This paper deals with weighted set systems (V, E , q), where V is a set of indices, E ⊂ 2 and the weight q is a nonnegative integer function on E . The basic idea of the paper is to apply weighted set systems to formulate restrictions on intersections. It is of interest to know whether a weighted set system can be represented by set intersections. An intersection representation of (V, E , q) is...

In real world environments the state is almost never completely known. Exploration is often expensive. The application of planning in these environments is consequently more difficult and less robust. In this paper we present an approach for predicting new information about a partially-known state. The state is translated into a partially-known multigraph, which can then be extended using machi...

In a recent paper by one of the authors it has been shown that there is a relationship between algebraic structures and labeled transition systems. Indeed, it has been shown that an algebraic structures can be viewed as labeled transition systems, which can also be viewed as multigraphs. In this paper, we extend this work by providing an estimation of the transition possibilities between vertic...

This article investigates a remarkable generalization of the generating function that enumerates partitions by area and number of parts. This generating function is given by the infinite product ∏ i≥1 1/(1 − tq ). We give uncountably many new combinatorial interpretations of this infinite product involving partition statistics that arose originally in the context of Hilbert schemes. We construc...

It is known that every multigraph with an even number of edges has an even orientation (i.e., all indegrees are even). We study parity constrained graph orientations under additional constraints. We consider two types of constraints for a multigraph G = (V,E): (1) an exact conflict constraint is an edge set C ⊆ E and a vertex v ∈ V such that C should not equal the set of incoming edges at v; (2...

This paper starts with a discussion of several old and new conjectures about choosability in graphs. In particular, the list-colouring conjecture, that ch′ = ′ for every multigraph, is shown to imply that if a line graph is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t. It is proved that ch(H )= (H ) for many “small” graphs H , including in9ations of all circuit...

Given a nite set V and a set function f : 2 V 7! Z, we consider the problem of constructing an undirected multigraph G = (V;E) such that the cut function c G : 2 V 7! Z of G and f together has value at least 2 for all non-empty and proper subsets of V . If f is intersecting submodular and posi-modular, and satis es the tripartite inequality, then we show that such a multigraph G with the minimu...

By a result of Holyer, unless P = NP, there does not exist a polynomial-time approximation algorithm to edge color a multigraph that always uses fewer than (f) x’ colors, where x’ is the optimal number of colors. This makes it appear that finding provably good edge colorings is extremely difficult. However, in this paper we present an algorithm to find an edge coloring of a multigraph that neve...

In single channel wireless networks, concurrent transmission at different links may interfere with each other. To improve system throughput, a scheduling algorithm is necessary to choose a subset of links at each time slot for data transmission. Throughput optimal link scheduling discipline in such a wireless network is generally an NPhard problem. In this paper, we develop a polynomial time al...

In non-partitional models of knowledge with objective and subjective state spaces, the issue of self-consistency arises. The present paper defines a multigraph Gj for each player j, and also a global multigraph G. The posteriors of player j are self-consistent if and only if all cycle equations associated with cycles in Gj are satisfied. Similarly, the posteriors of all players are consistent w...