Counting the number of independent sets is an important problem in graph theory, combinatorics, optimization, and social sciences. However, a polynomial-time exact calculation, or even a reasonably close approximation, is widely believed to be impossible, since their existence implies an efficient solution to various problems in the non-deterministic polynomial-time complexity class. To cope wi...

We present the first streaming algorithm for counting an arbitrary hypergraph H of constant size in a massive hypergraph G. Our algorithm can handle both edge-insertions and edge-deletions, and is applicable for the distributed setting. Moreover, our approach provides the first family of graph polynomials for the hypergraph counting problem. Because of the close relationship between hypergraphs...

In this paper we study the parameterized complexity of approximating the parameterized counting problems contained in the class #W [P ] ; the parameterized analogue of #P: We prove a parameterized analogue of a famous theorem of Stockmeyer claiming that approximate counting belongs to the second level of the polynomial hierarchy.

Counting the linear extensions of a partially ordered set (poset) is a fundamental problem with several applications. We present two exact algorithms that target sparse posets in particular. The first algorithm breaks the counting task into subproblems recursively. The second algorithm uses variable elimination via inclusion–exclusion and runs in polynomial time for posets with a cover graph of...

This paper outlines the use of rapidly mixing Markov Chains in randomized polynomial time algorithms to solve approximately certain counting problems. They fall into two classes : combinatorial problems like counting the number of perfect matchings in certain graphs and geometric ones like computing the volumes of convex sets.

One of the fundamental results of descriptive complexity theory, due to Immerman [12] and Vardi [17], says that a class of ordered finite structures is definable in fixed-point logic if, and only if, it is computable in polynomial time. Much effort has been spent on the problem of capturing polynomial time, that is, describing all polynomial time computable classes of not necessarily ordered fi...

In the face of an untrusted cloud infrastructure, outsourced data needs to be protected. Fully homomorphic encryption is one solution that also allows performing operations on outsourced data. However, the involved high overhead of today’s fully homomorphic encryption techniques outweigh cloud cost saving advantages, rendering it impractical. We present EPiC, a practical, efficient protocol for...

We make an attempt to understand the dominating set counting problem in graph classes from the viewpoint of polynomial-time computability. We give polynomial-time algorithms to count the number of dominating sets (and minimum dominating sets) in interval graphs and trapezoid graphs. They are based on dynamic programming. With the help of dynamic update on a binary tree, we further reduce the ti...

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...

Let I ⊂ R be a compact interval symmetric about 0. Let M and N , be compact oriented smooth manifolds of dimensions d and d+1 respectively. We suppose that we have an embedded copy of I ×M inside of N . (See Figure 1). Let N denote the complement of the hypersurface {0} ×M . We consider families ǫ → g(ǫ) of Riemannian metric (tensors) on N each of whose restriction to I ×M is a warped product o...