We study a cluster-identification type problem on networks motivated by applications in geographical analysis. Given a network N (a connected graph with positive edge lengths) together with a set of sites, which lie on the edges or vertices of N, we look for a connected subnetwork F of N of small total length that contains many sites. The edges of F can form parts of the edges of N. We consider different variants of this problem where N is either a general graph or restricted to a tree, and the subnetwork F that we are looking for is either a simple path, a path with self-intersections at vertices, or a tree. We give polynomial-time algorithms, NP-hardness and NP-completeness proofs, approximation algorithms, and also fixed-parameter tractable algorithms.
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