In this paper we give a simple algorithm for constructing sparse spanners for arbitrary weighted graphs. We then apply this algorithm to obtain specific results for planar graphs and Euclidean graphs. We discuss the optimality of our results and present several nearly matching lower bounds.

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we give a very simple algorithm for constructing sparse spanners for arbitrary weighted graphs. We then apply this algorithm to obtain specific results for planar graphs and Euclidean graphs. We discuss the optimality of our results and present several nearly matching lower bounds. I N T R O D U C T I O NLet G = (V, E) be a connected n-vertex graph with arbitrary positive edge weights.A subgraph G I --(V, E ~) is a t-spanner if, between any pair of vertices the distance in G ~ is at most t times longer than the distance in G. The vMue of t is the stretch factor a~ssociated with Gq We consider the problem of determining t-spanners for graphs where the spanners are sparse and t is a constant independent of the size of the graph.Sparsity will be measured according to two criteria. Let Weight(G) denote the sum of all edge weights of graph G, and Size(G) denote the number of edges. A graph is sparse in size if it has few edges. Similarly, a graph is sparse in weight if its total edge weight is small. Our results separate graphs into classes where spanners with linearly many edges achieve constant stretch factors, and classes where a non-linear number of edges are necessary.Problems of this type appear in numerous applications. Spanners appear to be the underlying graph structure in various constructions in distributed systems and communication networks [Aw~ PU1, PU]. They also appear in biology in the process of reconstructing phylogenetic trees from matrices, whose entries represent genetic distances among contemporary living species [BD]. Robotics researchers have studied spanners under the constraints of Euclidean geometry, where vertices of the graph are points in space, and edges are line segments joining pairs of points [C, DFS, D J, K, KG, LL].

Abstract. Simulated Annealing (SA) has become a very popular tool in combinatorial optimization since its introduction in 1982. Recently Dueck and Scheuer proposed another simple modification of local search which they called "Threshold Accepting" (TA). In this paper some convergence results for TA are presented. The proofs are not constructive and make use of the fact that in a certain sense "SA belongs to the convex hull of TA".

Graph realizations of finite metric spaces have widespread applications, for example, in biology, economics, and information theory. The main results of this paper are: 1. Finding optimal realizations of integral metrics (which means all distances are integral) is NP-complete. 2. There exist metric spaces with a continuum of optimal realizations. Furthermore, two conditions necessary for a weighted graph to be an optimal realization are given and an extremal problem arising in connection with the realization problem is investigated.