We consider a problem of allocating spatial units (SUs) to particular uses to form ''regions'' according to specified criteria, which is here called ''spatial unit allocation.'' Contiguity-the quality of a single region being connected-is one of the most frequently required criteria for this problem. This is also one that is difficult to model in algebraic terms for algorithmic solution. The purpose of this article is to propose a new exact formulation of contiguity that can be incorporated into any mixed integer programming model for SU allocation. The resulting model guarantees to enforce contiguity regardless of other included criteria such as compactness. Computational results suggest that problems involving a single region and fewer than about 200 SUs are optimally solved in fairly reasonable time, but that larger problems must rely on heuristics for approximate solutions. It is also found that a problem of any size can be formulated in a more tractable form when a fixed number of SUs are to be selected or when a certain SU is selected in advance.

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A classic problem in planning is districting, which aims to partition a given area into a specified number of subareas according to required criteria. Size, compactness, and contiguity are among the most frequently used districting criteria. While size and compactness may be interpreted differently in different contexts, contiguity is an unambiguous topological property. A district is said to be contiguous if all locations in it are ‘connected’—that is, one can travel between any two locations in the district without leaving it. This paper introduces a new integer-programming-based approach to districting modeling, which enforced contiguity constraints independently of any other criteria that might be additionally imposed. Three experimental models are presented, and tested with sample data on the forty-eight conterminous US states. A major implication of this paper is that the exact formulation of a contiguity requirement allows planners to address diverse sets of districting criteria.

Given a set of points in the plane and a constant t ≥ 1, a Euclidean t-spanner is a network in which, for any pair of points, the ratio of the network distance and the Euclidean distance of the two points is at most t. Such networks have applications in transportation or communication network design and have been studied extensively. In this paper we study 1-spanners under the Manhattan (or L 1-) metric. Such networks are called Manhattan networks. A Manhattan network for a set of points is a set of axis-parallel line segments whose union contains an x-and y-monotone path for each pair of points. It is not known whether it is NP-hard to compute minimum Manhattan networks (MMN), i.e. Manhattan networks of minimum total length. In this paper we present an approximation algorithm for this problem. Given a set P of n points, our algorithm computes in O(n log n) time and linear space a Manhattan network for P whose length is at most 3 times the length of an MMN of P. We also establish a mixed-integer programming formulation for the MMN problem. With its help we extensively investigate the performance of our factor-3 approximation algorithm on random point sets.

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