Vertex-disjoint Paths Problem Research Articles

Max Flows in Planar Graphs with Vertex Capacities

We consider the maximum flow problem in directed planar graphs with capacities on both vertices and arcs and with multiple sources and sinks. We present three algorithms when the capacities are integers. The first algorithm runs in O ( min { k 2 n , n log 3 n + kn }) time when all capacities are bounded by a constant, where n is the number of vertices in the graph, and k is the number of terminals. This algorithm is the first to solve the vertex-disjoint paths problem in linear time when k is fixed but larger than 2. The second algorithm runs in O ( k 5 Δ n polylog ( nU )) time, where each arc capacity and finite vertex capacity is bounded by U , and Δ is the maximum degree of the graph. Finally, when k = 3, we present an algorithm that runs in O ( n log n ) time; this algorithm works even when the capacities are arbitrary reals. Our algorithms improve on the fastest previously known algorithms when k and Δ are fixed and U is bounded by a polynomial in n . Prior to this result, the fastest algorithms ran in O ( n 4/3+ o (1) ) time for unit capacities; in the smallest of O ( n 3/2 log n log U ), Õ( n 10/7 U 1/7 ), O ( n 11/8+o(1) U 1/4 ), and O ( n 4/3 + o(1) U 1/3 ) time for integer capacities; and in O ( n 2 /log n ) time for real capacities, even when k = 3.

Physical Zero-Knowledge Proof for Numberlink Puzzle and k Vertex-Disjoint Paths Problem

Numberlink is a logic puzzle with an objective to connect all pairs of cells with the same number by non-crossing paths in a rectangular grid. In this paper, we propose a physical protocol of zero-knowledge proof for Numberlink using a deck of cards, which allows a prover to convince a verifier that he/she knows a solution without revealing it. In particular, the protocol shows how to physically count the number of elements in a list that are equal to a given secret value without revealing that value, the positions of elements in the list that are equal to it, or the value of any other element in the list. Finally, we show that our protocol can be modified to verify a solution of the well-known $k$ vertex-disjoint paths problem, both the undirected and directed settings.

Selecting vertex disjoint paths in plane graphs

We study variants of the vertex disjoint paths problem in plane graphs where paths have to be selected from given sets of paths. We investigate the problem as a decision, maximization, and routing-in-rounds problem. Although all considered variants are NP-hard in planar graphs, restrictions on the locations of the terminals on the outer face of the given planar embedding of the graph lead to polynomially solvable cases for the decision and maximization versions of the problem. For the routing-in-rounds problem, we obtain a p-approximation algorithm, where p is the maximum number of alternative paths for a terminal pair, when restricting the locations of the terminals to the outer face such that they appear in a counterclockwise traversal of the boundary as a sequence s1,s2,',sk,tπ1,tπ2,',tπk for some permutation π. © 2015 Wiley Periodicals, Inc.NETWORKS, Vol. 662, 136-144 2015

A Proposal for Reliable Routing in Communication Network

Multipath routing plays an important role in communication networks. It 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 built model for vertex-disjoint paths problem and then proposed algorithm for finding the shortest pair of vertex-disjoint paths. We consider in this paper the shortest-path problem in networks in which the length (or weight) of the edges are considered. We present algorithms for finding the shortest-path. In more restricted transit. Furthermore, we can extend this algorithm to find any k disjoint paths whose sum-weight is minimized.

Improved algorithms for finding length-bounded two vertex-disjoint paths in a planar graph and minmax k vertex-disjoint paths in a directed acyclic graph

This paper is composed of two parts. In the first part, an improved algorithm is presented for the problem of finding length-bounded two vertex-disjoint paths in an undirected planar graph. The presented algorithm requires O ( n 3 b min ) time and O ( n 2 b min ) space, where b min is the smaller of the two given length bounds. In the second part of this paper, we consider the minmax k vertex-disjoint paths problem on a directed acyclic graph, where k ⩾ 2 is a constant. An improved algorithm and a faster approximation scheme are presented. The presented algorithm requires O ( n k + 1 M k − 1 ) time and O ( n k M k − 1 ) space, and the presented approximation scheme requires O ( ( 1 / ϵ ) k − 1 n 2 k log k − 1 M ) time and O ( ( 1 / ϵ ) k − 1 n 2 k − 1 log k − 1 M ) space, where ϵ is the given approximation parameter and M is the length of the longest path in an optimal solution.

Vertex disjoint paths on clique-width bounded graphs

We show that the l vertex disjoint paths problem between l pairs of vertices can be solved in linear time for co-graphs but is NP-complete for graphs of clique-width at most 6 and NLC-width at most 4. The NP-completeness follows from the fact that the line graph of a graph of tree-width k has clique-width at most 2 k + 2 and NLC-width at most k + 2 , and a result by Nishizeki et al. [The edge-disjoint paths problem is NP-complete for series-parallel graphs, Discrete Appl. Math. 115 (2001) 177–186]. The vertex disjoint paths problem is the first graph problem shown to be NP-complete on graphs of bounded clique-width but solvable in linear time on co-graphs and graphs of bounded tree-width. Additionally, we show that the r vertex disjoint paths problem between each of l pairs of vertices can be solved in polynomial time for co-graphs, if l is given to the input, and for graphs of bounded clique-width, if l is fixed.

On the Inapproximability of Disjoint Paths and Minimum Steiner Forest with Bandwidth Constraints

In this paper, we study the inapproximability of several well-known optimization problems in network optimization. We show-that the max directed vertex-disjoint paths problem cannot be approximated within ratio 2log1−εn unless NP⊆DTIME[2polylogn], the max directed edge-disjoint paths problem cannot be approximated within ratio 2log1−εn unless NP⊆DTIME[2polylogn], the integer multicommodity flow problem in directed graphs cannot be approximated within ratio 2log1−εn unless NP⊆DTIME[2polylogn], the max undirected vertex-disjoint paths problem does not have a polynomial time approximation scheme unless P=NP, and the minimum Steiner forest with bandwidth constraints problem cannot be approximated within ratio exp(poly(n)) unless P=NP.

On the complexity of the disjoint paths problem

In this paper we consider the disjoint paths problem. Given a graphG and a subsetS of the edge-set ofG the problem is to decide whether there exists a family ℱ of disjoint circuits inG each containing exactly one edge ofS such that every edge inS belongs to a circuit inC. By a well-known theorem of P. Seymour the edge-disjoint paths problem is polynomially solvable for Eulerian planar graphsG. We show that (assumingP≠NP) one can drop neither planarity nor the Eulerian condition onG without losing polynomial time solvability. We prove theNP-completeness of the planar edge-disjoint paths problem by showing theNP-completeness of the vertex disjoint paths problem for planar graphs with maximum vertex-degree three. This disproves (assumingP≠NP) a conjecture of A. Schrijver concerning the existence of a polynomial time algorithm for the planar vertex-disjoint paths problem. Furthermore we present a counterexample to a conjecture of A. Frank. This conjecture would have implied a polynomial algorithm for the planar edge-disjoint paths problem. Moreover we derive a complete characterization of all minorclosed classes of graphs for which the disjoint paths problem is polynomially solvable. Finally we show theNP-completeness of the half-integral relaxation of the edge-disjoint paths problem. This implies an answer to the long-standing question whether the edge-disjoint paths problem is polynomially solvable for Eulerian graphs.

General vertex disjoint paths in series-parallel graphs

Let G = ( V, E ) be an undirected graph and let ( s i , t i ), 1 ≤ i ≤ k be k pairs of vertices in G . The vertex disjoint paths problem is to find k paths P 1 ,…, P k such that P i connects s i and t i and any two of these paths may intersect only at a common endpoint. This problem is NP-complete even for planar graphs. Robertson and Seymour proved that when k is a fixed integer this problem becomes polynomial. We present a linear time algorithm for solving the decision version of the general problem when the input graph is a series-parallel graph.