Cycle space

Abstract: 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 payoff to the edge player is $cost(T,e)$, defined 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 $T$. The main result is that the value of the game on any $n$-vertex graph is bounded above by $\exp(O(\sqrt{\log n \log\log n}))$. It is conjectured that the value of the game is $O(\log n)$. 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 $w(e)$. It is shown that, if the value of the game on $G$ is $Val(G,w)$, then there is a randomized strategy that achieves a competitive ratio of $k(1 + Val(G,w))$ against any oblivious adversary. Thus, on any $n$-vertex road network, there is a randomized algorithm for the $k$-server problem that is $k\cdot\exp(O(\sqrt{\log n \log\log n}))$ competitive against oblivious adversaries. At the heart of the analysis of the game is an algorithm that provides an approximate solution for the simple network design problem. Specifically, for any $n$-vertex weighted, connected multigraph, the algorithm constructs a spanning tree $T$ such that the average, over all edges $e$, of $cost(T,e)$ is less than or equal to $\exp(O(\sqrt{\log n \log\log n}))$. This result has potential application to the design of communication networks. It also improves substantially known estimates concerning the existence of a sparse basis for the cycle space of a graph.