Maximum Edge-disjoint Paths Problem Research Articles

Solving the maximum edge disjoint path problem using a modified Lagrangian particle swarm optimisation hybrid

This paper presents a new method to solve the Maximum Edge Disjoint Paths (MEDP) problem. Given a set of node pairs within a network, the MEDP problem is the task of finding the largest number of pairs that can be connected by paths, using each edge within the network at most once. We present a heuristic algorithm that builds a hybridisation of Lagrangian Relaxation and Particle Swarm Optimisation, referred to as LaPSO. This hybridisation is combined with a new repair heuristic, called Largest Violation Perturbation (LVP). We show that our LaPSO method produces better heuristic solutions than both current state-of-the-art heuristic methods, as well as the primal solution found by a standard Mixed Integer Programming (MIP) solver within a limited runtime. Significantly, when run with a limited runtime, our LaPSO method also produces strong bounds which are superior to a standard MIP solver for the larger instances tested, whilst being competitive for the remainder. This allows our LaPSO method to prove optimality for many instances and provide optimality gaps for the remainder, making it a “quasi-exact” method. In this way our LaPSO algorithm, which draws on ideas from both mathematical programming and evolutionary algorithms, is able to outperform both MIP and metaheuristic solvers that only use ideas from one of these areas.

All-or-Nothing Multicommodity Flow Problem with Bounded Fractionality in Planar Graphs

We study the following all-or-nothing multicommodity flow problem in planar graphs: Input: A graph $G$ with $n$ vertices and $k$ pairs of vertices $(s_1,t_1),(s_2,t_2),\dots, (s_k,t_k)$ in $G$. Find: A largest subset $W$ of $\{1,\dots,k\}$ such that for every $i$ in $W$, we can send one unit of flow between $s_i$ and $t_i$. This problem is different from the well-known maximum edge-disjoint paths problem in that we do not require integral flows for the pairs. This problem is APX-hard even for trees, and a 2-approximation algorithm is known for trees. For general graphs, Chekuri, Khanna, and Shepherd [SIAM J. Comput., 42 (2013), pp. 1467--1493] give a polylogarithmic factor approximation algorithm and show that a natural LP-relaxation has a polylogarithmic integrality gap. This result is in contrast with the integrality gap $\Omega(\sqrt{n})$ for the maximum edge-disjoint paths problem. Our main result considerably strengthens this result when an input graph is planar. Namely, for the all-or-nothing multicommodity flow problem in planar graphs, we give an $O(1)$-approximation algorithm and show that the integrality gap is $O(1)$. In particular, in polynomial time, we can find an index set $W$ with $|W| = \Omega({OPT})$ and eight $s_i$-$t_i$ paths for each $i \in W$ such that each edge is used at most eight times in these paths (with multiplicity), where OPT is the optimal value of the LP-relaxation of the all-or-nothing multicommodity flow problem. Our result can be compared to the result by Séguin-Charbonneau and Shepherd [Proceedings of FOCS, 2011, pp. 200--209], who give an $O(1)$-approximation algorithm for the maximum edge-disjoint paths problem in planar graphs with congestion 2 (but not implied by this result).

The Inapproximability of Maximum Single-Sink Unsplittable, Priority and Confluent Flow Problems

$ \newcommand{\nba}{\text{nba}} \newcommand{\np}{\textsf{NP}} $ While the maximum single-sink unsplittable and confluent flow problems have been studied extensively, algorithmic work has been primarily restricted to the case where one imposes the no-bottleneck assumption $(\nba)$ (that the maximum demand $d_{\max}$ is at most the minimum capacity $u_{\min}$). For instance, under the $\nba$ there is a factor-$4.43$ approximation algorithm due to Dinitz et al. (1999) for the unsplittable flow problem. Under the even stronger assumption of uniform capacities, there is a factor-$3$ approximation algorithm due to Chen et al. (2007) for the confluent flow problem. We show, however, that unlike the unsplittable flow problem, a constant-factor approximation algorithm cannot be obtained for the single-sink confluent flow problem even with the no-bottleneck assumption. Specifically, we prove that it is $\np$-hard to approximate single-sink confluent flow to within $O(\log^{1-\epsilon}(n))$, for any $\epsilon> 0$. The remainder of our results focus upon the setting without the no-bottleneck assumption. Using exponential-size demands, Azar and Regev prove a $\Omega(m^{1-\epsilon})$ inapproximability result for maximum cardinality single-sink unsplittable flow in directed graphs. We prove that this lower bound applies to undirected graphs, including planar networks (and for confluent flow). This is the first super-constant hardness known for undirected single-sink unsplittable flow. Furthermore, we show $\Omega(m^{1/2-\epsilon})$-hardness even if all demands and capacities lie within an arbitrarily small range $[1,1+\Delta]$, for $\Delta > 0$. This result is sharp in that if $\Delta=0$, then it becomes a single-sink maximum edge-disjoint paths problem which can be solved exactly via a maximum flow algorithm. This motivates us to study maximum priority flows for which we show the same inapproximability bound.

An Improved Approximation Algorithm for the Edge-Disjoint Paths Problem with Congestion Two

In the maximum edge-disjoint paths problem, we are given a graph and a collection of pairs of vertices, and the objective is to find the maximum number of pairs that can be routed by edge-disjoint paths. An r -approximation algorithm for this problem is a polynomial-time algorithm that finds at least OPT/ r edge-disjoint paths, where OPT denotes the maximum possible number of pairs that can be routed in a given instance. For a long time, an O ( n 1/2 -approximation algorithm has been best known for this problem even if a congestion of two is allowed, that is, each edge is allowed to be used in at most two of the paths. In this article, we give a randomized O ( n 3/7 ċ poly(log n ))-approximation algorithm with congestion two. This is the first result that breaks the O ( n 1/2 )-approximation algorithm. In particular, we prove the following: (1) If we have a (randomized) polynomial-time algorithm for finding Ω(OPT1/p /polylog( n )) edge-disjoint paths for some p > 1, then we can give a randomized O ( n 1/2 -α)-approximation algorithm for the edge-disjoint paths problem by using Rao-Zhou’s algorithm for some α > 0. (2) Based on the Chekuri-Khanna-Shepherd well-linked decomposition, we show that there is a randomized algorithm for finding Ω(OPT 1/4 /(log n ) 3/2 ) edge-disjoint paths connecting given terminal pairs with congestion two. Our framework for this algorithm is more general in the following sense. Indeed, the above two ingredients also work for the maximum edge-disjoint paths problem (with congestion one) if there is a (randomized) polynomial-time algorithm for finding Ω(OPT1/p) edge-disjoint paths connecting given terminal pairs for some p > 1.

A genetic algorithm for the maximum edge-disjoint paths problem

Abstract Optimization problems concerning edge-disjoint paths have attracted considerable attention for decades. These problems have a lot of applications in the areas of call admission control, real-time communication, VLSI (Very-large-scale integration) layout and reconfiguration, packing, etc. The maximum edge-disjoint paths problem (MEDP) seems to lie in the heart of these problems. Given an undirected graph G and a set of I connection requests, each request consists of a pair of nodes, MEDP is an NP-hard problem which determines the maximum number of accepted requests that can be routed by mutually edge-disjoint ( s i , t i ) paths. We propose a genetic algorithm (GA) to solve the problem. In comparison to the multi-start simple greedy algorithm (MSGA) and the ant colony optimization method (ACO), the proposed GA method performs better in most of the instances in terms of solution quality and time.

An O(log n)-Approximation Algorithm for the Edge-Disjoint Paths Problem in Eulerian Planar Graphs

In this article, we study an approximation algorithm for the maximum edge-disjoint paths problem. In this problem, we are given a graph and a collection of pairs of vertices, and the objective is to find the maximum number of pairs that can be connected by edge-disjoint paths. We give an O (log n )-approximation algorithm for the maximum edge-disjoint paths problem when an input graph is either 4-edge-connected planar or Eulerian planar. This improves an O (log 2 n )-approximation algorithm given by Kleinberg [2005] for Eulerian planar graphs. Our result also generalizes the result by Chekuri et al. [2004, 2005] who gave an O (log n )-approximation algorithm for the maximum edge-disjoint paths problem with congestion two when an input graph is planar.

Disjoint paths in sparse graphs

We generalize all the results obtained for maximum integer multiflow and minimum multicut problems in trees by Garg, Vazirani and Yannakakis [N. Garg, V.V. Vazirani, M. Yannakakis, Primal-dual approximation algorithms for integral flow and multicut in trees, Algorithmica 18 (1997) 3–20] to graphs with a fixed cyclomatic number, while this cannot be achieved for other classical generalizations of trees. We also introduce the k -edge-outerplanar graphs, a class of planar graphs with arbitrary (but bounded) tree-width that generalizes the cacti, and show that the integrality gap of the maximum edge-disjoint paths problem is bounded in these graphs.

Edge-Disjoint Paths in Planar Graphs with Constant Congestion

We study the maximum edge-disjoint paths problem in undirected planar graphs: given a graph G and node pairs (demands) $s_1t_1$, $s_2t_2$, $\dots$, $s_kt_k$, the goal is to maximize the number of demands that can be connected (routed) by edge-disjoint paths. The natural multicommodity flow relaxation has an $\Omega(\sqrt{n})$ integrality gap, where n is the number of nodes in G. Motivated by this, we consider solutions with small constant congestion $c>1$, that is, solutions in which up to c paths are allowed to use an edge (alternatively, each edge has a capacity of c). In previous work we obtained an $O(\log n)$ approximation with congestion 2 via the flow relaxation. This was based on a method of decomposing into well-linked subproblems. In this paper we obtain an $O(1)$ approximation with congestion 4. To obtain this improvement we develop an alternative decomposition that is specific to planar graphs. The decomposition produces instances that we call Okamura–Seymour (OS) instances. These have the property that all terminals lie on a single face. Another ingredient we develop is a constant factor approximation for the all-or-nothing flow problem on OS instances via the flow relaxation.

The maximum edge-disjoint paths problem in complete graphs

In this paper, we consider the undirected version of the well known maximum edge-disjoint paths problem, restricted to complete graphs. We propose an off-line 3.75-approximation algorithm and an on-line 6.47-approximation algorithm, improving the earlier 9-approximation algorithm proposed by Carmi, Erlebach, and Okamoto [P. Carmi, T. Erlebach, Y. Okamoto, Greedy edge-disjoint paths in complete graphs, in: Proc. 29th Workshop on Graph Theoretic Concepts in Computer Science, in: LNCS, vol. 2880, 2003, pp. 143–155]. Moreover, we show that for the general case, no on-line algorithm is better than a ( 2 − ε ) -approximation, for all ε > 0 . For the special case when the number of paths is within a linear factor of the number of vertices of the graph, it is established that the problem can be optimally solved in polynomial time by an off-line algorithm, but that no on-line algorithm is better than a ( 1.5 − ε ) -approximation. Finally, the proposed techniques are used to obtain off-line and on-line algorithms with a constant approximation ratio for the related problems of edge congestion routing and wavelength routing in complete graphs.

Edge-disjoint paths revisited

The approximability of the maximum edge-disjoint paths problem (EDP) in directed graphs was seemingly settled by an Ω( m 1/2 - ϵ)-hardness result of Guruswami et al. [2003], and an O (√ m ) approximation achievable via a natural multicommodity-flow-based LP relaxation as well as a greedy algorithm. Here m is the number of edges in the graph. We observe that the Ω( m 1/2 - ϵ)-hardness of approximation applies to sparse graphs, and hence when expressed as a function of n , that is, the number of vertices, only an Ω( n 1/2 - ϵ)-hardness follows. On the other hand, O (√ m )-approximation algorithms do not guarantee a sublinear (in terms of n ) approximation algorithm for dense graphs. We note that a similar gap exists in the known results on the integrality gap of the flow-based LP relaxation: an Ω(√ n ) lower bound and O (√ m ) upper bound. Motivated by this discrepancy in the upper and lower bounds, we study algorithms for EDP in directed and undirected graphs and obtain improved approximation ratios. We show that the greedy algorithm has an approximation ratio of O (min( n 2/3 , √ m )) in undirected graphs and a ratio of O (min( n 4/5 , √ m )) in directed graphs. For acyclic graphs we give an O (√ n ln n ) approximation via LP rounding. These are the first sublinear approximation ratios for EDP. The results also extend to EDP with weights and to the uniform-capacity unsplittable flow problem (UCUFP).

Finding Edge-disjoint Paths in Networks: An Ant Colony Optimization Algorithm

One of the basic operations in communication networks consists in establishing routes for connection requests between physically separated network nodes. In many situations, either due to technical constraints or to quality-of-service and survivability requirements, it is required that no two routes interfere with each other. These requirements apply in particular to routing and admission control in large-scale, high-speed and optical networks. The same requirements also arise in a multitude of other applications such as real-time communications, vlsi design, scheduling, bin packing, and load balancing. This problem can be modeled as a combinatorial optimization problem as follows. Given a graph G representing a network topology, and a collection T={(s1,t1)...(sk,tk)} of pairs of vertices in G representing connection request, the maximum edge-disjoint paths problem is an NP-hard problem that consists in determining the maximum number of pairs in T that can be routed in G by mutually edge-disjoint si−ti paths. We propose an ant colony optimization (aco) algorithm to solve this problem. aco algorithms are approximate algorithms that are inspired by the foraging behavior of real ants. The decentralized nature of these algorithms makes them suitable for the application to problems arising in large-scale environments. First, we propose a basic version of our algorithm in order to outline its main features. In a subsequent step we propose several extensions of the basic algorithm and we conduct an extensive parameter tuning in order to show the usefulness of those extensions. In comparison to a multi-start greedy approach, our algorithm generates in general solutions of higher quality in a shorter amount of time. In particular the run-time behaviour of our algorithm is one of its important advantages.

Path problems in generalized stars, complete graphs, and brick wall graphs

Path problems such as the maximum edge-disjoint paths problem, the path coloring problem, and the maximum path coloring problem are relevant for resource allocation in communication networks, in particular all-optical networks. In this paper, it is shown that maximum path coloring can be solved optimally in polynomial time for bidirected generalized stars, even in the weighted case. Furthermore, the maximum edge-disjoint paths problem is proved NP-hard for complete graphs (undirected or bidirected), a constant-factor off-line approximation algorithm is presented for the weighted case, and an on-line algorithm with constant competitive ratio is given for the unweighted case. Finally, an open problem concerning the existence of routings that simultaneously minimize the maximum load and the number of colors is solved: an example for a graph and a set of requests is given such that any routing that minimizes the maximum load requires strictly more colors for path coloring than a routing that minimizes the number of colors.

Edge disjoint paths and max integral multiflow/min multicut theorems in planar graphs

Abstract We generalize all the results obtained for integer multiflow and multicut problems in trees by Garg et al. [N. Garg, V.V. Vazirani and M. Yannakakis, Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18 (1997) 3–20] to planar graphs with a fixed number of faces, although other classical generalizations do not lead to such results. We also introduce the class of k-edge-outerplanar graphs and bound the integrality gap for the maximum edge-disjoint paths problem in these graphs.

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.