Combinatorial optimization with explicit delineation of the ground set by a collection of subsets

Moshe Dror, James B. Orlin

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

We examine a selective list of combinatorial optimization problems in NP with respect to inapproximability (Arora and Lund (1997)) given that the ground set of elements N has additional characteristics. For each problem in this paper, the set N is expressed explicitly by subsets of N either as a partition or in the form of a cover. The problems examined are generalizations of well-known classical graph problems and include the minimal spanning tree problem, a number of elementary machine scheduling problems, the bin-packing problem, and the travelling salesman problem (TSP). We conclude that for all these generalized problems the existence of a polynomial time approximation scheme (PTAS) is impossible unless P=NP. This suggests a partial characterization for a family of inapproximable problems. For the generalized Euclidean TSP we prove inapproximability even if the subsets are of cardinality 2.

Original languageEnglish (US)
Pages (from-to)1019-1034
Number of pages16
JournalSIAM Journal on Discrete Mathematics
Volume21
Issue number4
DOIs
StatePublished - 2007

Fingerprint

Combinatorial Optimization
Subset
Inapproximability
Travelling salesman problems
Minimal Spanning Tree
Bin Packing Problem
Machine Scheduling
Polynomial Time Approximation Scheme
Combinatorial Optimization Problem
Scheduling Problem
Cardinality
Euclidean
Partition
Cover
Partial
Graph in graph theory

Keywords

  • Approximation algorithms
  • Generalized bin packing
  • Generalized scheduling problems
  • Inapproximability
  • Subset travelling salesman

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Combinatorial optimization with explicit delineation of the ground set by a collection of subsets. / Dror, Moshe; Orlin, James B.

In: SIAM Journal on Discrete Mathematics, Vol. 21, No. 4, 2007, p. 1019-1034.

Research output: Contribution to journalArticle

@article{7dba04abb7a640dd88180b529cc413a2,
title = "Combinatorial optimization with explicit delineation of the ground set by a collection of subsets",
abstract = "We examine a selective list of combinatorial optimization problems in NP with respect to inapproximability (Arora and Lund (1997)) given that the ground set of elements N has additional characteristics. For each problem in this paper, the set N is expressed explicitly by subsets of N either as a partition or in the form of a cover. The problems examined are generalizations of well-known classical graph problems and include the minimal spanning tree problem, a number of elementary machine scheduling problems, the bin-packing problem, and the travelling salesman problem (TSP). We conclude that for all these generalized problems the existence of a polynomial time approximation scheme (PTAS) is impossible unless P=NP. This suggests a partial characterization for a family of inapproximable problems. For the generalized Euclidean TSP we prove inapproximability even if the subsets are of cardinality 2.",
keywords = "Approximation algorithms, Generalized bin packing, Generalized scheduling problems, Inapproximability, Subset travelling salesman",
author = "Moshe Dror and Orlin, {James B.}",
year = "2007",
doi = "10.1137/050636589",
language = "English (US)",
volume = "21",
pages = "1019--1034",
journal = "SIAM Journal on Discrete Mathematics",
issn = "0895-4801",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "4",

}

TY - JOUR

T1 - Combinatorial optimization with explicit delineation of the ground set by a collection of subsets

AU - Dror, Moshe

AU - Orlin, James B.

PY - 2007

Y1 - 2007

N2 - We examine a selective list of combinatorial optimization problems in NP with respect to inapproximability (Arora and Lund (1997)) given that the ground set of elements N has additional characteristics. For each problem in this paper, the set N is expressed explicitly by subsets of N either as a partition or in the form of a cover. The problems examined are generalizations of well-known classical graph problems and include the minimal spanning tree problem, a number of elementary machine scheduling problems, the bin-packing problem, and the travelling salesman problem (TSP). We conclude that for all these generalized problems the existence of a polynomial time approximation scheme (PTAS) is impossible unless P=NP. This suggests a partial characterization for a family of inapproximable problems. For the generalized Euclidean TSP we prove inapproximability even if the subsets are of cardinality 2.

AB - We examine a selective list of combinatorial optimization problems in NP with respect to inapproximability (Arora and Lund (1997)) given that the ground set of elements N has additional characteristics. For each problem in this paper, the set N is expressed explicitly by subsets of N either as a partition or in the form of a cover. The problems examined are generalizations of well-known classical graph problems and include the minimal spanning tree problem, a number of elementary machine scheduling problems, the bin-packing problem, and the travelling salesman problem (TSP). We conclude that for all these generalized problems the existence of a polynomial time approximation scheme (PTAS) is impossible unless P=NP. This suggests a partial characterization for a family of inapproximable problems. For the generalized Euclidean TSP we prove inapproximability even if the subsets are of cardinality 2.

KW - Approximation algorithms

KW - Generalized bin packing

KW - Generalized scheduling problems

KW - Inapproximability

KW - Subset travelling salesman

UR - http://www.scopus.com/inward/record.url?scp=56649093030&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=56649093030&partnerID=8YFLogxK

U2 - 10.1137/050636589

DO - 10.1137/050636589

M3 - Article

AN - SCOPUS:56649093030

VL - 21

SP - 1019

EP - 1034

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 4

ER -