin...

In this paper, we focus on computing the prices of securities represented by logical formulas in combinatorial prediction markets when the price function is represented by a Bayesian network. This problem turns out to be a natural extension of the weighted model counting (WMC) problem [15], which we call generalized weighted model counting (GWMC) problem. In GWMC, we are given a logical formula...

A long time ago, Yuri Gurevich made precise the problem of whether there is a logic capturing polynomial-time on arbitrary finite structures, and conjectured that no such logic exists. This conjecture is still open. Nevertheless, together with Andreas Blass and Saharon Shelah, he has also proposed what still seems to be the most promising candidate for a logic for polynomial time, namely Choice...

We describe a simpliication of a recent polynomial-time algorithm of A. I. Barvinok for counting the number of lattice points in a poly-hedron in xed dimension. In particular, we show that only very elementary properties of exponential sums are needed to develop a polynomial-time algorithm.

An attractive and challenging problem in computational number theory is to count in an e*cient manner the number of solutions to a multivariate polynomial equation over a -nite -eld. One desires an algorithm whose time complexity is a small polynomial function of some appropriate measure of the size of the polynomial. A natural measure of size is d logðqÞ for a polynomial of total degree d in n...