Approximating Disjoint-Path Problems Using Greedy Algorithms and Packing Integer Programs

  title={Approximating Disjoint-Path Problems Using Greedy Algorithms and Packing Integer Programs},
  author={Stavros G. Kolliopoulos and Clifford Stein},
In the edge(vertex)-disjoint path problem we are given a graph $G$ and a set ${\cal T}$ of connection requests. Every connection request in ${\cal T}$ is a vertex pair $(s_i,t_i),$ $1 \leq i \leq K.$ The objective is to connect a maximum number of the pairs via edge(vertex)-disjoint paths. The edge-disjoint path problem can be generalized to the multiple-source unsplittable flow problem where connection request $i$ has a demand $\rho_i$ and every edge $e$ a capacity $u_e.$ All these problems… 

Approximating disjoint-path problems using packing integer programs

Improved approximation algorithms for column-restricted packing integer programs are developed that are simple to implement and achieve good performance when the input has a special structure and are motivated by the disjoint paths applications.

Inapproximability of Edge-Disjoint Paths and low congestion routing on undirected graphs

This paper studies the hardness of EDPwC in undirected graphs and shows an Ω(log logV/log log log V) hardness of approximation for EDPWC and an Φ(loglogV/ log log logV) hardness-of- approximation for the Undirected congestion minimization problem.

Improved Bi-criteria Approximation for the All-or-Nothing Multicommodity Flow Problem in Arbitrary Networks

This paper is the first to achieve a constant approximation of the maximum throughput with an edge capacity violation ratio that is at most logarithmic in $n$, with high probability, and presents a derandomization of the algorithm that maintains the same approximation bounds.

Approximation Algorithms for the Edge-Disjoint Paths Problem via Raecke Decompositions

  • M. Andrews
  • Computer Science, Mathematics
    2010 IEEE 51st Annual Symposium on Foundations of Computer Science
  • 2010
This work presents a $(polylog(n), poly(\log\log n))-approximation, which means that if there exists a solution that routes X demands integrally on edge-disjoint paths (i.e. with congestion $1), then the approximation algorithm can route X/polylog (n) demands with congestion.

Exact and approximation algorithms for network flow and disjoint-path problems

This thesis focuses on efficient exact algorithms for network flow problems in P and on approximation algorithms for NP-hard variants such as disjoint paths and unsplittable flow, and explores the topic of approximating disJoint-path problems using polynomial-size packing integer programs.

Edge-Disjoint Paths and Unsplittable Flow

This chapter limits itself mostly to results on offline approximation algorithms for problems on general graphs as influenced from the network flow perspective.

Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems

It is shown that in directed networks, for any e>0, EDP is NP-hard to approximate within m1/2-e even in undirected networks, and design simple approximation algorithms that achieve essentially matching approximation guarantees for some generalizations of EDP.

Edge Disjoint Paths in Moderately Connected Graphs

This work shows a polylogarithmic approximation algorithm for the undirected EDP problem in general graphs with a moderate restriction on graph connectivity; it requires the global minimum cut of G to be Ω(log5n) and applies to graphs with high diameters and asymptotically large minors.

New Hardness Results for Undirected Edge Disjoint Paths

The hardness result of Andrews and Zhang as well as the first polylogarithmic integrality gaps and hardness results for undirected EDP when congestion is allowed are improved and it is shown that it is possible to obtain a hardness result that is comparable to the integrality gap.

Hardness of the undirected edge-disjoint paths problem with congestion

An improved hardness result for EDP is obtained, and the first polylogarithmic integrality gaps and hardness of approximation results for E DPwC are shown, and similar results can be obtained for the all-or-nothing flow (ANF) problem, a relaxation of EDP.



Short paths in expander graphs

This work shows that a greedy algorithm for approximating the maximum disjoint paths problem achieves a polylogarithmic approximation ratio in bounded-degree expanders, and develops new routing algorithms and structural results for bounded- degree expander graphs.

Disjoint paths in densely embedded graphs

This work considers the class of densely embedded, nearly-Eulerian graphs, which includes the two-dimensional mesh and other planar and locally planar interconnection networks, and obtains a constant-factor approximation algorithm for the maximum disjoint paths problem for this class of graphs.

Algorithms for Square Roots of Graphs

The $n$th power ($n \geq 1$) of a graph $G = (V, E)$, written $G^n$, is defined to be the graph having $V$ as its vertex set with two vertices $u, v$ adjacent in $G^n$ if and only if there exists a

Improved approximation algorithms for unsplittable flow problems

A generic framework for single-source unsplittable flow, that yields simpler algorithms and significant improvements upon the constant factors, applies to all optimization versions previously considered and treats in a unified manner directed and undirected graphs.

Single-source unsplittable flow

  • J. Kleinberg
  • Computer Science
    Proceedings of 37th Conference on Foundations of Computer Science
  • 1996
The max-flow min-cut theorem of Ford and Fulkerson is based on an even more foundational result, namely Menger's theorem on graph connectivity Menger's theorem provides a good characterization for

An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms

  • F. LeightonSatish Rao
  • Computer Science
    [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
  • 1988
The main result is an algorithm for performing the task provided that the capacity of each cut exceeds the demand across the cut by a Theta (log n) factor.

Improved approximation algorithms for shop scheduling problems

The authors give the first randomized and deterministic polynomial-time algorithms that yield polylogarithmic approximations to the optimal length schedule in the job shop scheduling problem.

Homotopic routing methods

is NP-complete, even for planar graphs, both in the vertex-disjoint and in the edgedisjoint case (Lynch [26)). In some special cases, however, there is a polynomialtime method for (1). These cases

Global wire routing in two-dimensional arrays

A central result of this paper is a “rounding algorithm” for obtaining integral approximations to solutions of linear equations for matrix A and real vector x.

Approximation algorithms for multicommodity flow and shop scheduling problems

This thesis gives efficient approximation algorithms for two classical combinatorial optimization problems: multicommodity flow problems and shop scheduling problem and shows that by allowing a small error in the solution of a problem, it is often possible to gain a significant reduction in the running time of an algorithm for that problem.