This paper considers a game between two players who choose from a collection of objects. The players make their choices alternately and Vi,j represents the value or amount that the ith player will gain if he selects the jth object. In relation to these values, the players may have various strategies or approaches to the game, and each of them constitutes a distinct theoretical problem. The paper formulates and solves three of these problems, each one having practical significance (for example, for the draft of professional football players).
In this paper, we consider two problems related to single-commodity flows on a directed network. In the first problem, for a given s −t flow, if an arc is destroyed, all the flow that is passing through that arc is destroyed. What is left flowing from s to t is the residual flow. The objective is to determine a flow pattern such that the residual flow is maximized. We provide a strongly polynomial algorithm for this problem, called the maximum residual flow problem, and consider various extensions of this basic model. In the second problem, known as the "most vital arc" problem, the objective is to remove an arc so that the maximal flow on the residual network is as small as possible. Results are also derived which help implement an efficient scheme for solving this problem.
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