Edge-disjoint Paths Problem Research Articles

An Approximation Algorithm for Fully Planar Edge-Disjoint Paths

We devise a constant-factor approximation algorithm for the maximization version of the edge-disjoint paths problem if the supply graph together with the demand edges form a planar graph. By planar duality this is equivalent to packing cuts in a planar graph such that each cut contains exactly one demand edge. We also show that the natural linear programming relaxations have constant integrality gap, yielding an approximate max-multiflow min-multicut theorem.

The Power of Cut-Based Parameters for Computing Edge-Disjoint Paths

This paper revisits the classical edge-disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our aim is to identify structural properties (parameters) of graphs which allow the efficient solution of EDP without restricting the placement of terminals in P in any way. In this setting, EDP is known to remain NP-hard even on extremely restricted graph classes, such as graphs with a vertex cover of size 3. We present three results which use edge-separator based parameters to chart new islands of tractability in the complexity landscape of EDP. Our first and main result utilizes the fairly recent structural parameter tree-cut width (a parameter with fundamental ties to graph immersions and graph cuts): we obtain a polynomial-time algorithm for EDP on every graph class of bounded tree-cut width. Our second result shows that EDP parameterized by tree-cut width is unlikely to be fixed-parameter tractable. Our final, third result is a polynomial kernel for EDP parameterized by the size of a minimum feedback edge set in the graph.

Terminal-pairability in complete bipartite graphs with non-bipartite demands: Edge-disjoint paths in complete bipartite graphs

We investigate the terminal-pairability problem in the case when the base graph is a complete bipartite graph, and the demand graph is a (not necessarily bipartite) multigraph on the same vertex set. In computer science, this problem is known as the edge-disjoint paths problem. We improve the lower bound on the maximum value of Δ(D) which still guarantees that the demand graph D has a realization in Kn,n. We also solve the extremal problem on the number of edges, i.e., we determine the maximum number of edges which guarantees that a demand graph is realizable in Kn,n.

The cavity approach for Steiner trees packing problems

The belief propagation approximation, or cavity method, has been recently applied to several combinatorial optimization problems in its zero-temperature implementation, the max-sum algorithm. In particular, recent developments to solve the edge-disjoint paths problem and the prize-collecting Steiner tree problem on graphs have shown remarkable results for several classes of graphs and for benchmark instances. Here we propose a generalization of these techniques for two variants of the Steiner trees packing problem where multiple ‘interacting’ trees have to be sought within a given graph. Depending on the interaction among trees we distinguish the vertex-disjoint Steiner trees problem, where trees cannot share nodes, from the edge-disjoint Steiner trees problem, where edges cannot be shared by trees but nodes can be members of multiple trees. Several practical problems of huge interest in network design can be mapped into these two variants, for instance, the physical design of very large scale integration (VLSI) chips.The formalism described here relies on two components edge-variables that allows us to formulate a massage-passing algorithm for the V-DStP and two algorithms for the E-DStP differing in the scaling of the computational time with respect to some relevant parameters. We will show that through one of the two formalisms used for the edge-disjoint variant it is possible to map the max-sum update equations into a weighted maximum matching problem over proper bipartite graphs. We developed a heuristic procedure based on the max-sum equations that shows excellent performance in synthetic networks (in particular outperforming standard multi-step greedy procedures by large margins) and on large benchmark instances of VLSI for which the optimal solution is known, on which the algorithm found the optimum in two cases and the gap to optimality was never larger than 4%.

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.

A Polylogarithmic Approximation Algorithm for Edge-Disjoint Paths with Congestion 2

In the Edge-Disjoint Paths with Congestion problem (EDPwC), we are given an undirected n -vertex graph G , a collection M ={ ( s 1 , t 1 ),… ,( s k , t k ) } of pairs of vertices called demand pairs, and an integer c . The goal is to connect the maximum possible number of the demand pairs by paths, so that the maximum edge congestion - the number of paths sharing any edge - is bounded by c . When the maximum allowed congestion is c = 1, this is the classical Edge-Disjoint Paths problem (EDP). The best current approximation algorithm for EDP achieves an O (√ n )-approximation by rounding the standard multi-commodity flow relaxation of the problem. This matches the Ω (√ n ) lower bound on the integrality gap of this relaxation. We show an O (poly log k )-approximation algorithm for EDPwC with congestion c = 2 by rounding the same multi-commodity flow relaxation. This gives the best possible congestion for a sub-polynomial approximation of EDPwC via this relaxation. Our results are also close to optimal in terms of the number of pairs routed, since EDPwC is known to be hard to approximate to within a factor of ~ Ω ((log n ) 1/( c +1) ) for any constant congestion c . Prior to our work, the best approximation factor for EDPwC with congestion 2 was Õ( n 3/7 ), and the best algorithm achieving a polylogarithmic approximation required congestion 14.

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.

An optimal adaptive stabilizing algorithm for two edge-disjoint paths problem

The problem of two edge-disjoint paths is to identify two paths $$Q_1$$Q1 and $$Q_2$$Q2 from source $$s \in V$$s?V to target $$t \in V$$t?V without any common arc in a directed connected graph $$G=(V, E)$$G=(V,E). In this paper, we present an adaptive stabilizing algorithm for finding a pair of edge-disjoint paths from s to t in G in O(D) rounds with state-space complexity of $$O(log\; n)$$O(logn) bits per process, where n is the number of nodes and D is the diameter of the graph. The proposed algorithm is optimal with respect to its time complexity, and the total length of the shortest paths. In addition, it can also be used to solve the problem for undirected graphs. Since the proposed algorithm is stabilizing, it does not require initialization and is capable of withstanding transient faults. We view a fault that perturbs the state of the system but not its program as a transient fault. In addition, the proposed algorithm is adaptive since it is capable of dealing with topology changes in the form of addition/removal of arcs and/or nodes as well as changes in the directions of arcs provided that two edge-disjoint paths between s and t exist after the topology change.

Edge-disjoint odd cycles in 4-edge-connected graphs

Finding edge-disjoint odd cycles is one of the most important problems in graph theory, graph algorithms and combinatorial optimization. One of the difficulties of this problem is that the Erdős–Pósa property does not hold for odd cycles in general. Motivated by this fact, we prove that for any positive integer k, there exists an integer f(k) satisfying the following. For any 4-edge-connected graph G=(V,E), either G has k edge-disjoint odd cycles or there exists an edge set F⊆E with |F|≤f(k) such that G−F is bipartite. We note that the 4-edge-connectivity is best possible in this statement.A similar approach can be applied to an algorithmic question. Suppose that the input graph G is a 4-edge-connected graph with n vertices. We show that, for any ε>0, if k=O((log⁡log⁡log⁡n)1/2−ε), then the k edge-disjoint odd cycle packing problem in G can be solved in polynomial time in n. This result implies the authors' algorithm for the k edge-disjoint paths problem in 4-edge-connected graphs [13].

The Edge-Disjoint Path Problem on Random Graphs by Message-Passing

We present a message-passing algorithm to solve a series of edge-disjoint path problems on graphs based on the zero-temperature cavity equations. Edge-disjoint paths problems are important in the general context of routing, that can be defined by incorporating under a unique framework both traffic optimization and total path length minimization. The computation of the cavity equations can be performed efficiently by exploiting a mapping of a generalized edge-disjoint path problem on a star graph onto a weighted maximum matching problem. We perform extensive numerical simulations on random graphs of various types to test the performance both in terms of path length minimization and maximization of the number of accommodated paths. In addition, we test the performance on benchmark instances on various graphs by comparison with state-of-the-art algorithms and results found in the literature. Our message-passing algorithm always outperforms the others in terms of the number of accommodated paths when considering non trivial instances (otherwise it gives the same trivial results). Remarkably, the largest improvement in performance with respect to the other methods employed is found in the case of benchmarks with meshes, where the validity hypothesis behind message-passing is expected to worsen. In these cases, even though the exact message-passing equations do not converge, by introducing a reinforcement parameter to force convergence towards a sub optimal solution, we were able to always outperform the other algorithms with a peak of 27% performance improvement in terms of accommodated paths. On random graphs, we numerically observe two separated regimes: one in which all paths can be accommodated and one in which this is not possible. We also investigate the behavior of both the number of paths to be accommodated and their minimum total length.

Min-Sum 2-Paths Problems

An orientation of an undirected graph G is a directed graph obtained by replacing each edge {u,v} of G by exactly one of the arcs (u,v) or (v,u). In the min-sumk-paths orientation problem, the input is an undirected graph G and ordered pairs (si,ti), where i?{1,2,?,k}. The goal is to find an orientation of G that minimizes the sum over all i?{1,2,?,k} of the distance from si to ti. In the min-sumkedge-disjoint paths problem, the input is the same, however the goal is to find for every i?{1,2,?,k} a path between si and ti so that these paths are edge-disjoint and the sum of their lengths is minimum. Note that, for every fixed k?2, the question of NP-hardness for the min-sum k-paths orientation problem and for the min-sum k edge-disjoint paths problem has been open for more than two decades. We study the complexity of these problems when k=2. We exhibit a PTAS for the min-sum 2-paths orientation problem. A by-product of this PTAS is a reduction from the min-sum 2-paths orientation problem to the min-sum 2 edge-disjoint paths problem. The implications of this reduction are: (i) an NP-hardness proof for the min-sum 2-paths orientation problem yields an NP-hardness proof for the min-sum 2 edge-disjoint paths problem, and (ii) any approximation algorithm for the min-sum 2 edge-disjoint paths problem can be used to construct an approximation algorithm for the min-sum 2-paths orientation problem with the same approximation guarantee and only an additive polynomial increase in the running time.

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.

The edge-disjoint paths problem in Eulerian graphs and 4-edge-connected graphs

In the edge-disjoint paths problem, we are given a graph and a set of k pairs of vertices, and we have to decide whether or not the graph has k edge-disjoint paths connecting given pairs of terminals. Robertson and Seymour's graph minor project gives rise to a polynomial time algorithm for this problem for any fixed k, but their proof of the correctness needs the whole Graph Minor project. We give a faster algorithm and a much simpler proof of the correctness for the edge-disjoint paths problem. Our results can be summarized as follows: 1.If an input graph is either 4-edge-connected or Eulerian, then our algorithm only needs to look for the following three simple reductions: (i) Excluding vertices of high degree. (ii) Excluding ≤3-edge-cuts. (iii) Excluding large clique minors.2.When an input graph is either 4-edge-connected or Eulerian, the number of terminals k is allowed to be a non-trivially super constant number, up to k=O((log log logn)½??) for any ? > 0. In addition, if an input graph is either 4-edge-connected planar or Eulerian planar, k is allowed to be O((logn½??) for any ? > 0.3.We also give our own algorithm for the edge-disjoint paths problem in general graphs. We basically follow the Robertson-Seymour's algorithm, but we cut half of the proof of the correctness for their algorithm. In addition, our algorithm is faster than Robertson and Seymour's.

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.

The All-or-Nothing Multicommodity Flow Problem

We consider the all-or-nothing multicommodity flow problem in general graphs. We are given a capacitated undirected graph $G=(V,E,u)$ and a set of $k$ node pairs $s_1 t_1, s_2t_2, \ldots ,s_kt_k$. Each pair has a unit demand. A subset $S$ of $\{1,2,\ldots,k\}$ is routable if there is a multicommodity flow in $G$ that simultaneously sends one unit of flow between $s_i$ and $t_i$ for each $i$ in $S$. Note that this differs from the edge-disjoint path problem (edp) in that we do not insist on integral flows for the pairs. The objective is to find a maximum routable subset $S$. When $G$ is a capacitated tree, the problem already generalizes $b$-matchings, and even in this case it is NP-hard and APX-hard to approximate. For trees, a $2$-approximation is known for the cardinality case and a $4$-approximation for the weighted case. In this paper we show that the natural linear programming relaxation for the all-or-nothing flow problem has a polylogarithmic integrality gap in general undirected graphs. This is in sharp contrast to edp, where the gap is known to be $\Theta(\sqrt{n})$; this ratio is also the best approximation ratio currently known for edp. Our algorithm extends to the case where each pair $s_it_i$ has a demand $d_i$ associated with it and we need to completely route $d_i$ to get credit for pair $i$; we assume that the maximum demand of the pairs is at most the minimum capacity of the edges. We also consider the online admission control version where pairs arrive online and the algorithm has to decide immediately on its arrival whether to accept it and the accepted pairs have to be routed. We obtain a randomized algorithm which has a polylogarithmic competitive ratio for maximizing throughput of the accepted requests if it is allowed to violate edge capacities by a $(2+\epsilon)$ factor.

LS(Graph): a constraint-based local search for constraint optimization on trees and paths

Constrained optimum tree (COT) and constrained optimum path (COP) problems arise in many real-life applications and are ubiquitous in communication networks. They have been traditionally approached by dedicated algorithms, which are often hard to extend with side constraints and to apply widely. This paper proposes a constraint-based local search framework for COT/COP applications, bringing the compositionality, reuse, and extensibility at the core of constraint-based local search and constraint programming systems. The modeling contribution is the ability to express compositional models for various COT/COP applications at a high level of abstraction, while cleanly separating the model and the search procedure. The main technical contribution is a connected neighborhood based on rooted spanning trees to find high-quality solutions to COP problems. This framework is applied to some COT/COP problems, e.g., the quorumcast routing problem, the edge-disjoint paths problem, and the routing and wavelength assignment with delay side constraints problem. Computational results show the potential importance of the approach.

The disjoint paths problem in quadratic time

We consider the following well-known problem, which is called the disjoint paths problem. For a given graph G and a set of k pairs of terminals in G, the objective is to find k vertex-disjoint paths connecting given pairs of terminals or to conclude that such paths do not exist. We present an O(n2) time algorithm for this problem for fixed k. This improves the time complexity of the seminal result by Robertson and Seymour, who gave an O(n3) time algorithm for the disjoint paths problem for fixed k. Note that Perković and Reed (2000) announced in [24] (without proofs) that this problem can be solved in O(n2) time. Our algorithm implies that there is an O(n2) time algorithm for the k edge-disjoint paths problem, the minor containment problem, and the labeled minor containment problem. In fact, the time complexity of all the algorithms with the most expensive part depending on Robertson and Seymourʼs algorithm can be improved to O(n2), for example, the membership testing for minor-closed class of graphs.

Upgrading edge-disjoint paths in a ring

In this paper, we introduce the upgrading problem of edge-disjoint paths. In the off-line upgrading problem, a supply graph G with integer capacities and two demand graphs H 1 and H 2 with unit demands are given on the same vertex set. Our task is to determine the maximum size of a set F ⊆ E ( H 1 ) ∩ E ( H 2 ) such that F has an integer routing in G which can be extended both to an integer routing of H 1 and to an integer routing of H 2 . In the online upgrading problem, we are given a supply graph G with integer capacities, a demand graph H with an integer routing, and another demand graph H 2 with unit demands such that E ( H ) ⊆ E ( H 2 ) . Our task is to determine the maximum size of a set F ⊆ E ( H ) such that the restriction of the given routing to F can be extended to an integer routing of H 2 . Thus, depending on whether the graphs are directed or undirected, we have four different versions. We give algorithmic proofs of minimax formulas for the case when G is a ring and the demand graphs are stars with the same center. All four versions are NP-complete for general graphs.

Inapproximability of Edge-Disjoint Paths and low congestion routing on undirected graphs

In the undirected Edge-Disjoint Paths problem with Congestion (EDPwC), we are given an undirected graph with V nodes, a set of terminal pairs and an integer c. The objective is to route as many terminal pairs as possible, subject to the constraint that at most c demands can be routed through any edge in the graph. When c = 1, the problem is simply referred to as the Edge-Disjoint Paths (EDP) problem. In this paper, we study the hardness of EDPwC in undirected graphs. Our main result is that for every ɛ > 0 there exists an α > 0 such that for 1 ⩽ c ⩽ $$ \frac{{\alpha log log V }} {{\log \log \log V}} $$ , it is hard to distinguish between instances where we can route all terminal pairs on edge-disjoint paths, and instances where we can route at most a $$ {1 \mathord{\left/ {\vphantom {1 {\left( {\log V} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\log V} \right)}}^{\frac{{1 - \varepsilon }} {{c + 2}}} $$ fraction of the terminal pairs, even if we allow congestion c. This implies a $$ \left( {\log V} \right)^{\frac{{1 - \varepsilon }} {{c + 2}}} $$ hardness of approximation for EDPwC and an Ω(log logV/log log logV) hardness of approximation for the undirected congestion minimization problem. These results hold assuming NP ⊊ ∪ d ZPTIME( $$ 2^{\log ^{d_n } } $$ ). In the case that we do not require perfect completeness, i.e. we do not require that all terminal pairs are routed for “yes-instances”, we can obtain a slightly better inapproximability ratio of $$ \left( {\log V} \right)^{\frac{{1 - \varepsilon }} {{c + 1}}} $$ . Note that by setting c=1 this implies that the regular EDP problem is $$ \left( {\log V} \right)^{\frac{1} {2} - \varepsilon } $$ hard to approximate. Using standard reductions, our results extend to the node-disjoint versions of these problems as well as to the directed setting. We also show a $$ \left( {\log V} \right)^{\frac{{1 - \varepsilon }} {{c + 1}}} $$ inapproximability ratio for the All-or-Nothing Flow with Congestion (ANFwC) problem, a relaxation of EDPwC, in which the flow unit routed between the source-sink pairs does not have to follow a single path, so the resulting flow is not necessarily integral.