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An algorithm is given which finds all optimal solutions, for a given set of criteria, to political redistricting problems. Using "population units" as indivisible elements, the first phase generates all feasible districts, where feasibility indicates contiguity, compactness and limited population deviation. The second phase finds that set of M feasible districts which "covers" each population unit exactly once, and minimizes the maximum deviation of any district population from the mean district population. Computational results indicate that states with 40 counties or fewer can be solved in less than 10 minutes on an IBM 7094. However, our attempt to solve a 55 county state was unsuccessful.

We deal with the problem of locating a point on a network so as to maximize the sum of its weighted distances to the nodes. It is shown that there is at least one optimal point in a finite set of points which can easily be generated. An algorithm is given for locating an optimal point in this set. When the network is a tree, this set consists of the “dangling” nodes.

A research theme involving location on networks, since its inception, has been the identification of a finite dominating set (FDS), or a finite set of points to which an optimal solution must belong. We attempt to unify and generalize results of this sort. We survey the literature and then prove some theorems that subsume most previous results and that are, at the same time, more general than previous results. The paper is aimed primarily at investigators who wish to know whether an FDS exists for a specific problem.

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