### Abstract

For an undirected graph G∈=∈(V, E), an edge cover is defined as a set of edges that covers all vertices of V. It is known that a minimum edge cover can be found in polynomial time and forms a collection of star graphs. In this paper, we consider the problem of finding a balanced edge cover where the degrees of star center vertices are balanced, which can be applied to optimize sensor network structures, for example. To this end, we formulate the problem as a minimization of the summation of strictly monotone increasing convex costs associated with degrees for covered vertices, and show that the optimality can be characterized as the non-existence of certain alternating paths. By using this characterization, we show that the optimal covers are also minimum edge covers, have the lexicographically smallest degree sequence of the covered vertices, and minimize the maximum degree of covered vertices. Based on the characterization we also present an O(|V||E|) time algorithm.

Original language | English |
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Title of host publication | Algorithms and Computation - 19th International Symposium, ISAAC 2008, Proceedings |

Pages | 246-257 |

Number of pages | 12 |

DOIs | |

Publication status | Published - Dec 1 2008 |

Event | 19th International Symposium on Algorithms and Computation, ISAAC 2008 - Gold Coast, QLD, Australia Duration: Dec 15 2008 → Dec 17 2008 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 5369 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 19th International Symposium on Algorithms and Computation, ISAAC 2008 |
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Country | Australia |

City | Gold Coast, QLD |

Period | 12/15/08 → 12/17/08 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithms and Computation - 19th International Symposium, ISAAC 2008, Proceedings*(pp. 246-257). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5369 LNCS). https://doi.org/10.1007/978-3-540-92182-0_24

**The balanced edge cover problem.** / Harada, Yuta; Ono, Hirotaka; Sadakane, Kunihiko; Yamashita, Masafumi.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Algorithms and Computation - 19th International Symposium, ISAAC 2008, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 5369 LNCS, pp. 246-257, 19th International Symposium on Algorithms and Computation, ISAAC 2008, Gold Coast, QLD, Australia, 12/15/08. https://doi.org/10.1007/978-3-540-92182-0_24

}

TY - GEN

T1 - The balanced edge cover problem

AU - Harada, Yuta

AU - Ono, Hirotaka

AU - Sadakane, Kunihiko

AU - Yamashita, Masafumi

PY - 2008/12/1

Y1 - 2008/12/1

N2 - For an undirected graph G∈=∈(V, E), an edge cover is defined as a set of edges that covers all vertices of V. It is known that a minimum edge cover can be found in polynomial time and forms a collection of star graphs. In this paper, we consider the problem of finding a balanced edge cover where the degrees of star center vertices are balanced, which can be applied to optimize sensor network structures, for example. To this end, we formulate the problem as a minimization of the summation of strictly monotone increasing convex costs associated with degrees for covered vertices, and show that the optimality can be characterized as the non-existence of certain alternating paths. By using this characterization, we show that the optimal covers are also minimum edge covers, have the lexicographically smallest degree sequence of the covered vertices, and minimize the maximum degree of covered vertices. Based on the characterization we also present an O(|V||E|) time algorithm.

AB - For an undirected graph G∈=∈(V, E), an edge cover is defined as a set of edges that covers all vertices of V. It is known that a minimum edge cover can be found in polynomial time and forms a collection of star graphs. In this paper, we consider the problem of finding a balanced edge cover where the degrees of star center vertices are balanced, which can be applied to optimize sensor network structures, for example. To this end, we formulate the problem as a minimization of the summation of strictly monotone increasing convex costs associated with degrees for covered vertices, and show that the optimality can be characterized as the non-existence of certain alternating paths. By using this characterization, we show that the optimal covers are also minimum edge covers, have the lexicographically smallest degree sequence of the covered vertices, and minimize the maximum degree of covered vertices. Based on the characterization we also present an O(|V||E|) time algorithm.

UR - http://www.scopus.com/inward/record.url?scp=58549112441&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=58549112441&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-92182-0_24

DO - 10.1007/978-3-540-92182-0_24

M3 - Conference contribution

AN - SCOPUS:58549112441

SN - 3540921818

SN - 9783540921813

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 246

EP - 257

BT - Algorithms and Computation - 19th International Symposium, ISAAC 2008, Proceedings

ER -