l ∞ -approximation via subdominants

Abstract Given a vector u and a certain subset K of a real vector space E , the problem of l ∞ -approximation involves determining an element u in K nearest to u in the sense of the l ∞ -error norm. The subdominant u ∗ of u is the upper bound (if it exists) of the set { x ∈ K  :  x ≺ u } (we let x ≺ y if all coordinates of x are smaller than or equal to the corresponding coordinates of y ). We present general conditions on K under which a simple relationship between the subdominant of u and a best l ∞ -approximation holds. We specify this result by taking as K the cone of isotonic functions defined on a poset ( X , ≺), the cone of convex functions defined on a subset of R N , the cone of ultrametrics on a set X , and the cone of tree metrics on a set X with fixed distances to a given vertex. This leads to simple optimal algorithms for the problem of best l ∞ -fitting of distances by ultrametrics and by tree metrics preserving the distances to a fixed vertex (the latter provides a 3-approximation algorithm for the problem of fitting a distance by a tree metric). This simplifies the recent results of Farach, Kannan, and Warnow (1995) and of Agarwala et al. (1996).