Edge-disjoint Paths Problem Research Articles

Conversion of coloring algorithms into maximum weight independent set algorithms

A general technique for converting approximation algorithms for the vertex coloring problem in a class of graphs into approximation algorithms for the maximum weight independent set problem (MWIS) in the same class of graphs is presented. The technique consists of solving an LP-relaxation of the MWIS problem with certain clique inequalities, constructing an instance of the vertex coloring problem from the LP solution, applying the coloring algorithm to this instance, and selecting the best resulting color class as the MWIS solution. The approximation ratio obtained is the product of the approximation ratio with which the LP formulation can be solved (usually equal to one) and the approximation ratio of the coloring algorithm with respect to the size of the largest relevant clique. Applying this technique, the best known approximation algorithms are obtained for the maximum weight edge-disjoint paths problem in bidirected trees and in bidirected two-dimensional meshes with row–column routing, and for time-constrained scheduling of weighted packets in the same topologies. These problems are also proved to be MAX SNP-hard.

Polynomial solvability of Mader's edge-disjoint paths problem

Polynomial solvability of Mader's edge-disjoint paths problem

Eulerian disjoint paths problem in grid graphs is NP-complete

We show that the edge disjoint paths problem is NP-complete in directed or undirected rectangular grids, even if the union G+ H of the supply and the demand graphs is Eulerian.

A note on the greedy algorithm for the unsplittable flow problem

In a recent paper Chekuri and Khanna improved the analysis of the greedy algorithm for the edge disjoint paths problem and proved the same bounds also for the related uniform capacity unsplittable flow problem. Here we show that their ideas can be used to get the same approximation ratio even for the more general Unsplittable Flow Problem with nonuniform edge capacities.

Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems

We study the approximability of edge-disjoint paths and related problems. In the edge-disjoint paths (EDP) problem, we are given a network G with source–sink pairs (s i,t i), 1⩽i⩽k , and the goal is to find a largest subset of source–sink pairs that can be simultaneously connected in an edge-disjoint manner. We show that in directed networks, for any ε>0, EDP is NP-hard to approximate within m 1/2− ε . We also design simple approximation algorithms that achieve essentially matching approximation guarantees for some generalizations of EDP. Another related class of routing problems that we study concerns EDP with the additional constraint that the routing paths be of bounded length. We show that, for any ε>0, bounded length EDP is hard to approximate within m 1/2− ε even in undirected networks, and give an O( m ) -approximation algorithm for it. For directed networks, we show that even the single source–sink pair case (i.e. find the maximum number of paths of bounded length between a given source–sink pair) is hard to approximate within m 1/2− ε , for any ε>0.

Scheduling multicasts on unit-capacity trees and meshes

This paper studies the multicast routing and admission control problem on unit-capacity tree and mesh topologies in the throughput model. The problem is a generalization of the edge-disjoint paths problem and is NP-hard both on trees and meshes. We study both the offline and the online version of the problem: In the offline setting, we give the first constant-factor approximation algorithm for trees, and an O(( log log n) 2) -factor approximation algorithm for meshes. In the online setting, we give the first polylogarithmic competitive online algorithm for tree and mesh topologies. No polylogarithmic-competitive algorithm is possible on general network topologies (Lower bounds for on-line graph problems with application to on-line circuits and optical routing, in: Proceedings of the 28th ACM Symposium on Theory of Computing, 1996, pp. 531–540) and there exists a polylogarithmic lower bound on the competitive ratio of any online algorithm on tree topologies (Making commitments in the face of uncertainity: how to pick a winner almost every time, in: Proceedings of the 28th Annual ACM Symposium on Theory of Computing, 1996, pp. 519–530). We prove the same lower bound for meshes.

Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs

Given a graph and terminal pairs $(s_i,t_i)$, $i\in[k]$, the edge-disjoint paths problem is to determine whether there exist $s_{i}t_{i}$ paths, $i\in[k]$, that do not share any edges. We consider this problem on acyclic digraphs. It is known to be NP-complete and solvable in time $n^{O(k)}$ where n is the number of nodes. It has been a long-standing open question whether it is fixed-parameter tractable in k, i.e., whether it admits an algorithm with running time of the form $f(k)\,n^{O(1)}$. We resolve this question in the negative: we show that the problem is $W[1]$-hard, hence unlikely to be fixed-parameter tractable. In fact it remains $W[1]$-hard even if the demand graph consists of two sets of parallel edges. On a positive side, we give an $O(m+k^{O(1)}\,k!\,n)$ algorithm for the special case when G is acyclic and $G+H$ is Eulerian, where H is the demand graph. We generalize this result (1) to the case when $G+H$ is “nearly” Eulerian, and (2) to an analogous special case of the unsplittable flow problem, a generalized version of disjoint paths that has capacities and demands.

Implementation of approximation algorithms for weighted and unweighted edge-disjoint paths in bidirected trees

Given a set of weighted directed paths in a bidirected tree, the maximum weight edge-disjoint paths problem (MWEDP) is to select a subset of the given paths such that the selected paths are edge-disjoint and the total weight of the selected paths is maximized. MWEDP is NP - hard for bidirected trees of unbounded degree, even if all weights are the same (the unweighted case). Three different approximation algorithms are implemented: a known combinatorial (5/3 + ε)-approximation algorithm A 1 for the unweighted case, a new combinatorial 2-approximation algorithm A 2 for the weighted case, and a known (5/3 + ε)-approximation algorithm A 3 for the weighted case that is based on linear programming. For algorithm A 1 , it is shown how efficient data structures can be used to obtain a worst-case running-time of O(m + n + 4 1/ε √n ċ m) for instances consisting of m paths in a tree with n nodes. Experimental results regarding the running-times and the quality of the solutions obtained by the three approximation algorithms are reported. Where possible, the approximate solutions are compared to the optimal solutions, which were computed by running CPLEX on an integer linear programming formulation of MWEDP.

The edge-disjoint paths problem is NP-complete for series–parallel graphs

Many combinatorial problems are NP-complete for general graphs. However, when restricted to series–parallel graphs or partial k-trees, many of these problems can be solved in polynomial time, mostly in linear time. On the other hand, very few problems are known to be NP-complete for series–parallel graphs or partial k-trees. These include the subgraph isomorphism problem and the bandwidth problem. However, these problems are NP-complete even for trees. In this paper, we show that the edge-disjoint paths problem is NP-complete for series–parallel graphs and for partial 2-trees although the problem is trivial for trees and can be solved for outerplanar graphs in polynomial time.

The Maximum Edge-Disjoint Paths Problem in Bidirected Trees

A bidirected tree is the directed graph obtained from an undirected tree by replacing each undirected edge by two directed edges with opposite directions. Given a set of directed paths in a bidirected tree, the goal of the maximum edge-disjoint paths problem is to select a maximum-cardinality subset of the paths such that the selected paths are edge-disjoint. This problem can be solved optimally in polynomial time for bidirected trees of constant degree but is APX-hard for bidirected trees of arbitrary degree. For every fixed $\varepsilon >0$, a polynomial-time $(5/3+\varepsilon)$-approximation algorithm is presented.

On the Inapproximability of Disjoint Paths and Minimum Steiner Forest with Bandwidth Constraints

In this paper, we study the inapproximability of several well-known optimization problems in network optimization. We show-that the max directed vertex-disjoint paths problem cannot be approximated within ratio 2log1−εn unless NP⊆DTIME[2polylogn], the max directed edge-disjoint paths problem cannot be approximated within ratio 2log1−εn unless NP⊆DTIME[2polylogn], the integer multicommodity flow problem in directed graphs cannot be approximated within ratio 2log1−εn unless NP⊆DTIME[2polylogn], the max undirected vertex-disjoint paths problem does not have a polynomial time approximation scheme unless P=NP, and the minimum Steiner forest with bandwidth constraints problem cannot be approximated within ratio exp(poly(n)) unless P=NP.

Finding Edge-Disjoint Paths in Partial k -Trees

For a given graph G and p pairs (s i ,t i ) , $1\leq i\leq p$ , of vertices in G , the edge-disjoint paths problem is to find p pairwise edge-disjoint paths P i , $1\leq i\leq p$ , connecting s i and t i . Many combinatorial problems can be efficiently solved for partial k -trees (graphs of treewidth bounded by a fixed integer k ), but the edge-disjoint paths problem is NP-complete even for partial 3 -trees. This paper gives two algorithms for the edge-disjoint paths problem on partial k -trees. The first one solves the problem for any partial k -tree G and runs in polynomial time if p=O( log n) and in linear time if p=O(1) , where n is the number of vertices in G . The second one solves the problem under some restriction on the location of terminal pairs even if $p\geq \log n$ .

A software package of algorithms and heuristics for disjoint paths in Planar Networks

We present a package for algorithms on planar networks. This package comes with a graphical user interface, which may be used for demonstrating and animating algorithms. Our focus so far has been on disjoint path problems. However, the package is intended to serve as a general framework, wherein algorithms for various problems on planar networks may be integrated and visualized. For this aim, the structure of the package is designed so that integration of new algorithms and even new algorithmic problems amounts to applying a short “recipe.” The package has been used to develop new variations of well-known disjoint path algorithms, which heuristically optimize additional NP -hard objectives such as the total length of all paths. We will prove that the problem of finding edge-disjoint paths of minimum total length in a planar graph is NP -hard, even if all terminals lie on the outer face, the Eulerian condition is fulfilled, and the maximum degree is four. Finally, as a demonstration how PlaNet can be used as a tool for developing new heuristics for NP -hard problems, we will report on results of experimental studies on efficient heuristics for this problem.

Optimal Construction of Edge-Disjoint Paths in Random Graphs

Given a graph G=(V,E) with n vertices, m edges, and a family of $\kappa$ pairs of vertices in $V$, we are interested in finding for each pair (ai, bi) a path connecting ai to bi such that the set of $\kappa$ paths so found is edge disjoint. (For arbitrary graphs the problem is ${\cal NP}$-complete, although it is in ${\cal P}$ if $\kappa$ is fixed.) We present a polynomial time randomized algorithm for finding the optimal number of edge disjoint paths (up to constant factors) in the random graph Gn,m for all edge densities above the connectivity threshold. (The graph is chosen first; then an adversary chooses the pairs of endpoints.) Our results give the first tight bounds for the edge-disjoint paths problem for any nontrivial class of graphs.

The parsimonious property of cut covering problems and its applications

We consider the analysis of linear programming relaxations of a large class of combinatorial problems that can be formulated as problems of covering cuts, including the Steiner tree, the travelling salesman, the vehicle routing, the matching, the T-join and the survivable network design problem, to name a few. We prove that all of the problems in the class satisfy a nice structural property, the parsimonious property, generalizing earlier work by Goemans and Bertsimas (1993). We utilize the parsimonious property to establish worst-case bounds between the gap of the IP and LP values for the class of 0–1 proper functions, leading to a new 2-approximation algorithm for this class of problems. We also extend the parsimonious property to a class of cut-covering problems that model certain instances of the edge-disjoint path problem.

NP-completeness of some edge-disjoint paths problems

We prove: The directed edge-disjoint paths problem is NP-complete, even if (a) the underlying graph G is acyclic, the demand graph H consists just of three sets of parallel edges and G + H is Eulerian, or (b) G + H is planar, or (c) G is planar and acyclic. (d) The undirected edge-disjoint paths problem is NP-complete, even if G + H is Eulerian and H consists just of three sets of parallel edges.

Preserving and Increasing Local Edge-Connectivity in Mixed Graphs

Generalizing and unifying earlier results of W. Mader, and A. Frank and B. Jackson, we prove two splitting theorems concerning mixed graphs. By invoking these theorems we obtain min-max formulae for the minimum number of new edges to be added to a mixed graph so that the resulting graph satisfies local edge-connectivity prescriptions. An extension of Edmonds's theorem on disjoint arborescences is also deduced along with a new sufficient condition for the solvability of the edge-disjoint paths problem in digraphs. The approach gives rise to strongly polynomial algorithms for the corresponding optimization problems.

A linear-time algorithm for edge-disjoint paths in planar graphs

In this paper we discuss the problem of finding edge-disjoint paths in a planar, undirected graph such that each path connects two specified vertices on the boundary of the graph. We will focus on the “classical” case where an instance additionally fulfills the so-calledevenness-condition. The fastest algorithm for this problem known from the literature requiresO (n 5/3(loglogn)1/3) time, wheren denotes the number of vertices. In this paper now, we introduce a new approach to this problem, which results in anO(n) algorithm. The proof of correctness immediately yields an alternative proof of the Theorem of Okamura and Seymour, which states a necessary and sufficient condition for solvability.

A Linear-Time Algorithm for the Homotopic Routing Problem in Grid Graphs

The paper considers the problem of finding edge-disjoint paths between pairs of vertices in a finite grid graph. The homotopy class for each path to be routed is prespecified. A very fast algorithm that guarantees to find a solution for any solvable homotopic routing problem is given.

On the complexity of the disjoint paths problem

In this paper we consider the disjoint paths problem. Given a graphG and a subsetS of the edge-set ofG the problem is to decide whether there exists a family ℱ of disjoint circuits inG each containing exactly one edge ofS such that every edge inS belongs to a circuit inC. By a well-known theorem of P. Seymour the edge-disjoint paths problem is polynomially solvable for Eulerian planar graphsG. We show that (assumingP≠NP) one can drop neither planarity nor the Eulerian condition onG without losing polynomial time solvability. We prove theNP-completeness of the planar edge-disjoint paths problem by showing theNP-completeness of the vertex disjoint paths problem for planar graphs with maximum vertex-degree three. This disproves (assumingP≠NP) a conjecture of A. Schrijver concerning the existence of a polynomial time algorithm for the planar vertex-disjoint paths problem. Furthermore we present a counterexample to a conjecture of A. Frank. This conjecture would have implied a polynomial algorithm for the planar edge-disjoint paths problem. Moreover we derive a complete characterization of all minorclosed classes of graphs for which the disjoint paths problem is polynomially solvable. Finally we show theNP-completeness of the half-integral relaxation of the edge-disjoint paths problem. This implies an answer to the long-standing question whether the edge-disjoint paths problem is polynomially solvable for Eulerian graphs.