### Abstract

Consider the conditional covering problem on an undirected graph, where each node represents a site that must be covered by a facility, and facilities may only be established at these nodes. Each facility can cover all sites that lie within some common covering radius, except the site at which it is located. Although this problem is difficult to solve on general graphs, there exist special structures on which the problem is easily solvable. In this paper, we consider the special case in which the graph is a simple path. For the case in which facility location costs do not vary based on the site, we derive characteristics of the problem that lead to a linear-time shortest path algorithm for solving the problem. When the facility location costs vary according to the site, we provide a more complex, but still polynomial-time, dynamic programming algorithm to find the optimal solution.

Original language | English (US) |
---|---|

Pages (from-to) | 293-301 |

Number of pages | 9 |

Journal | Naval Research Logistics |

Volume | 52 |

Issue number | 4 |

DOIs | |

State | Published - Jun 2005 |

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### Keywords

- Covering problems
- Dynamic programming
- Location theory

### ASJC Scopus subject areas

- Management Science and Operations Research

### Cite this

*Naval Research Logistics*,

*52*(4), 293-301. https://doi.org/10.1002/nav.20074

**Algorithms for solving the conditional covering problem on paths.** / Lunday, Brian J.; Smith, J. Cole; Goldberg, Jeffrey B.

Research output: Contribution to journal › Article

*Naval Research Logistics*, vol. 52, no. 4, pp. 293-301. https://doi.org/10.1002/nav.20074

}

TY - JOUR

T1 - Algorithms for solving the conditional covering problem on paths

AU - Lunday, Brian J.

AU - Smith, J. Cole

AU - Goldberg, Jeffrey B

PY - 2005/6

Y1 - 2005/6

N2 - Consider the conditional covering problem on an undirected graph, where each node represents a site that must be covered by a facility, and facilities may only be established at these nodes. Each facility can cover all sites that lie within some common covering radius, except the site at which it is located. Although this problem is difficult to solve on general graphs, there exist special structures on which the problem is easily solvable. In this paper, we consider the special case in which the graph is a simple path. For the case in which facility location costs do not vary based on the site, we derive characteristics of the problem that lead to a linear-time shortest path algorithm for solving the problem. When the facility location costs vary according to the site, we provide a more complex, but still polynomial-time, dynamic programming algorithm to find the optimal solution.

AB - Consider the conditional covering problem on an undirected graph, where each node represents a site that must be covered by a facility, and facilities may only be established at these nodes. Each facility can cover all sites that lie within some common covering radius, except the site at which it is located. Although this problem is difficult to solve on general graphs, there exist special structures on which the problem is easily solvable. In this paper, we consider the special case in which the graph is a simple path. For the case in which facility location costs do not vary based on the site, we derive characteristics of the problem that lead to a linear-time shortest path algorithm for solving the problem. When the facility location costs vary according to the site, we provide a more complex, but still polynomial-time, dynamic programming algorithm to find the optimal solution.

KW - Covering problems

KW - Dynamic programming

KW - Location theory

UR - http://www.scopus.com/inward/record.url?scp=18744388857&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=18744388857&partnerID=8YFLogxK

U2 - 10.1002/nav.20074

DO - 10.1002/nav.20074

M3 - Article

AN - SCOPUS:18744388857

VL - 52

SP - 293

EP - 301

JO - Naval Research Logistics

JF - Naval Research Logistics

SN - 0894-069X

IS - 4

ER -