The cut cone, L1 embeddability, complexity, and multicommodity flows

A finite metric (or more properly semimetric) on n points is a non-negative vector d = (dij) 1 ≤ i < j ≤ n that satisfies the triangle inequality: dij ≤ dik + d jk . The L (or Manhattan )distance | |x − y| |1 between two vectors x = (xi) and y = (yi) in R is given by | |x − y| |1 = 1≤i≤m Σ |xi − yi |. A metric d is L − embeddable if there exist vectors z1, z2, . . . , zn in R for some m, such that dij = | |zi − z j | |1 for 1 ≤ i < j ≤ n. A cut metric is a metric with all distances zero or one and corresponds to the incidence vector of a cut in the complete graph on n vertices. The cut cone Hn is the convex cone formed by taking all non-negative combinations of cut metrics. It is easily shown that a metric is L − embeddable if and only if it is contained in the cut cone. In this expository paper we provide a unified setting for describing a number of results related to L − embeddability and the cut cone. We collect and describe results on the facial structure of the cut cone and the complexity of testing the L − embeddability of a metric. One of the main sections of the paper describes the role of L − embeddability in the feasibility problem for multicommodity flows. The Ford and Fulkerson theorem for the existence of a single commodity flow can be restated as an inequality that must be valid for all cut metrics. A more general result, known as the Japanese theorem, gives a condition for the existence of a multicommodity flow. This theorem gives an inequality that must be satisfied by all metrics. For multicommodity flows involving a small number of terminals, it is known that the condition of the Japanese theorem can be replaced with one of the Ford-Fulkerson type. We review these results and show that the existence of such Ford-Fulkerson type conditions for flows with few terminals depends critically on * Research supported by the Natural Science and Engineering Research Council Grant number A3013 and the F.C.A.R. Grant number EQ1678.