22,312 research outputs found
Koszulness, Krull Dimension and Other Properties of Graph-Related Algebras
The algebra of basic covers of a graph G, denoted by \A(G), was introduced by
Juergen Herzog as a suitable quotient of the vertex cover algebra. In this
paper we show that if the graph is bipartite then \A(G) is a homogeneous
algebra with straightening laws and thus is Koszul. Furthermore, we compute the
Krull dimension of \A(G) in terms of the combinatorics of G. As a consequence
we get new upper bounds on the arithmetical rank of monomial ideals of pure
codimension 2. Finally, we characterize the Cohen-Macaulay property and the
Castelnuovo-Mumford regularity of the edge ideal of a certain class of graphs.Comment: 23 page
Cooperation through social influence
We consider a simple and altruistic multiagent system in which the agents are eager to perform a collective task but where their real engagement depends on the willingness to perform the task of other influential agents. We model this scenario by an influence game, a cooperative simple game in which a team (or coalition) of players succeeds if it is able to convince enough agents to participate in the task (to vote in favor of a decision). We take the linear threshold model as the influence model. We show first the expressiveness of influence games showing that they capture the class of simple games. Then we characterize the computational complexity of various problems on influence games, including measures (length and width), values (Shapley-Shubik and Banzhaf) and properties (of teams and players). Finally, we analyze those problems for some particular extremal cases, with respect to the propagation of influence, showing tighter complexity characterizations.Peer ReviewedPostprint (author’s final draft
Incremental complexity of a bi-objective hypergraph transversal problem
The hypergraph transversal problem has been intensively studied, from both a
theoretical and a practical point of view. In particular , its incremental
complexity is known to be quasi-polynomial in general and polynomial for
bounded hypergraphs. Recent applications in computational biology however
require to solve a generalization of this problem, that we call bi-objective
transversal problem. The instance is in this case composed of a pair of
hypergraphs (A, B), and the aim is to find minimal sets which hit all the
hyperedges of A while intersecting a minimal set of hyperedges of B. In this
paper, we formalize this problem, link it to a problem on monotone boolean
-- formulae of depth 3 and study its incremental complexity
- …