The plurality strategy on graphs

Goldman [Transportation Science 5 (1971), 212–221] proved the classical result on how to find the medians for a set of clients in a tree using majority rule. Here the clients are located at vertices of the tree, and a median is a vertex in the tree that minimizes the sum of the distances to the locations of the clients. The majority rule can be rephrased as the Majority Strategy: if we are at vertex v, then we move to neighbor w of v if a majority of the clients is closer to w than to v. This strategy can be applied in any connected graph. In Mulder [Discrete Applied Math. 80 (1997), 97–105] the question was answered for which connected graphs the Majority Strategy always produces the set of medians for any given set of clients: these are precisely the median graphs. This class of graphs has been well-studied in the literature. In this paper we relax the Majority Strategy: instead of requiring a majority of the clients to be ∗ This work was done under the DST, Govt. of India grant M/04–1999 awarded to this author. The financial support of the DST, New Delhi is gratefully acknowledged. 192 K. BALAKRISHNAN, M. CHANGAT AND H.M. MULDER closer to w than to v, we move to w if there are more vertices closer to w than to v (thus ignoring the clients at equal distance from v and w). The main result of the paper is that the Plurality Strategy always produces the median set for any given set of clients if and only if all median sets are connected. We prove a similar result for the Hill Climbing strategy and for the Steepest Ascent Hill Climbing strategy.