A Graph - Theoretic Game and its Application

This paper investigates a zero-sum game played on a weighted connected graph G between two players, the tree player and the edge player. At each play, the tree player chooses a spanning tree T and the edge player chooses an edge e. The payoo to the edge player is cost(T ; e), deened as follows: If e lies in the tree T then cost(T ; e) = 0; if e does not lie in the tree then cost(T ; e) = cycle(T ; e)=w(e), where w(e) is the weight of edge e and cycle(T ; e) is the weight of the unique cycle formed when edge e is added to the tree T. Our main result is that the value of the game on an n-vertex graph is bounded above by exp(O(p log n log log n)). The game arises in connection with the k-server problem on a road network ; i.e., a metric space that can be represented as a multigraph G in which each edge e represents a road of length w(e). We show that, if the value of the game on G is V al(G; w), then there is a randomized strategy that achieves a competitive ratio of k(1 + V al(G; w)) against any oblivious adversary. Thus, on any n-vertex road network, there is a random-ized algorithm for the k-server problem that is k exp(O(p log n log log n))-competitive against oblivious adversaries. At the heart of our analysis of the game is an algorithm that, for any n-vertex weighted, connected multigraph, constructs a spanning tree T such that the average, over all edges e, of cost(T ; e) is less than or equal to exp(O(p log n log log n)). This result has potential application to the design of communication networks.