We consider approximation algorithms for the metric labeling problem. This problem was introduced in a paper by Kleinberg and Tardos [J. ACM, 49 (2002), pp. 616--630] and captures many classification problems that arise in computer vision and related fields. They gave an O(log k log log k) approximation for the general case, where k is the number of labels, and a 2-approximation for the uniform metric case. (In fact, the bound for general metrics can be improved to O(log k) by the work of Fakcheroenphol, Rao, and Talwar [Proceedings of the 35th Annual ACM Symposium on Theory of Computing, 2003, pp. 448--455].) Subsequently, Gupta and Tardos [Proceedings of the 32nd Annual ACM Symposium on the Theory of Computing, 2000, pp. 652--658] gave a 4-approximation for the truncated linear metric, a metric motivated by practical applications to image restoration and visual correspondence. In this paper we introduce an integer programming formulation and show that the integrality gap of its linear relaxation either matches or improves the ratios known for several cases of the metric labeling problem studied until now, providing a unified approach to solving them. In particular, we show that the integrality gap of our linear programming (LP) formulation is bounded by O(log k) for a general k-point metric and 2 for the uniform metric, thus matching the known ratios. We also develop an algorithm based on our LP formulation that achieves a ratio of $2+\sqrt{2}\simeq 3.414$ for the truncated linear metric improving the earlier known ratio of 4. Our algorithm uses the fact that the integrality gap of the LP formulation is 1 on a linear metric.

A directed multiway cut separates a set of terminals T={s 1 , . . . , s k } in a directed capacitated graph G=(V,E). Finding a minimum directed multiway cut is an NP-hard problem. We give a polynomial-time algorithm that achieves an approximation factor of 2 for this problem. This improves the result of Garg, Vazirani, and Yannakakis [Proceedings of the 21st International Colloquium on Automata, Languages, and Programming, Jerusalem, Israel, 1994, pp. 487--498], who gave an algorithm that achieves an approximation factor of 2 log k. Our approximation algorithm uses a novel technique for relaxing a multiway flow function in order to find a directed multiway cut. It also implies that the integrality gap of the linear program for the directed multiway cut problem is at most 2.

We present an 8-approximation algorithm for the problem of finding a minimum weight subset feedback vertex set (or subset-fvs, in short). The input in this problem consists of an undirected graph G=(V,E) with vertex weights c(v) and a subset of vertices S called special vertices. A cycle is called interesting if it contains at least one special vertex. A subset of vertices is called a subset-fvs with respect to S if it intersects every interesting cycle. The goal is to find a minimum weight subset-fvs. The best previous algorithm for the general case provided only a logarithmic approximation factor. The minimum weight subset-fvs problem generalizes two NP-complete problems: the minimum weight feedback vertex set problem in undirected graphs and the minimum weight multiway vertex cut problem. The main tool that we use in our algorithm and its analysis is a new version of multicommodity flow, which we call relaxed multicommodity flow. Relaxed multicommodity flow is a hybrid of multicommodity flow and multiterminal flow.

Let G=(V,E) be a weighted undirected graph where all weights are at least one. We consider the following generalization of feedback set problems. Let $S \subset V$ be a subset of the vertices. A cycle is called interesting if it intersects the set S. A subset feedback edge (vertex) set is a subset of the edges (vertices) that intersects all interesting cycles. In minimum subset feedback problems the goal is to find such sets of minimum weight. This problem has a variety of applications, among them genetic linkage analysis and circuit testing. The case in which S consists of a single vertex is equivalent to the multiway cut problem, in which the goal is to separate a given set of terminals. Hence, the subset feedback problem is NP-complete and also generalizes the multiway cut problem. We provide a polynomial time algorithm for approximating the subset feedback edge set problem that achieves an approximation factor of two. This implies a $\Delta$-approximation algorithm for the subset feedback vertex set problem, where $\Delta$ is the maximum degree in G. We also consider the multicut problem and show how to achieve an $O(\log \tau^*)$ approximation factor for this problem, where $\tau^*$ is the value of the optimal fractional solution. To achieve the $O(\log \tau^*)$ factor we employ a bootstrapping technique.

A directed multiway cut separates a set of terminals s/sub 1/,...,s/sub /spl kappa// in a directed capacitated graph G=(V, E). Finding a minimum capacity directed multiway cut is an NP-complete problem. We give a polynomial-time algorithm that achieves an approximation factor of 2 for this problem. This improves the result of Garg, Vazirani and Yannakakis (1994) who gave an algorithm that achieves an approximation factor of 2 log /spl kappa/. Our approximation algorithm uses a novel technique for relaxing a multiway flow function in order to find a directed multiway cut. It also implies that the integrality gap of the linear program for the directed multiway cut problem is at most 2.

We present an 8-approximation algorithm for the problem of finding a minimum weight subset feedback vertex set. The input in this problem consists of an undirected graph G=(V,E) with vertex weights w(v) and a subset of vertices S called special vertices. A cycle is called interesting if it contains at least one special vertex. A subset of vertices is called a subset feedback vertex set with respect to S if it intersects every interesting cycle The goal is to find a minimum weight subset feedback vertex set. The best pervious algorithm for the general case provided only a logarithmic approximation factor. The minimum weight subset feedback vertex set problem generalizes two NP-Complete problems: the minimum weight feedback vertex set problem in undirected graphs and the minimum weight multiway vertex cut problem. The main tool that we use in our algorithm and its analysis is a new version of multi-commodity flow which we call relaxed multi-commodity flow. Relaxed multi-commodity flow is a hybrid of multi-commodity flow and multi-terminal flow.