Inapproximability of Edge-Disjoint Paths and low congestion routing on undirected graphs (original) (raw)

Hardness of the Undirected Edge-Disjoint Paths Problem with Congestion

2005

In the Edge-Disjoint Paths problem with Congestion (EDPwC), we are given a graph with n nodes, a set of terminal pairs and an integer c. The objective is to route as many terminal pairs as possible, subject to the constraint that at most c demands can be routed through any edge in the graph. When c = 1, the problem is simply referred to as the Edge-Disjoint Paths (EDP) problem. In this paper, we study the hardness of EDPwC in undirected graphs.

Routing with congestion in acyclic digraphs

Information Processing Letters, 2019

We study the version of the k-disjoint paths problem where k demand pairs (s 1 , t 1),. .. , (s k , t k) are specified in the input and the paths in the solution are allowed to intersect, but such that no vertex is on more than c paths. We show that on directed acyclic graphs the problem is solvable in time n O(d) if we allow congestion k − d for k paths. Furthermore, we show that, under a suitable complexity theoretic assumption, the problem cannot be solved in time f (k)n o(d/ log d) for any computable function f. Digital Object Identifier 10.4230/LIPIcs...

Valid Inequalities for a Shortest-Path Routing Optimization Problem

In autonomous systems of the Internet packets are routed on shortest paths to their destinations, for example according to the ECMP principle. The problem of finding a feasible traffic routing configuration realized on paths which can be generated by a system of weights assigned to IP links is NP-hard. This problem can be formulated as a mixed-integer program and attempted with a branch-and-cut algorithm if effective cuts (valid inequalities) can be derived. In this paper we present exact and approximate LP-and MIPbased methods for generating valid inequalities that separate fractional solutions of the basic problem. Besides, a family of complementary valid inequalities, generated with a shortest-path algorithm, related to combinatorial properties of feasible traffic routes is introduced to speed up the cut generation process. Results of a numerical study illustrating computational issues are discussed.

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.

New inequalities for the General Routing Problem

European Journal of Operational Research, 1997

A large new class of valid inequalities is introduced for the General Routing Problem which properly contains the class of 'K-C constraints'. These are also valid for the Rural Postman Problem. A separation algorithm is given for a subset of these inequalities which runs in polynomial time.

Routing Symmetric Demands in Directed Minor-Free Graphs with Constant Congestion

ArXiv, 2019

The problem of routing in graphs using node-disjoint paths has received a lot of attention and a polylogarithmic approximation algorithm with constant congestion is known for undirected graphs [Chuzhoy and Li 2016] and [Chekuri and Ene 2013]. However, the problem is hard to approximate within polynomial factors on directed graphs, for any constant congestion [Chuzhoy, Kim and Li 2016]. Recently, [Chekuri, Ene and Pilipczuk 2016] have obtained a polylogarithmic approximation with constant congestion on directed planar graphs, for the special case of symmetric demands. We extend their result by obtaining a polylogarithmic approximation with constant congestion on arbitrary directed minor-free graphs, for the case of symmetric demands.

Constrained Routing Problem

International Journal of Computer Applications, 2013

In this paper, we have worked on a problem in the domain of graph theory and geometry .Objective of our problem is to find out the shortest path with forbidden pairs in a graphs. Given a graph G=(V, E) and set of pairs P={a i, b i |a i ϵ Vᴧ b i ϵ V}, we have to find out the shortest path between two given vertices s and t, s.t. a i b i both do not occur on the path for any i. We reduce SAT to this problem and thus claim that this problem is NP-hard.

Optimal algorithms for restricted single row routing problems

Some restricted single-row routing problem are considered. A graph-theoretic approach is used to obtain restricted classes of single-row routing problems. Optimal street congestion algorithms are proposed for single-row routing problems that have overlap graphs isomorphic to path, binary tree, and clique

Shortest-Path Routing in Arbitrary Networks

Journal of Algorithms, 1999

We introduce an on{line protocol which routes any set of N packets along shortest paths with congestion C and dilation D through an arbitrary network in O(C +D +log N) steps, with high probability. This time bound is optimal up to the additive log N, and it has previously only been reached for bounded{degree leveled networks. Further, we show that the above bound holds also for random routing problems with C denoting the maximum expected congestion over all links. Based on this result, we give applications for random routing in Cayley networks, general node symmetric networks, edge symmetric networks, and de Bruijn networks. Finally, we examine the problems arising when our approach is applied to routing along non{shortest paths, deterministic routing, or routing with bounded bu ers.