### Abstract

In this paper, we propose new exact and approximation algorithms for the weighted matroid intersection problem. Our exact algorithm is faster than previous algorithms when the largest weight is relatively small. Our approximation algorithm delivers a (1 - ϵ)-approximate solution with a running time significantly faster than known exact algorithms. The core of our algorithms is a decomposition technique: we decompose an instance of the weighted matroid intersection problem into a set of instances of the unweighted matroid intersection problem. The computational advantage of this approach is that we can make use of fast unweighted matroid intersection algorithms as a black box for designing algorithms. Precisely speaking, we prove that we can solve the weighted matroid intersection problem via solving W instances of the unweighted matroid intersection problem, where W is the largest given weight. Furthermore, we can find a (1 - ϵ)-approximate solution via solving O(ϵ^{-1} log r) instances of the unweighted matroid intersection problem, where r is the smallest rank of the given two matroids. Our algorithms are simple and flexible: they can be adapted to special cases of the weighted matroid intersection problem, using specialized unweighted matroid intersection algorithms. In this paper, we will show the following results. 1. Given two general matroids, using Cunningham's algorithm, we can solve the weighted matroid intersection problem exactly in O(τWnr^{1.5}) time and (1 - ϵ)-approximately in O(τϵ^{-1}nr^{1.5} logr) time, where n is the size of the ground set and r is the time complexity of an independence oracle call. 2. Given two graphic matroids, using the algorithm of Gabow and Xu, we can solve the weighted matroid intersection problem exactly in O(W√rn log r) time and (1 - ϵ)-approximately in O(ϵ^{-1} √rn log^{2}r) time. 3. Given two linear matroids (in the form of two r-by-n matrices), using the algorithm of Cheung, Kwok, and Lau, we can solve the weighted matroid intersection problem exactly in O(nr log r∗ + Wnrω∗^{-1}) time and (1 - ϵ)-approximately in O(nr logr∗ + ϵ^{-1}nrω∗^{-1} logr∗) time, where ω is the exponent of the matrix multiplication time and r∗ is the maximum size of a common independent set. Finally, we give a further application of our decomposition technique. We use our technique to solve efficiently the rankmaximal matroid intersection problem, a problem motivated by matching problems under preferences.

Original language | English |
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Title of host publication | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 |

Publisher | Association for Computing Machinery |

Pages | 430-444 |

Number of pages | 15 |

Volume | 1 |

ISBN (Electronic) | 9781510819672 |

Publication status | Published - 2016 |

Externally published | Yes |

Event | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 - Arlington, United States Duration: 2016 Jan 10 → 2016 Jan 12 |

### Other

Other | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 |
---|---|

Country | United States |

City | Arlington |

Period | 16/1/10 → 16/1/12 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016*(Vol. 1, pp. 430-444). Association for Computing Machinery.

**Exact and approximation algorithms for weighted matroid intersection.** / Huang, Chien Chung; Kakimura, Naonori; Kamiyama, Naoyuki.

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

*27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016.*vol. 1, Association for Computing Machinery, pp. 430-444, 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, United States, 16/1/10.

}

TY - GEN

T1 - Exact and approximation algorithms for weighted matroid intersection

AU - Huang, Chien Chung

AU - Kakimura, Naonori

AU - Kamiyama, Naoyuki

PY - 2016

Y1 - 2016

N2 - In this paper, we propose new exact and approximation algorithms for the weighted matroid intersection problem. Our exact algorithm is faster than previous algorithms when the largest weight is relatively small. Our approximation algorithm delivers a (1 - ϵ)-approximate solution with a running time significantly faster than known exact algorithms. The core of our algorithms is a decomposition technique: we decompose an instance of the weighted matroid intersection problem into a set of instances of the unweighted matroid intersection problem. The computational advantage of this approach is that we can make use of fast unweighted matroid intersection algorithms as a black box for designing algorithms. Precisely speaking, we prove that we can solve the weighted matroid intersection problem via solving W instances of the unweighted matroid intersection problem, where W is the largest given weight. Furthermore, we can find a (1 - ϵ)-approximate solution via solving O(ϵ-1 log r) instances of the unweighted matroid intersection problem, where r is the smallest rank of the given two matroids. Our algorithms are simple and flexible: they can be adapted to special cases of the weighted matroid intersection problem, using specialized unweighted matroid intersection algorithms. In this paper, we will show the following results. 1. Given two general matroids, using Cunningham's algorithm, we can solve the weighted matroid intersection problem exactly in O(τWnr1.5) time and (1 - ϵ)-approximately in O(τϵ-1nr1.5 logr) time, where n is the size of the ground set and r is the time complexity of an independence oracle call. 2. Given two graphic matroids, using the algorithm of Gabow and Xu, we can solve the weighted matroid intersection problem exactly in O(W√rn log r) time and (1 - ϵ)-approximately in O(ϵ-1 √rn log2r) time. 3. Given two linear matroids (in the form of two r-by-n matrices), using the algorithm of Cheung, Kwok, and Lau, we can solve the weighted matroid intersection problem exactly in O(nr log r∗ + Wnrω∗-1) time and (1 - ϵ)-approximately in O(nr logr∗ + ϵ-1nrω∗-1 logr∗) time, where ω is the exponent of the matrix multiplication time and r∗ is the maximum size of a common independent set. Finally, we give a further application of our decomposition technique. We use our technique to solve efficiently the rankmaximal matroid intersection problem, a problem motivated by matching problems under preferences.

AB - In this paper, we propose new exact and approximation algorithms for the weighted matroid intersection problem. Our exact algorithm is faster than previous algorithms when the largest weight is relatively small. Our approximation algorithm delivers a (1 - ϵ)-approximate solution with a running time significantly faster than known exact algorithms. The core of our algorithms is a decomposition technique: we decompose an instance of the weighted matroid intersection problem into a set of instances of the unweighted matroid intersection problem. The computational advantage of this approach is that we can make use of fast unweighted matroid intersection algorithms as a black box for designing algorithms. Precisely speaking, we prove that we can solve the weighted matroid intersection problem via solving W instances of the unweighted matroid intersection problem, where W is the largest given weight. Furthermore, we can find a (1 - ϵ)-approximate solution via solving O(ϵ-1 log r) instances of the unweighted matroid intersection problem, where r is the smallest rank of the given two matroids. Our algorithms are simple and flexible: they can be adapted to special cases of the weighted matroid intersection problem, using specialized unweighted matroid intersection algorithms. In this paper, we will show the following results. 1. Given two general matroids, using Cunningham's algorithm, we can solve the weighted matroid intersection problem exactly in O(τWnr1.5) time and (1 - ϵ)-approximately in O(τϵ-1nr1.5 logr) time, where n is the size of the ground set and r is the time complexity of an independence oracle call. 2. Given two graphic matroids, using the algorithm of Gabow and Xu, we can solve the weighted matroid intersection problem exactly in O(W√rn log r) time and (1 - ϵ)-approximately in O(ϵ-1 √rn log2r) time. 3. Given two linear matroids (in the form of two r-by-n matrices), using the algorithm of Cheung, Kwok, and Lau, we can solve the weighted matroid intersection problem exactly in O(nr log r∗ + Wnrω∗-1) time and (1 - ϵ)-approximately in O(nr logr∗ + ϵ-1nrω∗-1 logr∗) time, where ω is the exponent of the matrix multiplication time and r∗ is the maximum size of a common independent set. Finally, we give a further application of our decomposition technique. We use our technique to solve efficiently the rankmaximal matroid intersection problem, a problem motivated by matching problems under preferences.

UR - http://www.scopus.com/inward/record.url?scp=84962800110&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84962800110&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84962800110

VL - 1

SP - 430

EP - 444

BT - 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016

PB - Association for Computing Machinery

ER -