### 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 language | English (US) |
---|---|

Pages (from-to) | 1019-1034 |

Number of pages | 16 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 21 |

Issue number | 4 |

DOIs | |

State | Published - 2007 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*21*(4), 1019-1034. https://doi.org/10.1137/050636589

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

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 21, no. 4, pp. 1019-1034. https://doi.org/10.1137/050636589

}

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 -