In this chapter, we think about problems involving metrics. A metric (V, d) on a set of vertices V gives a distance duv for each pair of vertices u, v ∈ V such that three properties are obeyed: (1) duv = 0 if and only if v = u; (2) duv = duv for all u, v ∈ V; and (3) duv ≤ duw + dwv for all u, v, w ∈ V. The final property is sometimes called the triangle inequality. We will sometimes simply refer to the metric d instead of (V, d) if the set of vertices V is clear from the context. A concept related to a metric is a semimetric (V, d), in which properties (2) and (3) are obeyed, but not necessarily (1), so that if duv = 0, then possibly u ≠ v (a semimetric maintains that duu = 0). We may sometimes ignore this distinction between metrics and semimetrics, and call them both metrics.

Metrics turn out to be a useful way of thinking about graph problems involving cuts. Many important problems in discrete optimization require finding cuts in graphs of various types. To see the connection between cuts and metrics, note that for any cut S ⊆ V, we can define d where duv = 1 if u ∈ S and v ∉ S, and duv = 0 otherwise. Note that (V, d) is then a semimetric; it is sometimes called the cut semimetric associated with S.