The metric bridge partition problem : partitioning of a metric space into two subspaces linked by an edge in any optimal realization

Hertz, Alain ; Varone, Sacha

In: Journal of classification, 2007, vol. 24, no. 2, p. 235-249

Let G = (V,E,w) be a graph with vertex and edge sets V and E, respectively, and w:E → R + a function which assigns a positive weight or length to each edge of G. G is called a realization of a finite metric space (M,d), with M = { 1,...,n} if and only if { 1,...,n} ⫅ V and d(i,j) is equal to the length of the shortest chain linking i and j in G ∀ i,j = 1,...,n. A realization G of (M,d), is... Plus

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    Summary
    Let G = (V,E,w) be a graph with vertex and edge sets V and E, respectively, and w:E → R + a function which assigns a positive weight or length to each edge of G. G is called a realization of a finite metric space (M,d), with M = { 1,...,n} if and only if { 1,...,n} ⫅ V and d(i,j) is equal to the length of the shortest chain linking i and j in G ∀ i,j = 1,...,n. A realization G of (M,d), is said optimal if the sum of its weights is minimal among all the realizations of (M,d). Consider a partition of M into two nonempty subsets K and L, and let e be an edge in a realization G of (M,d); we say that e is a bridge linking K with L if e belongs to all chains in G linking a vertex of K with a vertex of L. The Metric Bridge Partition Problem is to determine if the elements of a finite metric space (M,d) can be partitioned into two nonempty subsets K and L such that all optimal realizations of (M,d) contain a bridge linking K with L. We prove in this paper that this problem is polynomially solvable. We also describe an algorithm that constructs an optimal realization of (M,d) from optimal realizations of (K,d|K) and (L,d|L).