11,935 research outputs found
Coalition structure generation over graphs
We give the analysis of the computational complexity of coalition structure generation over graphs. Given an undirected graph G = (N,E) and a valuation function v : P(N) → R over the subsets of nodes, the problem is to find a partition of N into connected subsets, that maximises the sum of the components values. This problem is generally NP-complete; in particular, it is hard for a defined class of valuation functions which are independent of disconnected members — that is, two nodes have no effect on each others marginal contribution to their vertex separator. Nonetheless, for all such functions we provide bounds on the complexity of coalition structure generation over general and minor free graphs. Our proof is constructive and yields algorithms for solving corresponding instances of the problem. Furthermore, we derive linear time bounds for graphs of bounded treewidth. However, as we show, the problem remains NP-complete for planar graphs, and hence, for any Kk minor free graphs where k ≥ 5. Moreover, a 3-SAT problem with m clauses can be represented by a coalition structure generation problem over a planar graph with O(m2) nodes. Importantly, our hardness result holds for a particular subclass of valuation functions, termed edge sum, where the value of each subset of nodes is simply determined by the sum of given weights of the edges in the induced subgraph
Amenable hyperbolic groups
We give a complete characterization of the locally compact groups that are
non-elementary Gromov-hyperbolic and amenable. They coincide with the class of
mapping tori of discrete or continuous one-parameter groups of compacting
automorphisms. We moreover give a description of all Gromov-hyperbolic locally
compact groups with a cocompact amenable subgroup: modulo a compact normal
subgroup, these turn out to be either rank one simple Lie groups, or
automorphism groups of semi-regular trees acting doubly transitively on the set
of ends. As an application, we show that the class of hyperbolic locally
compact groups with a cusp-uniform non-uniform lattice, is very restricted.Comment: 41 pages, no figure. v2: revised version (minor changes
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