Disjoint Paths Problem Research Articles

Node-to-Node and Node-to-Set Disjoint Paths Problems in Bicubes

Node-to-Node and Node-to-Set Disjoint Paths Problems in Bicubes

Set-to-set disjoint paths in a folded hypercube

The hypercube is a very popular topology for the interconnection networks of parallel systems, and the folded hypercube is a variant of the hypercube. The folded hypercube attains much higher performance by introducing one additional link to each processing element. Therefore, there are many research activities regarding the folded hypercube. We focus on this topology and address an unresolved problem, that is, the set-to-set disjoint paths problem in it. In this paper, we show an algorithm that solves the problem in a folded hypercube in polynomial time. We prove the correctness of the algorithm. Moreover, we show that the time complexity of the algorithm is O(ν3log⁡ν) and the maximum length of the paths is 2ν+2 if the algorithm is applied to a ν-dimensional folded hypercube.

Node-to-Set Disjoint Paths Problem in Cross-Cubes

Node-to-Set Disjoint Paths Problem in Cross-Cubes

Almost disjoint paths and separating by forbidden pairs

By Menger's theorem, the maximum number of arc-disjoint paths from a vertex s to a vertex t in a directed graph equals the minimum number of arcs needed to disconnect s and t, i.e., the minimum size of an s-t-cut. The max-flow problem in a network with unit capacities is equivalent to the arc-disjoint paths problem. Moreover, the max-flow and min-cut problems form a strongly dual pair. We relax the disjointedness requirement on the paths, allowing them to be almost disjoint, meaning they may share up to one arc. The resulting almost disjoint paths problem (ADP) asks for k s-t-paths such that any two of them are almost disjoint. The separating by forbidden pairs problem (SFP) is the corresponding dual problem and calls for a set of k arc pairs such that every s-t-path contains both arcs of at least one such pair.In this paper, we explore these two problems, showing that they have an unbounded duality gap in general and analyzing their complexity. We prove that ADP is NP-complete when k is part of the input and that SFP is Σ2P-complete, even for acyclic graphs. Furthermore, we efficiently solve ADP when k≤2 is fixed and present a polynomial time algorithm based on dynamic programming for ADP when k is constant.

Excluding a planar matching minor in bipartite graphs

The notion of matching minors is a specialisation of minors fit for the study of graphs with perfect matchings. Matching minors have been used to give a structural description of bipartite graphs on which the number of perfect matchings can be computed efficiently, based on a result of Little, by McCuaig et al. in 1999.In this paper we generalise basic ideas from the graph minor series by Robertson and Seymour to the setting of bipartite graphs with perfect matchings. We introduce a version of Erdős-Pósa property for matching minors and find a direct link between this property and planarity. From this, it follows that a class of bipartite graphs with perfect matchings has bounded perfect matching width if and only if it excludes a planar matching minor. We also present algorithms for bipartite graphs of bounded perfect matching width for a matching version of the disjoint paths problem, matching minor containment, and for counting the number of perfect matchings. From our structural results, we obtain that recognising whether a bipartite graph G contains a fixed planar graph H as a matching minor, and that counting the number of perfect matchings of a bipartite graph that excludes a fixed planar graph as a matching minor are both polynomial time solvable.

First-order Logic with Connectivity Operators

First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem parameterized by solution size. However, FO cannot express the very simple algorithmic question whether two vertices are connected. We enrich FO with connectivity predicates that are tailored to express algorithmic graph problems that are commonly studied in parameterized algorithmics. By adding the atomic predicates conn k ( x,y,z_1,..., z k ) that hold true in a graph if there exists a path between (the valuations of) x and y after (the valuations of) z 1 ,..., z k have been deleted, we obtain separator logic FO + conn. We show that separator logic can express many interesting problems, such as the feedback vertex set problem and elimination distance problems to first-order definable classes. Denote by FO + conn k the fragment of separator logic that is restricted to connectivity predicates with at most k + 2 variables (that is, at most k deletions), we show that FO + conn k + 1 is strictly more expressive than FO + conn k for all k ≥ 0 . We then study the limitations of separator logic and prove that it cannot express planarity, and, in particular, not the disjoint paths problem. We obtain the stronger disjoint-paths logic FO + DP by adding the atomic predicates disjoint-paths k [( x 1 , y 1 ),..., ( x k , y k ) that evaluate to true if there are internally vertex-disjoint paths between (the valuations of) x i and y i for all 1 ≤ i ≤ k . Disjoint-paths logic can express the disjoint paths problem, the problem of (topological) minor containment, the problem of hitting (topological) minors, and many more. Again, we show that the fragments FO + DP k that use predicates for at most k disjoint paths form a strict hierarchy of expressiveness. Finally, we compare the expressive power of the new logics with that of transitive-closure logics and monadic second-order logic.

Induced Disjoint Paths and Connected Subgraphs for H-Free Graphs

Paths P^1,ldots ,P^k in a graph G=(V,E) are mutually induced if any two distinct P^i and P^j have neither common vertices nor adjacent vertices. The Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices (s_i,t_i) contains k mutually induced paths P^i such that each P^i starts from s_i and ends at t_i. This is a classical graph problem that is NP-complete even for k=2. We introduce a natural generalization, Induced Disjoint Connected Subgraphs: instead of connecting pairs of terminals, we must connect sets of terminals. We give almost-complete dichotomies of the computational complexity of both problems for H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. Finally, we give a complete classification of the complexity of the second problem if the number k of terminal sets is fixed, that is, not part of the input.

The widestk-set of disjoint paths problem

Finding disjoint and widest paths are key problems in telecommunication networks. In this paper, we study the Widestk-set of Disjoint Paths Problem (WKDPP), anNP-Hard optimization problem that considers both aspects. Given a digraphG = (N,A), WKDPP consists of computingkarc-disjoint paths between two nodes such that the sum of its minimum capacity arcs is maximized. We present three mathematical formulations for WKDPP, a symmetry-breaking inequality set, and propose two heuristic algorithms. Computational experiments compare the proposed heuristics with another from the literature to show the effectiveness of the proposed methods.

Few induced disjoint paths for H-free graphs

Paths P1,…,Pk in a graph G=(V,E) are mutually induced if any two distinct Pi and Pj have neither common vertices nor adjacent vertices. For a fixed integer k, the k-Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices (si,ti) contains k mutually induced paths Pi such that each Pi starts from si and ends at ti. Whereas the non-induced version is well-known to be polynomial-time solvable for every fixed integer k, a classical result from the literature states that even 2-Induced Disjoint Paths is NP-complete. We prove new complexity results for k-Induced Disjoint Paths if the input is restricted to H-free graphs, that is, graphs without a fixed graph H as an induced subgraph. We compare our results with a complexity dichotomy for Induced Disjoint Paths, the variant where k is part of the input.

Maximum weight disjoint paths in outerplanar graphs via single-tree cut approximators

Since 1997 there has been a steady stream of advances for the maximum disjoint paths problem. Achieving tractable results has usually required focusing on relaxations such as: (i) to allow some bounded edge congestion in solutions, (ii) to only consider the unit weight (cardinality) setting, (iii) to only require fractional routability of the selected demands (the all-or-nothing flow setting). For the general form (no congestion, general weights, integral routing) of edge-disjoint paths (edp) even the case of unit capacity trees which are stars generalizes the maximum matching problem for which Edmonds provided an exact algorithm. For general capacitated trees, Garg, Vazirani, Yannakakis showed the problem is APX-Hard and Chekuri, Mydlarz, Shepherd provided a 4-approximation. This is essentially the only setting where a constant approximation is known for the general form of edp. We extend their result by giving a constant-factor approximation algorithm for general-form edp in outerplanar graphs. A key component for the algorithm is to find a single-tree O(1) cut approximator for outerplanar graphs. Previously O(1) cut approximators were only known via distributions on trees and these were based implicitly on the results of Gupta, Newman, Rabinovich and Sinclair for distance tree embeddings combined with results of Anderson and Feige.

Set-to-Set Disjoint Paths Problem in Möbius Cubes

The set-to-set disjoint paths problem is to find <i>n</i> vertex-disjoint paths <i>s_i</i> &#x021DD; <i>t_{j_i}</i> (1 &#x2264; <i>i</i> &#x2264; <i>n</i>, &#x007B;<i>j</i>₁,<i>j</i>₂,...,<i>j</i>_<i>n</i>&#x007D; = &#x007B;1,2,...,<i>n</i>&#x007D;) between two sets of vertices <i>S</i> = &#x007B;<i>s</i>₁,<i>s</i>₂,...,<i>s_n</i>&#x007D; and <i>T</i> = &#x007B;<i>t</i>₁,<i>t</i>₂,...,<i>tn</i>&#x007D; in a graph whose connectivity is equal to <i>n</i>. It is a very important problem as well as the vertex-to-vertex disjoint paths problem and the vertex-to-set disjoint paths problem. In this paper, we propose an algorithm that generates <i>n</i> vertex-disjoint paths between two vertex sets in an <i>n</i>-M&#x00F6;bius cube. Much attention has been attracted by a M&#x00F6;bius cube because its diameter is almost half of that of a hypercube while it can interconnect the same number of vertices as the hypercube. We also give a proof of correctness of the algorithm and estimate that the time complexity of the algorithm is <i>O</i>(<i>n</i>⁶) and the maximum length of the paths generated by the algorithm is 2<i>n</i> - 2.

A relaxation of the Directed Disjoint Paths problem: A global congestion metric helps

In the Directed Disjoint Paths problem, we are given a digraph D and a set of requests {(s1,t1),…,(sk,tk)}, and the task is to find a collection of pairwise vertex-disjoint paths {P1,…,Pk} such that each Pi is a path from si to ti in D. This problem is NP-complete for fixed k=2 and W[1]-hard with parameter k in DAGs. A few positive results are known under restrictions on the input digraph, such as being planar or having bounded directed tree-width, or under relaxations of the problem, such as allowing for vertex congestion. Positive results are scarce, however, for general digraphs. In this article we propose a novel global congestion metric for the problem: we only require the paths to be “disjoint enough”, in the sense that they must behave properly not in the whole graph, but in an unspecified part of size prescribed by a parameter. Namely, in the Disjoint Enough Directed Paths problem, given an n-vertex digraph D, a set of k requests, and non-negative integers d and s, the task is to find a collection of paths connecting the requests such that at least d vertices of D occur in at most s paths of the collection. We study the parameterized complexity of this problem for a number of choices of the parameter, including the directed tree-width of D. Among other results, we show that the problem is W[1]-hard in DAGs with parameter d and, on the positive side, we give an algorithm in time O(nd+2⋅kd⋅s) and a kernel of size d⋅2k−s⋅(ks)+2k in general digraphs. This latter result has consequences for the Steiner Network problem: we show that it is FPT parameterized by the number k of terminals and p, where p=n−q and q is the size of the solution.

Induced Disjoint Paths in AT-free graphs

Paths P1,…,Pk are mutually induced if any two distinct Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices (si,ti) contains k mutually induced paths Pi such that each Pi connects si and ti. This problem is NP-complete even for k=2. We prove that it can be solved in polynomial time for AT-free graphs even when k is part of the input. Consequently, the problem of deciding if an AT-free graph contains a fixed graph H as an induced topological minor admits a polynomial-time algorithm. We show that such an algorithm is essentially optimal by proving that the problem is W[1]-hard with parameter |VH|, even for a subclass of AT-free graphs, namely cobipartite graphs. We also show that the problems k-in-a-Path and k-in-a-Tree are polynomial-time solvable on AT-free graphs even if k is part of the input.

Disjoint paths and connected subgraphs for H-free graphs

The well-known Disjoint Paths problem is to decide if a graph contains k pairwise disjoint paths, each connecting a different terminal pair from a set of k distinct vertex pairs. We determine, with an exception of two cases, the complexity of the Disjoint Paths problem for H-free graphs. If k is fixed, we obtain the k-Disjoint Paths problem, which is known to be polynomial-time solvable on the class of all graphs for every k≥1. The latter does no longer hold if we need to connect vertices from terminal sets instead of terminal pairs. We completely classify the complexity of k-Disjoint Connected Subgraphs for H-free graphs, and give the same almost-complete classification for Disjoint Connected Subgraphs for H-free graphs as for Disjoint Paths. Moreover, we give exact algorithms for Disjoint Paths and Disjoint Connected Subgraphs on graphs with n vertices and m edges that have running times of O(2nn2k) and O(3nkm), respectively.

Node-to-set disjoint paths problem in cross-cubes

Node-to-set disjoint paths problem in cross-cubes

On Structural Parameterizations of the Edge Disjoint Paths Problem

In this paper we revisit 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 focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or FPT) algorithms. As our first result, we answer an open question stated in Fleszar et al. (Proceedings of the ESA, 2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain NP-hard even for treewidth two, a result by Zhou et al. (Algorithmica 26(1):3--30, 2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an FPT-algorithm has remained open since then. We show that this is highly unlikely by establishing the W[1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an FPT-algorithm for EDP by exploiting a novel structural parameter of the augmented graph.

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.

Pairwise Disjoint Paths Routing in Tori

The pairwise disjoint paths problem is to construct c disjoint paths s <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> → d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> (1 ≤ i ≤ c) between given pairs of nodes (s <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ) (1 ≤ i ≤ c) in a graph whose connectivity is no less than 2c. It is a major problem for interconnection networks, together with the node-to-node disjoint paths problem, the node-to-set disjoint paths problem, and the set-to-set disjoint paths problem. In this paper, we propose an algorithm that solves this problem for a torus. In a previous work, Bossard and Kaneko have developed an algorithm that constructs disjoint paths between c pairs of nodes in a k-ary n-dimensional torus, where c ≤ n. The time complexity of their algorithm is O(c <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> n+kcn), and the maximum path length is [k/2jn+2k(c-1). However, the algorithm proposed in this paper achieves a time complexity of O(c <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> n+kcn), and its maximum path length is [k/2jn + ([3k/21 - 2)(c - 1), which are both improvements over the previous algorithm. We also conducted an evaluation experiment to show that the average path lengths are proportional to k if n is fixed, and ton if k is fixed. The theoretical maximum path lengths were not attained by the paths constructed by our algorithm, and the average execution time was proportional to k if n is fixed, and to n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> if k is fixed. Additional experimental results show that, compared to the previous algorithm by Bossard and Kaneko, our algorithm achieves a better performance with respect to the maximum path lengths, but both algorithms achieve a similar level of performance with respect to the average maximum path lengths. Also, it is shown that the average execution time of our algorithm is about O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ), which is better than the average execution time of the previous algorithm O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) in the experimental framework.

Dichotomies for Evaluating Simple Regular Path Queries

Regular path queries (RPQs) are a central component of graph databases. We investigate decision and enumeration problems concerning the evaluation of RPQs under several semantics that have recently been considered: arbitrary paths, shortest paths, paths without node repetitions (simple paths), and paths without edge repetitions (trails). Whereas arbitrary and shortest paths can be dealt with efficiently, simple paths and trails become computationally difficult already for very small RPQs. We study RPQ evaluation for simple paths and trails from a parameterized complexity perspective and define a class of simple transitive expressions that is prominent in practice and for which we can prove dichotomies for the evaluation problem. We observe that, even though simple path and trail semantics are intractable for RPQs in general, they are feasible for the vast majority of RPQs that are used in practice. At the heart of this study is a result of independent interest: the two disjoint paths problem in directed graphs is W[1]-hard if parameterized by the length of one of the two paths.

A lower bound on the tree-width of graphs with irrelevant vertices

For their famous algorithm for the disjoint paths problem, Robertson and Seymour proved that there is a function f such that if the tree-width of a graph G with k pairs of terminals is at least f(k), then G contains a solution-irrelevant vertex (Robertson and Seymour (2012) [13]). We give a single-exponential lower bound on f. This bound even holds for planar graphs.