Dartmouth College Computer Science Technical Report series CS home TR home TR search TR listserv By author: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z By number: 2017, 2016, 2015, 2014, 2013, 2012, 2011, 2010, 2009, 2008, 2007, 2006, 2005, 2004, 2003, 2002, 2001, 2000, 1999, 1998, 1997, 1996, 1995, 1994, 1993, 1992, 1991, 1990, 1989, 1988, 1987, 1986

Approximating Disjoint-Path Problems Using Greedy Algorithms and Packing Integer Programs
Stavros G. Kolliopoulos, Clifford Stein
Dartmouth PCS-TR97-325

Abstract:

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 are NP-hard and have a multitude of applications in areas such as routing, scheduling and bin packing.

Given the hardness of the problem, we study polynomial-time approximation algorithms. In this context, a $\rho$-approximation algorithm is able to route at least a $1/\rho$ fraction of the connection requests. Although the edge- and vertex-disjoint path problems, and more recently the unsplittable flow generalization, have been extensively studied, they remain notoriously hard to approximate with a bounded performance guarantee. For example, even for the simple edge-disjoint path problem, no $o(\sqrt{|E|})$-approximation algorithm is known. Moreover some of the best existing approximation ratios are obtained through sophisticated and non-standard randomized rounding schemes.

In this paper we introduce techniques which yield algorithms for a wide range of disjoint-path and unsplittable flow problems. For the general unsplittable flow problem, even with weights on the commodities, our techniques lead to the first approximation algorithm and obtain an approximation ratio that matches, to within logarithmic factors, the $O(\sqrt{|E|})$ approximation ratio for the simple edge-disjoint path problem. In addition to this result and to improved bounds for several disjoint-path problems, our techniques simplify and unify the derivation of many existing approximation results.

We use two basic techniques. First, we propose simple greedy algorithms for edge- and vertex-disjoint paths and second, we propose the use of a framework based on packing integer programs for more general problems such as unsplittable flow. A packing integer program is of the form maximize $c^{T}\cdot x,$ subject to $Ax \leq b,$ $A,b,c \geq 0.$ As part of our tools we develop improved approximation algorithms for a class of packing integer programs, a result that we believe is of independent interest.

Note: Revised November 1997.

PDF (1931KB)

Bibliographic citation for this report: [plain text] [BIB] [BibTeX] [Refer]

Or copy and paste:
Stavros G. Kolliopoulos and Clifford Stein, "Approximating Disjoint-Path Problems Using Greedy Algorithms and Packing Integer Programs ." Dartmouth Computer Science Technical Report PCS-TR97-325, October 1997.

To receive paper copy of a report, by mail, send your address and the TR number to reports AT cs.dartmouth.edu

Copyright notice: The documents contained in this server are included by the contributing authors as a means to ensure timely dissemination of scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the authors or by other copyright holders, notwithstanding that they have offered their works here electronically. It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may not be reposted without the explicit permission of the copyright holder.

Technical reports collection maintained by David Kotz.