l∞-Approximation via Subdominants

• Published in: Journal of mathematical psychology Vol. 44; no. 4; pp. 600 - 616
• Main Authors: Chepoi, Victor, Fichet, Bernard
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 01-12-2000
• ISSN: 0022-2496; 1096-0880
• DOI: 10.1006/jmps.1999.1270

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 RN, 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).