In this paper we study two problems that can be viewed as on-line games on a dynamic bipartite graph. The first problem is on-line load balancing with preemption. A centralized scheduler must assign tasks to servers, processing on-line a sequence of task arrivals and departures. Each task is restricted to run on some subset of the servers. The scheduler attempts to keep the load well-balanced. If preemptive reassignments are disallowed, Azar et al. proved a lower bound of Ω(√(n)) on the ratio between the maximum load achieved by an on-line algorithm and the optimum off-line maximum load. We show that this ratio can be greatly reduced by an efficient scheduler using only a small amount of rescheduling. We then apply these ideas to network flow. Cheriyan and Hagerup introduced an on-line game on a bipartite graph as a fundamental step in improving algorithms for computing the maximum flow in networks. They described a randomized strategy to play the game. King et al. studied a modified version of this game, called ‘node kill’, and gave a deterministic strategy. We obtain an improved deterministic algorithm for the node kill game and hence for maximum flow) in all but the sparsest graphs. The running time is significantly less than King et al.'s. These problems combine a demand for good competitive ratios with more traditional requirements of implementation efficiency. Our solutions deal with the tradeoffs between these measures.
