Approximation Algorithms for Steiner and Directed Multicuts
نویسندگان
چکیده
In this paper we consider the steiner multicut problem. This is a generalization of the minimum multicut problem where instead of separating node pairs, the goal is to find a minimum weight set of edges that separates all given sets of nodes. A set is considered separated if it is not contained in a single connected component. We show an O(log3(kt)) approximation algorithm for the steiner multicut problem, where k is the number of sets and t is the maximum cardinality of a set. This improves the O(t logk) bound that easily follows from the previously known multicut results. We also consider an extension of multicuts to directed case, namely the problem of finding a minimum-weight set of edges whose removal ensures that none of the strongly connected components includes one of the prespecified k node pairs. In this paper we describe an O(log2 k) approximation algorithm for this directed multicut problem. If k n, this represents and an improvement over the O(logn log logn) approximation algorithm that is implied by the technique of Seymour. Research supported by NSF PYI award CCR-9157620, together with PYI matching funds from Honeywell Corporation, Thinking Machines Corporation, and Xerox Corporation. Additional support provided by ARPA contract N00014-91-J-4052 ARPA Order No. 8225. yResearch supported by U.S. Army Research Office Grant DAAL-03-91-G-0102 and by a grant from Mitsubishi Electric Laboratories. zResearch supported in part by a Packard Fellowship, an NSF PYI award, by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550, and by NEC.
منابع مشابه
Minimum multicuts and Steiner forests for Okamura-Seymour graphs
We study the problem of finding minimum multicuts for an undirected planar graph, where all the terminal vertices are on the boundary of the outer face. This is known as an Okamura-Seymour instance. We show that for such an instance, the minimum multicut problem can be reduced to the minimum-cost Steiner forest problem on a suitably defined dual graph. The minimum-cost Steiner forest problem ha...
متن کاملGreedy approximation algorithms for directed multicuts
The Directed Multicut (DM) problem is: given a simple directed graph G = (V ,E) with positive capacities ue on the edges, and a set K ⊆ V × V of ordered pairs of nodes of G, find a minimum capacity K-multicut; C ⊆ E is a K-multicut if in G − C there is no (s, t)-path for any (s, t) ∈ K. In the uncapacitated case (UDM) the goal is to find a minimum size K-multicut. The best approximation ratio k...
متن کاملApproximating Directed Multicuts
The seminal paper of Leighton and Rao (1988) and subsequent papers presented approximate minmax theorems relating multicommodity flow values and cut capacities in undirected networks, developed the divide-and-conquer method for designing approximation algorithms, and generated novel tools for utilizing linear programming relaxations. Yet, despite persistent research efforts, these achievements ...
متن کاملApproximating directed multicuts
The Directed Multicut (DM) problem is: given a simple directed graph G = (V,E) with positive capacities ue on the edges, and a set K ⊆ V × V of ordered pairs of nodes of G, find a minimum capacity K-multicut; C ⊆ E is a K-multicut if in G− C there is no (s, t)-path for every (s, t) ∈ K. In the uncapacitated case (UDM) the goal is to find a minimum size K-multicut. The best approximation ratio k...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- J. Algorithms
دوره 22 شماره
صفحات -
تاریخ انتشار 1997