We give a trichotomy theorem for the complexity of approximately counting the number of satisfying assignments of a Boolean CSP instance. Such problems are parameterised by a constraint language specifying the relations that may be used in constraints. If every relation in the constraint language is affine then the number of satisfying assignments can be exactly counted in polynomial time. Othe...

Holant problems capture a class of Sum-of-Product computations such as counting matchings. It is inspired by holographic algorithms and is equivalent to tensor networks, with counting CSP being a special case. A classification for Holant problems is more difficult to prove, not only because it implies a classification for counting CSP, but also due to the deeper reason that there exist more int...

We study the complexity of computing the sign of the Tutte polynomial of a graph. As there are only three possible outcomes (positive, negative, and zero), this seems at first sight more like a decision problem than a counting problem. Surprisingly, however, there are large regions of the parameter space for which computing the sign of the Tutte polynomial is actually #P-hard. As a trivial cons...

This paper addresses the problems of counting proof trees (as introduced by Venkateswaran and Tompa) and counting proof circuits, a related but seemingly more natural question. These problems lead to a common generalization of straight-line programs which we call polynomial replacement systems. We contribute a classification of these systems and we investigate their complexity. Diverse problems...

We introduce a class of counting problems that arise naturally in equational matching and study their computational complexity. If E is an equational theory, then #E-Matching is the problem of counting the number of complete minimal E-matchers of two given terms. #E-Matching is a well-deened algorithmic problem for every nitary equational theory. Moreover, it captures more accurately the comput...

We use production matrices to count several classes of geometric graphs. present novel for non-crossing partitions, connected graphs, and k -angulations, which provide another, simple elegant, way counting the number such objects. Counting graphs is then equivalent calculating powers a matrix. Applying technique Riordan Arrays these matrices, we establish new formulas numbers as well combinator...

The n Birkhoff polytope is the set of all doubly stochastic n × n matrices, that is, those matrices with nonnegative real coefficients in which every row and column sums to one. A wide open problem concerns the volumes of these polytopes, which have been known for n ≤ 8. We present a new, complex-analytic way to compute the Ehrhart polynomial of the Birkhoff polytope, that is, the function coun...

We present two algorithms that, given a prime l and an elliptic curve E/Fq, directly compute the polynomial Φl(j(E), Y ) ∈ Fq[Y ] whose roots are the j-invariants of the elliptic curves that are l-isogenous to E. We do not assume that the modular polynomial Φl(X, Y ) is given. The algorithms may be adapted to handle other types of modular polynomials, and we consider applications to point count...

We show that there exists a a fully polynomial randomized approximation scheme for counting the number of Hamilton cycles in almost all directed graphs.

Inclusion/exclusion branching is a way to branch on requirements imposed on problems, in contrast to the classical branching on parts of the solution. The technique turned out to be useful for finding and counting (minimum) dominating sets (van Rooij et al., ESA 2009). In this paper, we extend the technique to the setting where one is given a set of properties and seeks (or wants to count) solu...