No abstract

Abstract. In this paper, we define a one-vertex-extension tree for a distance-hereditary graph and show how to build it. We then give a unified approach to designing efficient dynamic programming algorithms for distance-hereditary graphs based upon the one-vertex-extension tree. We give linear time algorithms for the weighted vertex cover and weighted independent domination problems and give an O(n 2) time algorithm to compute a minimum fill-in and the treewidth for a distance-hereditary graph. IntroductionThe distance da (u, v) between two vertices u and v of a connected graph G is the minimum length of a u-v path in G. A graph is distance-hereditary if each pair of vertices are equidistant in every connected induced subgraph containing them. Distance-hereditary graphs form a subclass of perfect graphs [7,11,12] that are graphs G in which the maximum clique size equals the chromatic number for every induced subgraph of G. Properties and optimization problems in distance-hereditary graphs have been extensively studied during the past two decades [2,3, 4,7,8,9,11,12,17,19,20] which result in sequential algorithms that solve quite a few graph-theoretical problems on this special class of graphs. However, some problems remain unresolved on distance-hereditary graphs. For example, the independent domination problem and the vertex cover problem, the minimum fill-in and treewidth problem, and the edge domination problem etc. Most of the known polynomial time algorithms for distance-hereditary graphs utilize ad hoc techniques. In this paper, we first define a new data structure, called one-vertex-exlension tree, to represent a distance-hereditary graph in the form of a rooted tree. We give a new recursive definition for this class of graphs. We then show how to design dynamic programming algorithms based on this recursive definition. We present linear time algorithms to solve the weighted vertex cover problem and weighted independent domination problem and present a O(n 2) time algorithm to compute the minimum fill-in and treewidth for a distance-hereditary graph. The above three problems are NP-complete for general graphs [1,5,14], and the vertex cover problem is known to be MAX SNP-

No abstract

In the context of investment analysis, we formulate an abstract online computing problem called a planning game and develop general tools for solving such a game. We then use the tools to investigate a practical buy-and-hold trading problem faced by long-term investors in stocks. We obtain the unique optimal static online algorithm for the problem and determine its exact competitive ratio. We also compare this algorithm with the popular dollar averaging strategy using actual market data.

The star graph Sn has been recognized as an attractive alternative to the hypercube. Since S1; S2, and S3 have trivial structures, we focus our attention on Sn with n¿4 in this paper. Let Fv denote the set of faulty vertices in Sn. We show that when |Fv|6n − 5; Sn with n¿6 contains a fault-free path of length n! − 2|Fv| − 2 (n! − 2|Fv| − 1) between arbitrary two vertices of even (odd) distance. Since Sn is bipartite with two partite sets of equal size, the path is longest for the worst-case scenario. The situation of n¿4 and |Fv| ¿ n − 5 is also discussed.