### Abstract

An independence system is one of the most fundamental combinatorial concepts, which includes a variety of objects in graphs and hypergraphs such as matchings, stable sets, and matroids. We discuss the robustness for independence systems, which is a natural generalization of the greedy property of matroids. For a real number α > 0, a set is said to be α-robust if for any k, it includes an α-approximation of the maximum k-independent set, where a set Y in is called k-independent if the size |Y| is at most k. In this paper, we show that every independence system has a -robust independent set, where denotes the exchangeability of . Our result contains a classical result for matroids and the ones of Hassin and Rubinstein,[12] for matchings and Fujita, Kobayashi, and Makino,[7] for matroid 2-intersections, and provides better bounds for the robustness for many independence systems such as b-matchings, hypergraph matchings, matroid p-intersections, and unions of vertex disjoint paths. Furthermore, we provide bounds of the robustness for nonlinear weight functions such as submodular and convex quadratic functions. We also extend our results to independence systems in the integral lattice with separable concave weight functions.

Original language | English |
---|---|

Title of host publication | Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Proceedings |

Pages | 367-378 |

Number of pages | 12 |

Volume | 6755 LNCS |

Edition | PART 1 |

DOIs | |

Publication status | Published - 2011 |

Externally published | Yes |

Event | 38th International Colloquium on Automata, Languages and Programming, ICALP 2011 - Zurich, Switzerland Duration: 2011 Jul 4 → 2011 Jul 8 |

### Publication series

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

Number | PART 1 |

Volume | 6755 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 38th International Colloquium on Automata, Languages and Programming, ICALP 2011 |
---|---|

Country | Switzerland |

City | Zurich |

Period | 11/7/4 → 11/7/8 |

### Fingerprint

### Keywords

- exchangeability
- independence systems
- matroids
- robustness

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Proceedings*(PART 1 ed., Vol. 6755 LNCS, pp. 367-378). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6755 LNCS, No. PART 1). https://doi.org/10.1007/978-3-642-22006-7_31

**Robust independence systems.** / Kakimura, Naonori; Makino, Kazuhisa.

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

*Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Proceedings.*PART 1 edn, vol. 6755 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), no. PART 1, vol. 6755 LNCS, pp. 367-378, 38th International Colloquium on Automata, Languages and Programming, ICALP 2011, Zurich, Switzerland, 11/7/4. https://doi.org/10.1007/978-3-642-22006-7_31

}

TY - GEN

T1 - Robust independence systems

AU - Kakimura, Naonori

AU - Makino, Kazuhisa

PY - 2011

Y1 - 2011

N2 - An independence system is one of the most fundamental combinatorial concepts, which includes a variety of objects in graphs and hypergraphs such as matchings, stable sets, and matroids. We discuss the robustness for independence systems, which is a natural generalization of the greedy property of matroids. For a real number α > 0, a set is said to be α-robust if for any k, it includes an α-approximation of the maximum k-independent set, where a set Y in is called k-independent if the size |Y| is at most k. In this paper, we show that every independence system has a -robust independent set, where denotes the exchangeability of . Our result contains a classical result for matroids and the ones of Hassin and Rubinstein,[12] for matchings and Fujita, Kobayashi, and Makino,[7] for matroid 2-intersections, and provides better bounds for the robustness for many independence systems such as b-matchings, hypergraph matchings, matroid p-intersections, and unions of vertex disjoint paths. Furthermore, we provide bounds of the robustness for nonlinear weight functions such as submodular and convex quadratic functions. We also extend our results to independence systems in the integral lattice with separable concave weight functions.

AB - An independence system is one of the most fundamental combinatorial concepts, which includes a variety of objects in graphs and hypergraphs such as matchings, stable sets, and matroids. We discuss the robustness for independence systems, which is a natural generalization of the greedy property of matroids. For a real number α > 0, a set is said to be α-robust if for any k, it includes an α-approximation of the maximum k-independent set, where a set Y in is called k-independent if the size |Y| is at most k. In this paper, we show that every independence system has a -robust independent set, where denotes the exchangeability of . Our result contains a classical result for matroids and the ones of Hassin and Rubinstein,[12] for matchings and Fujita, Kobayashi, and Makino,[7] for matroid 2-intersections, and provides better bounds for the robustness for many independence systems such as b-matchings, hypergraph matchings, matroid p-intersections, and unions of vertex disjoint paths. Furthermore, we provide bounds of the robustness for nonlinear weight functions such as submodular and convex quadratic functions. We also extend our results to independence systems in the integral lattice with separable concave weight functions.

KW - exchangeability

KW - independence systems

KW - matroids

KW - robustness

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UR - http://www.scopus.com/inward/citedby.url?scp=79959971775&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-22006-7_31

DO - 10.1007/978-3-642-22006-7_31

M3 - Conference contribution

AN - SCOPUS:79959971775

SN - 9783642220050

VL - 6755 LNCS

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

SP - 367

EP - 378

BT - Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Proceedings

ER -