1-Approximation algorithm for bottleneck disjoint path matching (original) (raw)
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Algorithmica, 1992
The notion of matching in graphs is generalized in this paper to a set of paths rather than to a set of edges. The generalized problem, which we call the path-matching problem, is to pair the vertices of an undirected weighted graph such that the paths connecting each pair are subject to certain objectives and/or constraints. This paper concentrates on the case where the paths are required to be edge-disjoint and the objective is to minimize the maximal cost of a path in the matching (i.e., the bottleneck version). Other variations of the problem are also mentioned. Two algorithms are presented to find the best matching under the constraints listed above for trees. Their worst-case running times are O(n log d log w), where d is the maximal degree of a vertex, w is the maximal cost of an edge, and n is the size of the tree, and O(n2), respectively. The problem is shown to be NP-complete for general graphs. Applications of these problems are also discussed.
Length‐constrained path‐matchings in graphs
2002
Abstract The path-matching problem is to find a set of vertex-or edge-disjoint paths with length constraints in a given graph with a given set of endpoints. This problem has several applications in broadcasting and multicasting in computer networks. In this paper, we study the algorithmic complexity of different cases of this problem. In each case, we either provide a polynomial-time algorithm or prove that the problem is NP-complete.© 2002 Wiley Periodicals, Inc.
Parameterized and approximation algorithms for finding two disjoint matchings
Theoretical Computer Science, 2014
We first present a randomized fixed-parameter algorithm for the NP-hard problem of deciding if there are two matchings M 1 and M 2 in a given graph G such that |M 1 | + |M 2 | is a given number k. The algorithm runs in O (2 k k(m + n)) expected time and can be derandomized to run in O (2 2k+12 log 2 (2k) kn(m + n)) time, where n (respectively, m) is the number of vertices (respectively, edges) in G. We then extend the algorithm to the weighted version of the problem. We further present a combinatorial approximation algorithm for the NP-hard problem of finding two disjoint matchings in a given edgeweighted graph G so that their total weight is maximized. The algorithm achieves an approximation ratio close to 0.76 and runs in O (m + n 3 α(n)) time, where α is the inverse Ackermann function.
Operations Research Letters, 2007
Using flow and matching algorithms to solve the problem of finding disjoint paths through a given node, and with a technique of Chekuri and Khanna, we give an O( √ n) approximation for the edge-disjoint paths problem in undirected graphs, directed acyclic graphs and directed graphs with edge capacity at least 2.
Finding disjoint paths with different path‐costs: Complexity and algorithms
Networks, 1992
Consider a network G = (V,E) with distinguished vertices s and t, and with k different costs on every edge. We consider the problem of finding k disjoint paths from s to t such that the total cost of the paths is minimized, where the jth edge‐cost is associated with the jth path. The problem has several variants: The paths may be vertex‐disjoint or arc‐disjoint and the network may be directed or undirected. We show that all four versions of the problem are strongly NP‐complete even for k = 2. We describe polynomial time heuristics for the problem and a polynomial time algorithm for the acyclic directed case.
A Heuristic for Dijkstra's Algorithm with Many Targets and Its Use in Weighted Matching Algorithms
Algorithmica, 2003
We consider the single-source many-targets shortest-path (SSMTSP) problem in directed graphs with non-negative edge weights. A source node s and a target set T is specified and the goal is to compute a shortest path from s to a node in T . Our interest in the shortest path problem with many targets stems from its use in weighted bipartite matching algorithms. A weighted bipartite matching in a graph with n nodes on each side reduces to n SSMTSP problems, where the number of targets varies between n and 1.
A quick method for finding shortest pairs of disjoint paths
Networks, 1984
Let G be a directed graph containing n vertices, one of which is a distinguished source s, and m edges, each with a non-negative cost. We consider the problem of finding, for each possible sink vertex u, a pair of edge-disjoint paths from s to u of minimum total edge cost. Suurballe has given an O(n2 1ogn)-time algorithm for this problem. We give an implementation of Suurballe's algorithm that runs in O(m log(, +,+)n) time and O(m) space. Our algorithm builds an implicit representation of the n pairs of paths; given this representation, the time necessary to explicitly construct the pair of paths for any given sink is O(1) per edge on the paths.
Efficient Algorithms for Variants of Weighted Matching and Assignment Problems
Mathematics in Computer Science, 2008
Obtaining a matching in a graph satisfying a certain objective is an important class of graph problems. Matching algorithms have received attention for several decades. However, while there are efficient algorithms to obtain a maximum weight matching, not much is known about the maximum weight maximum cardinality, and maximum cardinality maximum weight matching problems for general graphs. Our contribution in this work is to show that for bounded weight input graphs one can obtain an algorithm for both maximum weight maximum cardinality (for real weights), and maximum cardinality maximum weight matching (for integer weights) by modifying the input and running the existing maximum weight matching algorithm. Also, given the current state of the art in maximum weight matching algorithms, we show that, for bounded weight input graphs, both maximum weight maximum cardinality, and maximum cardinality maximum weight matching have algorithms of similar complexities to that of maximum weight matching. Subsequently, we also obtain approximation algorithms for maximum weight maximum cardinality, and maximum cardinality maximum weight matching.