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 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 (assumingPNP) 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 (assumingPNP) 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.Supported by Sonderforschungsbereich 303 (DFG)     

On the disjoint paths problem

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 approximation for the edge-disjoint paths problem in undirected graphs, directed acyclic graphs and directed graphs with edge capacity at least 2.     

Path problems in generalized stars, complete graphs, and brick wall graphs

Path problems such as the maximum edge-disjoint paths problem, the path coloring problem, and the maximum path coloring problem are relevant for resource allocation in communication networks, in particular all-optical networks. In this paper, it is shown that maximum path coloring can be solved optimally in polynomial time for bidirected generalized stars, even in the weighted case. Furthermore, the maximum edge-disjoint paths problem is proved NP-hard for complete graphs (undirected or bidirected), a constant-factor off-line approximation algorithm is presented for the weighted case, and an on-line algorithm with constant competitive ratio is given for the unweighted case. Finally, an open problem concerning the existence of routings that simultaneously minimize the maximum load and the number of colors is solved: an example for a graph and a set of requests is given such that any routing that minimizes the maximum load requires strictly more colors for path coloring than a routing that minimizes the number of colors.     

The hardness of routing two pairs on one face

We prove the NP-completeness of the integer multiflow problem in planar graphs, with the following restrictions: there are only two classes of parallel demand edges, both lying on the infinite face of the routing graph. This was one of the open challenges concerning disjoint paths, explicitly asked by Müller (Math Program 105 (2–3):275–288, 2006). It also strengthens Schw?rzler’s recent proof of one of the open problems of Schrijver’s book (Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin 2003), about the complexity of the edge-disjoint paths problem with terminals on the outer boundary of a planar graph. We also give a directed acyclic reduction. This proves that the arc-disjoint paths problem is NP-complete in directed acyclic graphs, even with only two classes of demand arcs.     

Covering partially directed graphs with directed paths

We consider graphs which contain both directed and undirected edges (partially directed graphs). We show that the problem of covering the edges of such graphs with a minimum number of edge-disjoint directed paths respecting the orientations of the directed edges is polynomially solvable. We exhibit a good characterization for this problem in the form of a min-max theorem. We introduce a more general problem including weights on possible orientations of the undirected edges. We show that this more general weighted formulation is equivalent to the weighted bipartite b-factor problem. This implies the existence of a strongly polynomial algorithm for this weighted generalization of Euler's problem to partially directed graphs (compare this with the negative results for the mixed Chinese postman problem). We also provide a compact linear programming formulation for the weighted generalization that we propose.     

Edge disjoint paths and max integral multiflow/min multicut theorems in planar graphs

We generalize all the results obtained for integer multiflow and multicut problems in trees by Garg et al. [N. Garg, V.V. Vazirani and M. Yannakakis, Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18 (1997) 3–20] to planar graphs with a fixed number of faces, although other classical generalizations do not lead to such results. We also introduce the class of k-edge-outerplanar graphs and bound the integrality gap for the maximum edge-disjoint paths problem in these graphs.     

Disjoint paths in sparse graphs

We generalize all the results obtained for maximum integer multiflow and minimum multicut problems in trees by Garg, Vazirani and Yannakakis [N. Garg, V.V. Vazirani, M. Yannakakis, Primal-dual approximation algorithms for integral flow and multicut in trees, Algorithmica 18 (1997) 3–20] to graphs with a fixed cyclomatic number, while this cannot be achieved for other classical generalizations of trees. We also introduce thek-edge-outerplanar graphs, a class of planar graphs with arbitrary (but bounded) tree-width that generalizes the cacti, and show that the integrality gap of the maximum edge-disjoint paths problem is bounded in these graphs.     

A well-quasi-order for tournaments

A digraph H is immersed in a digraph G if the vertices of H are mapped to (distinct) vertices of G, and the edges of H are mapped to directed paths joining the corresponding pairs of vertices of G, in such a way that the paths are pairwise edge-disjoint. For graphs the same relation (using paths instead of directed paths) is a well-quasi-order; that is, in every infinite set of graphs some one of them is immersed in some other. The same is not true for digraphs in general; but we show it is true for tournaments (a tournament is a directed complete graph).     

The Klein bottle and multicommodity flows

LetG be an eulerian graph embedded on the Klein bottle. Then the maximum number of pairwise edge-disjoint orientation-reversing circuits inG is equal to the minimum number of edges intersecting all orientation-reversing circuits. This generalizes a theorem of Lins for the projective plane. As consequences we derive results on disjoint paths in planar graphs, including theorems of Okamura and of Okamura and Seymour.     

The reduction of graph families closed under contraction

Let be a family of graphs. Suppose there is a nontrivial graph H such that for any supergraph G of H, G is in if and only if the contraction G/H is in . Examples of such an : graphs with a spanning closed trail; graphs with at least k edge-disjoint spanning trees; and k-edge-connected graphs (k fixed). We give a reduction method using contractions to find when a given graph is in and to study its structure if it is not in . This reduction method generalizes known special cases.     

Path and cycle decompositions of complete equipartite graphs: 3 and 5 parts

In 1998 Cavenagh [N.J. Cavenagh, Decompositions of complete tripartite graphs into k-cycles, Australas. J. Combin. 18 (1998) 193-200] gave necessary and sufficient conditions for the existence of an edge-disjoint decomposition of a complete equipartite graph with three parts, into cycles of some fixed length k. Here we extend this to paths, and show that such a complete equipartite graph with three partite sets of size m, has an edge-disjoint decomposition into paths of length k if and only if k divides 3m² and k<3m. Further, extending to five partite sets, we show that a complete equipartite graph with five partite sets of size m has an edge-disjoint decomposition into cycles (and also into paths) of length k with k?3 if and only if k divides 10m² and k?5m for cycles (or k<5m for paths).     

Maximum edge-disjoint paths in planar graphs with congestion 2

Mathematical Programming - We study the maximum edge-disjoint path problem (medp) in planar graphs $$G=(V,E)$$ with edge capacities u(e). We are given a set of terminal pairs $$s_it_i$$ , $$i=1,2...     

Lambda-fold 2-perfect 6-cycle systems in equipartite graphs

A 6-cycle system of a graph G is an edge-disjoint decomposition of G into 6-cycles. Graphs G, for which necessary and sufficient conditions for existence of a 6-cycle system have been found, include complete graphs and complete equipartite graphs. A 6-cycle system of G is said to be 2-perfect if the graph formed by joining all vertices distance 2 apart in the 6-cycles is again an edge-disjoint decomposition of G, this time into 3-cycles, since the distance 2 graph in any 6-cycle is a pair of disjoint 3-cycles.Necessary and sufficient conditions for existence of 2-perfect 6-cycle systems of both complete graphs and complete equipartite graphs are known, and also of λ-fold complete graphs. In this paper, we complete the problem, giving necessary and sufficient conditions for existence of λ-fold 2-perfect 6-cycle systems of complete equipartite graphs.     

A note on packing paths in planar graphs

Seymour (1981) proved that the cut criterion is necessary and sufficient for the solvability of the edge-disjoint paths problem when the union of the supply graph and the demand graph is planar and Eulerian. When only planarity is required, Middendorf and Pfeiffer (1993) proved the problem to be NP-complete. For this case, Korach and Penn (1992) proved that the cut criterion is sufficient for the existence of a near-complete packing of paths. Here we generalize this result by showing how a natural strengthening of the cut criterion yields better packings of paths.Analogously to Seymour's approach, we actually prove a theorem on packing cuts in an arbitrary graph and then the planar edge-disjoint paths case is obtained by planar dualization. The main result is derived from a theorem of Seb (1990) on the structure of ±1 weightings of a bipartite graph with no negative circuits.Research partially supported by the Hungarian National Foundation for Scientific Research Grants OTKA 2118 and 4271.     

Lower bounds on two-terminal network reliability

Computing the probability that two nodes in a probabilistic network are connected is a well-known computationally difficult problem. Two strategies are devised for obtaining lower bounds on the connection probability for two terminals. The first improves on the Kruskal-Katona bound by using efficient computations of small pathsets. The second strategy employs efficient algorithms for finding edge-disjoint paths. The resulting bounds are compared; while the edge-disjoint path bounds typically outperform the Kruskal-Katona bounds, they do not always do so. Finally, a method is outlined for developing a uniform bound which combines both strategies.     

On a packing problem for infinite graphs and independence spaces

In this paper several infinite extensions of the well-known results for packing bases in finite matroids are considered. A counterexample is given to a conjecture of Nash-Williams on edge-disjoint spanning trees of countable graphs, and a sufficient condition is proved for the packing problem in independence spaces over a countably infinite set.     

On the complexity of the planar edge-disjoint paths problem with terminals on the outer boundary

It is shown that both the undirected and the directed edge-disjoint paths problem are NP-complete, if the supply graph is planar and all edges of the demand graph are incident with vertices lying on the outer boundary of the supply graph. In the directed case, the problem remains NP-complete, if in addition the supply graph is acyclic. The undirected case solves open problem no. 56 of A. Schrijver’s book Combinatorial Optimization.     

On a problem of a. kotzig concerning factorizations of 4-regular graphs

The paper gives a solution of a problem of A. Kotzǐg. This problem concerns the 4-regular graphs G with the property that in every decomposition of G into two edge-disjoint 2-regular factors at least one factor is a Hamiltonian circuit in G.     

Small Edge Sets Meeting all Triangles of a Graph

It was conjectured in 1981 by the third author that if a graph G does not contain more than t pairwise edge-disjoint triangles, then there exists a set of at most 2t edges that shares an edge with each triangle of G. In this paper, we prove this conjecture for odd-wheel-free graphs and for ‘triangle-3-colorable’ graphs, where the latter property means that the edges of the graph can be colored with three colors in such a way that each triangle receives three distinct colors on its edges. Among the consequences we obtain that the conjecture holds for every graph with chromatic number at most four. Also, two subclasses of K ₄-free graphs are identified, in which the maximum number of pairwise edge-disjoint triangles is equal to the minimum number of edges covering all triangles. In addition, we prove that the recognition problem of triangle-3-colorable graphs is intractable.     

Minimum Dominating Trail Set for Two-Terminal Series Parallel Graphs

Given a graph G, the Minimum Dominating Trail Set (MDTS) problem consists in finding a minimum cardinality collection of pairwise edge-disjoint trails such that each edge of G has at least one endvertex on some trail. The MDTS problem is NP–hard for general graphs. In this paper an algorithmic approach for the MDTS problem on (two-terminal) series parallel graphs is presented.