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) = 0. This problem is known to be NP-hard if H = K3, (as being equivalent to the minimum 3-terminal cut problem), while it is polynomially solvable if H = K2 (the minimum cut problem) or H = K2,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. © 1998 Academic Press Limited.