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

2010, Combinatorica

https://doi.org/10.1007/S00493-010-2455-9

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Abstract

In the undirected Edge-Disjoint Paths problem with Congestion (EDPwC), we are given an undirected graph with V 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,

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.

On general routing problems: Comments

Networks, 1976

The n o t e , "On General Routing Problems" by J. K. Lenstra and A . H . G . Rinnooy Kan [ I ] , c o r r e c t s two e r r o r s i n r e c e n t papers by t h e author [2,31 on General Routing Problems, b u t i t tends t o obscure t h e r e a l i s s u e i n determining t h e complexity, o r d i f f i c u l t y , i n s o l v i n g a c t u a l r o u t i n g problems. The RPP, TSP, and GRP a r e a l l polynomial complete r o u t i n g problems, and t h a t i s p r e c i s e l y why t h e y r e q u i r e branch and bound ( s u b t o u r e l i m i n a t i o n ) a l g o r i t h m s , a s given i n [ 2 1 , which a r e n o t polynomial bounded. For such problems, an important approach i s t o reduce t h e complexity o f t h e problem a s much a s p o s s i b l e . The complexity of G R P problems depends n o t only on t h e number o f odd nodes and r e q u i r e d nodes (which may, f o r example, be t h e same f o r a TSP and CPP) b u t more i m p o r t a n t l y , on t h e number of disconnected components i n t h e problem ( t h e TSP has N while t h e CPP has only o n e ) . I t i s recommended i n [21 t h a t , a s f a r a s p o s s i b l e , r e q u i r e d nodes be converted t o r eq u i r e d a r c s because t h a t reduces t h e number of disconnected components i n t h e problem (because every r e q u i r e d node i s a disconnected component, while r e q u i r e d a r c s t e n d t o be conn e c t e d ) . The theorem p r e s e n t e d i n 111 , t h a t t h e RPP i s polynomial complete, i s t r u e b u t misleading. The RPP i s fundament a l l y more d i f f i c u l t than t h e CPP because it has disconnected components. The theorem i s obvious, s i n c e any TSP can be conv e r t e d t o an e q u i v a l e n t RPP by expanding any TSP node j t o a r e q u i r e d arc ( j , j') where j ' i s an a r t i f i c i a l d u p l i c a t e of j . Of course, t h i s t r a n s f o r m a t i o n from TSP t o RPP i s of no b e n e f i t , except a 2-matching problem on N nodes i s converted t o an e q u i v a l e n t 1-matching problem on 2 N nodes. On t h e o t h e r hand, an RPP w i t h only 2 disconnected components i s n o t much more d i f f i c u l t t o s o l v e than a CPP ( a t most one branch i n t h e branch and bound procedure i s r e q u i r e d ) . I n complexity o r d i f f i c u l t y , t h e RPP is somewhere between t h e CPP and t h e TSP, where t h e c r i t i c a l f a c t o r i s t h e number of disconnected components. Networks, 6: 281-284

Inapproximability Bounds for Shortest-Path Network Interdiction Problems

2005

We consider two network interdiction problems: one where a network user tries to traverse a network from a starting vertex s to a target vertex t along the shortest path while an interdictor tries to eliminate all short s-t paths by destroying as few vertices (arcs) as possible, and one where the network user, as before, tries to traverse the network from s to t along the shortest path while the interdictor tries to destroy a fixed number of vertices (arcs) so as to cause the biggest increase in the shortest s-t path. The latter problem is known as the Most Vital Vertices (Arcs) Problem. In this paper we provide inapproximability bounds for several variants of these problems.

A linear programming formulation of Mader's edge-disjoint paths problem

Journal of Combinatorial Theory, Series B, 2006

We give a dual pair of linear programs for a min-max result of Mader describing the maximum number of edge-disjoint T-paths in a graph G = (V , E) with T ⊆ V . We conclude that there exists a polynomial-time algorithm (based on the ellipsoid method) for finding the maximum number of Tpaths in a capacitated graph, where the number of T-paths using an edge does not exceed the capacity of that edge.

Multicommodity flow problems with a bounded number of paths: A flow deviation approach

Networks, 2007

We propose a modified version of the Flow Deviation method of Fratta, Gerla and Kleinrock to solve multicommodity problems with minimal congestion and a bounded number of active paths. We discuss the approximation of the min-max objective function by a separable convex potential function and give a mixed-integer non linear model for the constrained routing problem. A heuristic control of the path generation is then embedded in the original algorithm based on the concepts of cleaning the quasi inactive paths and the reduction of the flow width for each commodity. Numerical experiments show the validity of the approach for realistic medium-size networks associated with routing problems in broadband communication networks with multiple protocols and label-switched paths (MPLS).

A Simple Approximation for a Hard Routing Problem

2018

We consider a routing problem which plays an important role in several applications, primarily in communication network planning and VLSI layout design. The original underlying graph algorithmic task is called Disjoint Paths problem. In most applications, one can encounter its capacitated generalization, which is known as the Unsplitting Flow problem. These algorithmic tasks are very hard in general, but various efficient (polynomial-time) approximate solutions are known. Nevertheless, the approximations tend to be rather complicated, often rendering them impractical in large, complex networks. Our goal is to present a solution that provides a simple, efficient algorithm for the unsplittable flow problem in large directed graphs. The simplicity is achieved by sacrificing a small part of the solution space. This also represents a novel paradigm of approximation: rather than giving up finding an exact solution, we restrict the solution space to its most important subset and exclude th...

Lower bounds for the mixed capacitated arc routing problem

Computers & Operations Research, 2010

Capacitated Arc Routing Problems (CARP) arise in distribution or collecting problems where activities are performed by vehicles, with limited capacity, and are continuously distributed along some pre-defined links of a network. The CARP is defined either as an undirected problem or as a directed problem depending on whether the required links are undirected or directed. The Mixed Capacitated Arc Routing Problem (MCARP) models a more realistic scenario since it considers directed as well as undirected required links in the associated network. We present a compact flow based model for the MCARP. Due to its large number of variables and constraints, we have created an aggregated version of the original model. Although this model is no longer valid, we show that it provides the same linear programming bound than the original model. Different sets of valid inequalities are also derived. The quality of the models is tested on benchmark instances with quite promising results.

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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...

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.

Edge-disjoint paths revisited

ACM Transactions on Algorithms, 2007

The approximability of the maximum edge-disjoint paths problem (EDP) in directed graphs was seemingly settled by an Ω( m 1/2 - ϵ)-hardness result of Guruswami et al. [2003], and an O (√ m ) approximation achievable via a natural multicommodity-flow-based LP relaxation as well as a greedy algorithm. Here m is the number of edges in the graph. We observe that the Ω( m 1/2 - ϵ)-hardness of approximation applies to sparse graphs, and hence when expressed as a function of n , that is, the number of vertices, only an Ω( n 1/2 - ϵ)-hardness follows. On the other hand, O (√ m )-approximation algorithms do not guarantee a sublinear (in terms of n ) approximation algorithm for dense graphs. We note that a similar gap exists in the known results on the integrality gap of the flow-based LP relaxation: an Ω(√ n ) lower bound and O (√ m ) upper bound. Motivated by this discrepancy in the upper and lower bounds, we study algorithms for EDP in directed and undirected graphs and obtain improved appr...

Degree-constrained network flows

Proceedings of the thirty-ninth annual ACM symposium on Theory of computing - STOC '07, 2007

A d-furcated flow is a network flow whose support graph has maximum outdegree d. Take a single-sink multicommodity flow problem on any network and with any set of routing demands. Then we show that the existence of feasible fractional flow with node congestion one implies the existence of a d-furcated flow with congestion at most 1 + 1 d−1 , for d ≥ 2. This result is tight, and so the congestion gap for d-furcated flows is bounded and exactly equal to 1 + 1 d−1 . For the case d = 1 (confluent flows), it is known that the congestion gap is unbounded, namely Θ(log n). Thus, allowing single-sink multicommodity network flows to increase their maximum outdegree from one to two virtually eliminates this previously observed congestion gap.

Solving the edge-disjoint paths problem using a two-stage method

International Transactions in Operational Research

There exists a wide variety of network problems where several connection requests occur simultaneously. In general, each request is attended by finding a route in the network, where the origin and destination of such a route are those hosts that wish to establish a connection for information exchange. As is well documented in the related literature, the exchange of information through disjoint routes increases the effective bandwidth, velocity, and the probability of receiving the corresponding information. The definition of disjoint paths may refer to nodes, edges, or both. One of the most studied variants is the one where disjointness implies not to share edges. This optimization problem is usually known as the maximum edge-disjoint paths problem. This N Phard optimization problem has applications in real-time communications, very large scale integration design, scheduling, bin packing, or load balancing. The proposed approach hybridizes an integer linear programming formulation of the problem with an evolutionary algorithm. Empirical results using 174 previously reported instances show that the proposed procedure compares favorably to previous metaheuristics for this problem. We confirm the significance of the results by conducting nonparametric statistical tests.

Routing-Proofness in Congestion-Prone Networks

Games

We consider the problem of sharing the cost of connecting a large number of atomless agents in a network. The centralized agency elicits the target nodes that agents want to connect, and charges agents based on their demands. We look for a cost-sharing mechanism that satisfies three desirable properties: efficiency which charges agents based on the minimum total cost of connecting them in a network, stand-alone core stability which requires charging agents not more than the cost of connecting by themselves directly, and limit routing-proofness which prevents agents from profitable reporting as several agents connecting from A to C to B instead of A to B. We show that these three properties are not always compatible for any set of cost functions and demands. However, when these properties are compatible, a new egalitarian mechanism is shown to satisfy them. When the properties are not compatible, we find a rule that meets stand-alone core stability, limit routing-proofness and minimi...

Restricted Constraints in the Problem of Maximum Attainable Flow.pdf

In this paper we have considered a version of restricted constraints in the problem of maximum attainable flow (capacity) in minimum cost (distance) in a network (Ahmed, Das, & Purusotham, 2012b). By restricted constraints one means that the link(s) (cities or stations or nodes) are completed (or visited) in such a way that a particular link is to be preceded (completed or visited) by another link (precedence relation need not be immediate). Here the aim is to obtained an optimal route of a more realistic situation as to scheduling a restricted constraints to a maximum flows at a minimum cost from a source to a destination. The distance (cost) and arc capacity between any two stations are given.