Disjoint paths and connected subgraphs for H-free graphs

Many fundamental problems in the area of distributed systems such as security, reliable routing, network survivability and broadening available bandwidth can be addressed through the use of disjoint paths between communication endpoints. Therefore, distributed solutions to the disjoint paths problem are of crucial importance to these fundamental problems. Disjoint paths naturally exist in traditional networks; however, these networks provide no guarantees regarding the presence and the number of available disjoint paths. On the other hand, network topologies structured based on Caley graphs such as hypercube, star networks and their variations possess many desirable properties related to disjoint paths. Therefore, it is anticipated that backbones of future networks will be structured following some Caley graph topologies. In this paper, we present a simple novel stabilizing and inherently stabilizing algorithm to route messages over all node-disjoint paths of optimal length from one non-faulty process to another in at most n+1 rounds in an n-dimensional hypercube network in the presence of crash failures. The proposed algorithm can tolerate up to 2n-⌈log (n+1)⌉ process/link failures in the network and a maximum of ⌊(L-2)/3⌋ process/link failures on each disjoint path, where L is the number of processes on a disjoint path and the distance between any two failed processes/links is at least three. The proposed algorithm tolerates a large number of process and link failures, while delivering all n messages over optimal-length disjoint paths in the presence of maximum permissible process/link failures on each path. This is achieved by a simple uniform distributed algorithm using only local knowledge of failure locations.

We consider the Maximum Node Disjoint Paths (MNDP) problem in undirected graphs. The input consists of an undirected graph G = (V, E) and a collection {(s1, t1), …, (sk, tk)} of k source-sink pairs. The goal is to select a maximum cardinality subset of pairs that can be routed/connected via node-disjoint paths. A relaxed version of MNDP allows up to c paths to use a node, where c is the congestion parameter. We give a polynomial time algorithm that routes Ω(OPT/poly log k) pairs with O(1) congestion, where OPT is the value of an optimum fractional solution to a natural multicommodity flow relaxation. Our result builds on the recent breakthrough of Chuzhoy [17] who gave the first poly-logarithmic approximation with constant congestion for the Maximum Edge Disjoint Paths (MEDP) problem.

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

Consider the problem of routing from a single source node to multiple target nodes with the additional condition that these disjoint paths be the shortest. This problem is harder than the standard one-to-many routing in that such paths do not always exist. Various sufficient and necessary conditions have been found to determine when such paths exist for some interconnection networks. And when these conditions do hold, the problem of finding such paths can be reduced to the problem of finding a disjoint ordering of sets. We study the problem of finding a disjoint ordering of sets $X_1$, $X_2$, $X_s$ where $X_i \subseteq \{1, 2, \cdots, n\}$ and $s \leq n$. We present an $O(n^3)$ algorithm for doing so, under certain conditions, thus improving the previously known $O(n^4)$ algorithm, and consequently, improving the corresponding one-to-many routing algorithms for finding disjoint and shortest paths.

Automation production systems and computer integrated manufacturing: Mikell P. Groover

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, Mnich, and Spoerhase (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. (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.

Given a set of sources and a set of sinks in the two dimensional grid of size n, the disjoint paths (DP) problem is to connect every source to a distinct sink by a set of edge-disjoint paths. Let v be the total number of sources and sinks. In [3], Chan and Chin showed that without loss of generality we can assume v ≤ n ≤ 4v2. They also showed how to compress the grid optimally to a dynamic network (structure of the network may change depending on the paths found currently) of size O(√nv), and solve the problem in O(√nv3/2) time using augmenting path method in maximum flow. In this paper, we improve the time complexity of solving the DP problem to O(n3/4v3/4). The factor of improvement is as large as √v when n is Θ(v), while it is at least 4√v for n is Θ(v2).

Given a source node and a node set, the node-to-set disjoint paths problem is to find disjoint paths from the source node to all nodes from the node set. In this paper, we propose an improved algorithm for the node-to-set disjoint paths problem on bi-rotator graphs. Our result improves the complexity of previous result from O(n5) to O(n3logn), where n is the dimension of the input graph.

A new polynomial time algorithm is presented for finding two vertex-disjoint paths between two specified pairs of vertices on an undirected graph. An application of the two disjoint path algorithm to the automatic wire routing is also discussed.

Nowadays, the development of robots has created problems whose solution should be found. One of these problems is the search for disjoint paths on polygons with a minimum number of links. The article describes cases when there are several sources and targets at the polygon. The special cases of 2×2 and 3×3 links are considered. The intersection solution in the case of 2×2 is achieved by minimizing the sum of the Euclidean link lengths. If the paths with minimal Euclidean distances intersect, then there is no solution, otherwise solution exists. This method can be extended to the 2×N case. The 3×3 case is solved using an hourglass model. First and third links are taut-string links. The middle link is selected within the side walls of the hourglass. For the case of M×N links, a heuristic algorithm for finding disjoint paths is given. The article also describes an algorithm for constructing forbidden regions that are used in each case.

A rotator graph was proposed as a topology for interconnection networks of parallel computers, and it is promising because of its small diameter and small degree. However, a rotator graph is a directed graph that sometimes behaves harmful when it is applied to actual problems. A bi-rotator graph is obtained by making each edge of a rotator graph bi-directional. A bi-rotator has a Hamilton cycle and it is also pan-cyclic. In this paper, we give an algorithm for the node-to-set disjoint paths problem in bi-rotator graphs with its evaluation results. The solution achieves some fault tolerance such as file distribution based information dispersal technique. The algorithm is of polynomial order of n for an n-bi-rotator graph. It is based on recursion and divided into three cases according to the distribution of destination nodes in classes into which all the nodes in a bi-rotator graph are categorized. The sum of lengths of paths obtained and the time complexity of the algorithm are estimated. Average performance of the algorithm is also evaluated by computer experiments.

The disjoint paths problem in quadratic time

SummaryFinding link/node‐disjoint paths between a pair of nodes is capable of providing Quality of Service and reliable routing which is very critical for mesh‐connected Network‐on‐Chip. State‐of‐art works usually aim at random topologies and multiple constraints. Therefore, it is difficult to optimize their time complexity. In this paper, MDPPIM (Manhattan‐distance‐constrained Disjoint Path Pair Problem in Incomplete Mesh) problem is presented based on Network‐on‐Chip application scenarios. Then, PCDP (Path‐Counting Disjoint Path) algorithm is proposed to solve the MDPPIM problem with low time complexity. Compared with previous disjoint path algorithms, PCDP algorithm does not have to use Dijkstra's algorithm to find a shortest path. It is optimized according to the feature of incomplete mesh such as the regularity of mesh and Manhattan‐distance constraint. Therefore, it is with low time complexity. Numerical results demonstrate the proposed PCDP algorithm's effectiveness and low time complexity.

The metacube interconnection network introduced a few years ago has some very interesting properties: it has a short diameter similar to the hypercube, and its degree is much lower than that of a hypercube of the same size. In this paper, we describe an efficient algorithm for finding disjoint paths between one source node and at most m+k target nodes in a metacube MC(k, m) excluding MC(*, 1), MC(2, 2), MC(3, 2) and MC(3, 3). We show that we can find m+k disjoint paths between the source node and the m+k targets of length at most metacube diameter plus (k + 4) with time complexity of order of metacube degree times its diameter.

Given four distinct vertices s1,s2,t1, and t2 of a graph G, the 2-disjoint paths problem is to determine two disjoint paths, p1 from s1 to t1 and p2 from s2 to t2, if such paths exist. Disjoint can mean vertex- or edge-disjoint. Both, the edge- and the vertex-disjoint version of the problem, are NP-hard in the case of directed graphs. For undirected graphs, we show that the O(mn)-time algorithm of Shiloach can be modified to solve the 2-vertex-disjoint paths problem in only O(n + mα(m,n)) time, where m is the number of edges in G, n is the number of vertices in G, and where α denotes the inverse of the Ackermann function. Our result also improves the running time for the 2-edge-disjoint paths problem on undirected graphs as well as the running times for the 2-vertex- and the 2-edge-disjoint paths problem on dags.