Minimum 0-Extensions of Graph Metrics

Abstract

Let H = (T, U ) be a connected graph, V ⊇ T a set, and c a non-negative function on the unordered pairs of elements of V . In the minimum 0-extension problem ( * ), one is asked to minimize the inner product c · m over all metrics m on V such that (i) m coincides with the distance function of H within T ; and (ii) each v ∈ V is at zero distance from some s ∈ T , i.e. m (v, s) This problem is known to be NP-hard if H = K 3 (as being equivalent to the minimum 3-terminal cut problem), while it is polynomially solvable if H = K 2 (the minimum cut problem) or H = K 2,r (the minimum (2, r )-metric problem). We study problem ( * ) for all fixed H . More precisely, we consider the linear programming relaxation ( * * ) of ( * ) that is obtained by dropping condition (ii) above, and call H minimizable if the minima in ( * ) and ( * * ) coincide for all V and c. Note that for such an H problem ( * ) is solvable in strongly polynomial time.Our main theorem asserts that H is minimizable if and only if H is bipartite, has no isometric circuit with six or more nodes, and is orientable in the sense that H can be oriented so that nonadjacent edges of any 4-circuit are oppositely directed along this circuit. The proof is based on a combinatorial and topological study of tight and extreme extensions of graph metrics.Based on the idea of the proof of the NP-hardness for the minimum 3-terminal cut problem in [4], we then show that the minimum 0-extension problem is strongly NP-hard for many non-minimizable graphs H . Other results are also presented.

Approximability of Robust Network Design

“…Remark 1. Karzanov [22] shows (with a different algorithm) polynomial solvability of a larger class of 0-extension problems that includes trees as a special case.…”

“…In this section, we describe an exact algorithm for the following hierarchical hubbing problem which is very similar to the zero-extension problem (Karzanov [22], Calinescu et al [4]) on a tree. Given our capacitated tree T , find a mapping h V T → V G such that h w = w for each leaf w, and subject to this, we wish to minimize e=wz∈E T b e d G h w h z .…”

On Lipschitz extension from finite subsets

“…The 0-Extension problem is a clustering framework for finite graphs that was introduced by Karzanov [Kar98]. The input of the 0-Extension problem is a graph G with edge weights w : E G → [0, ∞), and a subset T ⊆ V G equipped with a metric d T : T × T → [0, ∞).…”

“…The first is the metric relaxation, which was formulated in [Kar98] and studied extensively in [CKR05]. Given an instance of 0-Extension, i.e., an n-vertex graph G, a metric…”

Half-integrality of node-capacitated multiflows and tree-shaped facility locations on trees

“…Thus (1.2) reduces to a point-location problem on a tree: The location problem of this type has been well studied in the literature [28]. In the multiflow theory (of edge-only-capacitated), the corresponding duality relationship has been discovered by Karzanov [16,17], and further developed by the author [7,8,9], for more general weights beyond tree distances. So our main theorem is a node-capacitated variation of these results.…”