An improved approximation algorithm for the 0-extension problem Jittat Fakcharoenphol * Chris Harrelson † Satish Rao ‡ Kunal Talwar § Abstract Given a graph G =(V,E), a set of terminals T ⊆ V , and a metric D on T , the 0-extension problem is to assign vertices in V to terminals, so that the sum, over all edges e, of the distance (under D) between the terminals to which the end points of e are assigned, is minimized. This problem was first studied by Karzanov. Calinescu, Karloff and Rabani gave an O(log k) approximation algorithm based on a linear programming relaxation for the problem, where k is the number of terminals. We improve on this bound, and give an O(log k/ log log k) approximation algorithm for the problem. 1 Introduction In the 0-extension problem, we are given an undirected graph G =(V,E) with costs c(u, v) on edges, a set of terminals T ⊆ V among the vertices of G, and a metric D on the terminals. The goal is to find an assignment f of every vertex to a terminal so that ∑ (u,v)∈E c(u, v)D(f (u),f (v)) is minimized. 1 We will call such an f a 0-extension of D to V , or simply a 0-extension for short. In order to understand this optimization function, consider the special case where D(t, t ′ ) = 1 for all terminals t = t ′ , and D(t, t) = 0 for all terminals t. In this case the problem amounts to minimizing the sum of the costs of all edges whose endpoints end up at different terminals. This is just the multiway cut problem studied by Dahlhaus, Johnson, Papadimitriou, Seymour and Yannakakis [DJP + 92, DJP + 94]. Hence the 0-extension problem is a natural generalization of the multiway cut problem, where the cost of cutting an * CS Division, UC Berkeley. Supported in part by the DPST scholarship and by NSF grant CCR-0105533. Email: jittat@cs.berkeley.edu † CS Division, UC Berkeley. Supported in part by a GAANN fellowship and by NSF grant CCR-0105533. Email: chrishtr@cs.berkeley.edu ‡ CS Division, UC Berkeley. Supported in part by NSF grant CCR-0105533. Email: satishr@cs.berkeley.edu § CS Division, UC Berkeley. Supported in part by NSF grants CCR-0105533 and CCR-9820897. Email: kunal@cs.berkeley.edu 1 Thus, effectively, we want to extend the metric D on T to a metric δ on all of V , such that for every v ∈ V , there is some terminal t ∈ T at distance 0. Hence the name 0-extension. edge now depends on the distance between the terminals that its endpoints are assigned to. The 0-extension problem was originally posed by Karzanov [Kar98], who introduced the linear program (1.1) we will study in this paper, studied various special cases with a constant number of terminals, and charac- terized graphs for which the linear program optima are integral. Calinescu, Karloff and Rabani [CKR01] gave an O(log k) approximation algorithm for general graphs based on randomized rounding. For planar graphs, they gave a constant factor approximation, but showed an Ω( √ log k) lower bound on the integrality gap of the lin- ear program for general graphs. For the special case of the multiway cut prob- lem, this linear program yields a (2 − 2 k ) approxi- mation (see [CKR98]). In fact, a combinatorial al- gorithm with the same approximation guarantee was also given by Dahlhaus, Johnson, Papadimitriou, Sey- mour and Yannakakis [DJP + 94]. For the multiway cut problem, a different relaxation gives significantly better approximation guarantees; see Calinescu, Karloff and Rabani [CKR98], Cunningham and Tang [CT99] and Karger, Klein, Stein, Thorup and Young [KKS + 99]. However, there is no obvious way of generalizing the linear program considered in the above papers to 0- extension. The 0-extension problem is also a special case of the metric labeling problem studied by Kleinberg and Tardos [KT99]. In essence the metric labeling problem is the 0-extension problem where, in addition to having costs for edges to be cut, there are assignment costs C(u, t) to assign vertex u to terminal t. Kleinberg and Tardos gave an O(log k log log k) approximation algorithm for the metric labelling problem. They further show that the uniform case (where D(t, t ′ )= 1 for all terminals t = t ′ ) and the case where D can be represented by a hierarchically well-separated tree metric (in the sense of Bartal [Bar98]) have O(1) approximation algorithms. Since metric labeling is a generalization of the 0-extension problem, these results apply to 0-extension as well. In this paper, we improve on the O(log k) bound of [CKR01]. We give a polynomial time algorithm that takes a fractional solution to the linear program- ming relaxation, and rounds it to a 0-extension, such