Free Energy Approximations for CSMA networks

In this paper we study how to estimate the back-off rates in an idealized CSMA network consisting of $n$ links to achieve a given throughput vector using free energy approximations. More specifically, we introduce the class of region-based free energy approximations with clique belief and present a closed form expression for the back-off rates based on the zero gradient points of the free energy approximation (in terms of the conflict graph, target throughput vector and counting numbers). Next we introduce the size $k_{max}$ clique free energy approximation as a special case and derive an explicit expression for the counting numbers, as well as a recursion to compute the back-off rates. We subsequently show that the size $k_{max}$ clique approximation coincides with a Kikuchi free energy approximation and prove that it is exact on chordal conflict graphs when $k_{max} = n$. As a by-product these results provide us with an explicit expression of a fixed point of the inverse generalized belief propagation algorithm for CSMA networks. Using numerical experiments we compare the accuracy of the novel approximation method with existing methods

The parameterised complexity of counting even and odd induced subgraphs

We consider the problem of counting, in a given graph, the number of induced k-vertex subgraphs which have an even number of edges, and also the complementary problem of counting the k-vertex induced subgraphs having an odd number of edges. We demonstrate that both problems are #W[1]-hard when parameterised by k, in fact proving a somewhat stronger result about counting subgraphs with a property that only holds for some subset of k-vertex subgraphs which have an even (respectively odd) number of edges. On the other hand, we show that each of the problems admits an FPTRAS. These approximation schemes are based on a surprising structural result, which exploits ideas from Ramsey theory

Galois correspondence for counting quantifiers

We introduce a new type of closure operator on the set of relations, max-implementation, and its weaker analog max-quantification. Then we show that approximation preserving reductions between counting constraint satisfaction problems (#CSPs) are preserved by these two types of closure operators. Together with some previous results this means that the approximation complexity of counting CSPs is determined by partial clones of relations that additionally closed under these new types of closure operators. Galois correspondence of various kind have proved to be quite helpful in the study of the complexity of the CSP. While we were unable to identify a Galois correspondence for partial clones closed under max-implementation and max-quantification, we obtain such results for slightly different type of closure operators, k-existential quantification. This type of quantifiers are known as counting quantifiers in model theory, and often used to enhance first order logic languages. We characterize partial clones of relations closed under k-existential quantification as sets of relations invariant under a set of partial functions that satisfy the condition of k-subset surjectivity. Finally, we give a description of Boolean max-co-clones, that is, sets of relations on {0,1} closed under max-implementations.Comment: 28 pages, 2 figure