Finding Two Disjoint Paths Between Two Pairs of Vertices in a Graph (original) (raw)

1978, Journal of the ACM

https://doi.org/10.1145/322047.322048

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Abstract

^SSTnXcr. Gwen a graph G = (V, E) and four verttces s~, tx, s~, and t2, the problem of finding two disjoint paths, P~ from s~ to tt and P2 from s2 to t2, is considered This problem may arise as a transportation network problem and m printed clrcmts routing The relations between several vemons of the problem are discussed Efficient algorithms are gwen for the following special cases-acyche directed graphs and 3-connected planar and chordal graphs.

Edge-disjoint paths in planar graphs

Journal of Combinatorial Theory, Series B, 1985

Given a planar graph G = (V, E), find k edge-disjoint paths in G connecting k pairs of terminals specified on the outer face of G. Generalizing earlier results of Okamura and Seymour (J. Combin. Theory Ser. B 31 (1981) 75-81) and of the author (Combinatorics 2, No. 4 (1982) 361-371) we solve this problem when each node of G not on the outer face has even degree. The solution involves a good characterization for the solvability and the proof gives rise to an algorithm of complexity 0(1 F'1310gl I'(). In particular, the integral multicommodity flow problem is proved to belong to the problem class P when the underlying graph is outerplanar.

A Fast Algorithm for Optimally Finding Partially Disjoint Shortest Paths

Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence, 2018

The classical disjoint shortest path problem has recently recalled interests from researchers in the network planning and optimization community. However, the requirement of the shortest paths being completely vertex or edge disjoint might be too restrictive and demands much more resources in a network. Partially disjoint shortest paths, in which a bounded number of shared vertices or edges is allowed, balance between degree of disjointness and occupied network resources. In this paper, we consider the problem of finding k shortest paths which are edge disjoint but partially vertex disjoint. For a pair of distinct vertices in a network graph, the problem aims to optimally find k edge disjoint shortest paths among which at most a bounded number of vertices are shared by at least two paths. In particular, we present novel techniques for exactly solving the problem with a runtime that significantly improves the current best result. The proposed algorithm is also validated by computer experiments on both synthetic and real networks which demonstrate its superior efficiency of up to three orders of magnitude faster than the state of the art.

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.

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

Combinatorica, 2014

We consider the following well-known problem, which is called the edge-disjoint paths problem. Input: A graph G with n vertices and m edges, k pairs of vertices (s1, t1), (s2, t2),. .. , (s k , t k) in G. Output : Edge-disjoint paths P1, P2,. .. , P k in G such that Pi joins si and ti for i = 1, 2,. .. , k. Robertson and Seymour's graph minor project gives rise to an O(m 3) algorithm for this problem for any fixed k, but their proof of the correctness needs the whole Graph Minor project, spanning 23 papers and at least 500 pages proof. We give a faster algorithm and a simpler proof of the correctness for the edge-disjoint paths problem for any fixed k. Our results can be summarized as follows: 1. If an input graph G 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 G is either 4-edge-connected or Eulerian, the number of terminals k is allowed to be non-trivially superconstant number, up to k = O((log log log n) 1 2 −ε) for any ε > 0. Thus our hidden constant in this case is dramatically smaller than Robertson-Seymour's. In addition, if an input graph G is either 4-edge-connected planar or Eulerian planar, k is allowed to be O((log n) 1 2 −ε) for any ε > 0. The same thing holds for bounded genus graphs. Moreover, if an input graph is either 4-edge-connected H-minor-free or Eulerian H-minor-free for fixed graph H, k is allowed to be O((log log n) 1 2 −ε) for any ε > 0. 3. We also give our own algorithm for the edge-disjoint paths problem in general graphs. We basically follow Robertson-Seymour's algorithm, but we cut half of the proof of the correctness for their algorithm. In addition, the time complexity of our algorithm is O(n 2), which is faster than Robertson and Seymour's.

Edge-disjoint paths revisited

ACM Transactions on Algorithms, 2007

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 appr...

Note on the problem of partially link disjoint paths

This paper discusses the problem of partially link disjoint paths in communication networks. The problem is situated in between the well-known shortest path generation and the synthesis of two disjoint paths. The contributions of this paper are twofold: firstly, topologies and relevant graph properties are investigated and the degree of divergence is defined as a metric to distinguish the degree of commonality for the two partially disjoint paths. Secondly, heuristic approaches are introduced that solve the problem of generating partly disjoint paths.

Algorithms for determining a node-disjoint path pair visiting specified nodes

Optical Switching and Networking, 2016

The problem of calculating the shortest path that visits a given set of nodes is at least as difficult as the traveling salesman problem, and it has not received much attention. Nevertheless an efficient integer linear programming (ILP) formulation has been recently proposed for this problem. That ILP formulation is firstly adapted to include the constraint that the obtained path can be protected by a node-disjoint path, and secondly to obtain a pair of node disjoint paths, of minimal total additive cost, each having to visit a given set of specified nodes. Computational experiments show that these approaches, namely in large networks, may fail to obtain solutions in a reasonable amount of time. Therefore heuristics are proposed for solving those problems, that may arise due to network management constraints. Extensive computational results show that they are capable of finding a solution in most cases, and that the calculated solutions present an acceptable relative error regarding the cost of the obtained path or pair of paths. Further the CPU time required by the heuristics is significantly smaller than the required by the used ILP solver.

CONSTRAINED DISJOINT PATHS IN GEOMETRIC NETWORKS

We address the problem of constructing a pair of node disjoint paths connecting two given nodes in a geometric network. We propose a simple algorithm for constructing such a path that does not use any complicated data structure which can be implemented very easily. We also consider two variations of disjoint path pair problems which we call narrow corridor problem and width bounded corridor problem and present efficient algorithms for solving them

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References (11)

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An Algorithm for Finding Two Node-Disjoint Paths in Arbitrary Graphs

Journal of Computing and Information Technology

Given two distinct vertices (nodes) source s and target t of a graph G = (V, E), the two node-disjoint paths problem is to identify two node-disjoint paths between s ∈ V and t ∈ V. Two paths are node-disjoint if they have no common intermediate vertices. In this paper, we present an algorithm with O(m)-time complexity for finding two node-disjoint paths between s and t in arbitrary graphs where m is the number of edges. The proposed algorithm has a wide range of applications in ensuring reliability and security of sensor, mobile and fixed communication networks.

On the disjoint paths problem

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

Combinatorica, 1995

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-called evenness-condition. The fastest algorithm for this problem known from the literature requires O (nb/3(loglogn)l/3) time, where n denotes the number of vertices. In this paper now, we introduce a new approach to this problem, which results in an O(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.

Finding Arc and Vertex-Disjoint Paths in Networks

2009 Eighth IEEE International Conference on Dependable, Autonomic and Secure Computing, 2009

Multipath Routing plays an important role in communication networks. Multiple disjoint paths can increase the effective bandwidth between pairs of vertices, avoid congestion in a network and reduce the probability of dropped packets. In this paper, we build mathematical models for arc-disjoint paths problem and vertex-disjoint paths problem respectively, and prove that they are both Linear Programming Problem. Then we propose polynomial algorithms for finding the shortest pair of arc and vertex-disjoint paths, both with the time complexity of () Om . Furthermore, we extend these algorithms to find any k disjoint paths, whose sum-weight is minimized in time () O km .

Constrained Disjoint Paths in Geometric Network

International Journal of Computational Intelligence and Applications, 2009

A linear programming formulation of Mader's edge-disjoint paths problem

Journal of Combinatorial Theory, Series B, 2006

We give a dual pair of linear programs for a min-max result of Mader describing the maximum number of edge-disjoint T-paths in a graph G = (V , E) with T ⊆ V . We conclude that there exists a polynomial-time algorithm (based on the ellipsoid method) for finding the maximum number of Tpaths in a capacitated graph, where the number of T-paths using an edge does not exceed the capacity of that edge.

Vertex Disjoint Paths in Upward Planar Graphs

Lecture Notes in Computer Science, 2014

The k-vertex disjoint paths problem is one of the most studied problems in algorithmic graph theory. In 1994, Schrijver proved that the problem can be solved in polynomial time for every fixed k when restricted to the class of planar digraphs and it was a long standing open question whether it is fixed-parameter tractable (with respect to parameter k) on this restricted class. Only recently, [13]. achieved a major breakthrough and answered the question positively. Despite the importance of this result (and the brilliance of their proof), it is of rather theoretical importance. Their proof technique is both technically extremely involved and also has at least double exponential parameter dependence. Thus, it seems unrealistic that the algorithm could actually be implemented. In this paper, therefore, we study a smaller class of planar digraphs, the class of upward planar digraphs, a well studied class of planar graphs which can be drawn in a plane such that all edges are drawn upwards. We show that on the class of upward planar digraphs the problem (i) remains NP-complete and (ii) the problem is fixed-parameter tractable. While membership in FPT follows immediately from [13]'s general result, our algorithm has only single exponential parameter dependency compared to the double exponential parameter dependence for general planar digraphs. Furthermore, our algorithm can easily be implemented, in contrast to the algorithm in [13].

The complexity of finding two disjoint paths with min-max objective function

Discrete Applied Mathematics, 1990

Given a network G = (V, E) and two vertices s and t, we consider the problem of finding two disjoint paths from s to t such that the length of the longer path is minimized. 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 as well as some related problems are strongly NPcomplete. We also give a pseudo-polynomial-time algorithm for the acyclic directed case.

Hardness of the undirected edge-disjoint paths problem

2005

We show that there is no log 1 3 −ε M approximation for the undirected Edge-Disjoint Paths problem unless N P ⊆ ZP T IM E(n polylog(n) ), where M is the size of the graph and ε is any positive constant. This hardness result also applies to the undirected All-or-Nothing Multicommodity Flow problem and the undirected Node-Disjoint Paths problem.

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The directed subgraph homeomorphism problem

Theoretical Computer Science, 1980

The set of pattern graphs for which the fixed directed subgraph homeomorphism problem is NP-complete is characterized. A polynomial time algorithm is given for the remaining cases. The restricted problem where the input graph is a directed acyclic graph is in polynomial ti. ie for all pattern graphs and an algorithm is giver?.

A Minimax Method for Finding the k Best “Differentiated” Paths

Geographical Analysis, 2010

A Minimax Method for Finding the k Best "Differentiated '' Paths In real-world applications, the k-shortest-paths between a pair of nodes on a network will often be slight variations of one another. This could be a problem f o r many path-based models, particularly those on capacitated networks where different routing alternatives are needed that are less likely to encounter the same capacity constraints. This paper develops a method to solve f o r k diferentiated paths that are relatively short and yet relatively different f r o m one another, but not necessarily disjoint. Our method utilizes the sum of a path's distance plus some fraction of its shared distance with each other path. A minimax algorithm is used to select the path whose largest sum of length, plus shared length vis-6-vis each previously selected path, is as small as possible. W e present computational results f o r the Chinese railway system, comparing the paths generated b y a standard k-shortest-path algorithm with those f r o m our new model. This work was supported by the World Bank, East Asia Department-and particularly Mr. Philip Anderson-and by the Chinese Ministry of Railways. Arizona State University provided sabbatical support to Michael Kuby and hosted the Chinese team. Barbara Trapido-Lurie of ASU made the maps based on the Railway GIS by Ma Jun of Tsinghua University. The algorithm incorporated a k shortest path generator developed earlier by Wang Xuesheng of MOR. Frank Southworth provided useful feedback on our idea and previous research in the area.

Bridging Theory and Practice with Query Log Analysis

ACM SIGMOD Record

Since large structured query logs have recently become available, we have a new opportunity to gain insights in the types of queries that users ask. Even though such logs can be quite volatile, there are various new observations that can be made about the structure of queries inside them, on which we report here. Furthermore, building on an extensive analysis that has been done on such logs, we were able to provide a theoretical explanation why regular path queries in graph database applications behave better than worst-case complexity results suggest at first sight.

Resilience Bounds of Network Clock Synchronization with Fault Correction

ACM Transactions on Sensor Networks, 2020

Naturally occurring disturbances and malicious attacks can lead to faults in synchronizing the clocks of two network nodes. In this article, we investigate the fundamental resilience bounds of network clock synchronization for a system of N nodes against the peer-to-peer synchronization faults. Our analysis is based on practical synchronization algorithms with time complexity down to O ( N 3 ) that attempt to correct the faults by checking the consistency among the following three types of data: (1) the estimated faults, (2) the estimated clock offsets among the nodes, and (3) the measured clock offsets from the potentially faulty peer-to-peer synchronization sessions. Our analysis gives the following three major results. First, the maximum number of faults that can be corrected by the algorithms has a tight bound of ⌊ N /2 ⌋ − 1 when every node pair performs a synchronization session. Second, by converting the fault resilience problem to a graph-theoretic edge connectivity problem ...

A Polynomial Solution to the Undirected Two Paths Problem

Journal of the ACM, 1980

P2 connecting s, with t, and s2 with t2, respectively. This problem is solved by an O(n.m) algorithm (n = IVJ, m = JElL An important by-product of the paper is a theorem stating that if G is 4-connected and nonplanar, then such paths P, and P2 exist for any choice of s,, s2, h, and t2 (as conjectured by Watkins).

Computational complexity of multicommodity flow problems with uniform assignment of flow

Electronics and Communications in Japan (Part III: Fundamental Electronic Science), 1995

This paper treats multicommodity flow problems with a flow assignment constraint. General multicommodity flow problems require both the path set in which the commodities are assigned and the volume to which the commodities are assigned for each path. However, in this paper, we treat only problems wherein paths are required. The flow is assigned to the paths uniformly. This problem appears in state-and-timedependent routing (STR), which is a kind of dynamic routing scheme in telecommunication networks. We show that this problem is NP-complete and remains NP-complete if the lengths (number of edges) of paths are restricted to less than or equal to two but can be solved in polynomial time if the path lengths are restricted to be less than or equal to two and the number of commodities is fixed.