### Abstract

In this paper, we study the sparse linear complementarity problem, denoted by k-LCP: the coefficient matrix has at most k nonzero entries per row. It is known that 1-LCP is solvable in linear time, while 3-LCP is strongly NP-hard. We show that 2-LCP is strongly NP-hard, while it can be solved in O(n^{3} logn) time if it is sign-balanced, i.e., each row has at most one positive and one negative entries, where n is the number of constraints. Our second result matches with the currently best known complexity bound for the corresponding sparse linear feasibility problem. In addition, we show that an integer variant of sign-balanced 2-LCP is weakly NP-hard and pseudo-polynomially solvable, and the generalized 1-LCP is strongly NP-hard.

Original language | English |
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Title of host publication | Algorithms and Complexity - 8th International Conference, CIAC 2013, Proceedings |

Pages | 358-369 |

Number of pages | 12 |

Volume | 7878 LNCS |

DOIs | |

Publication status | Published - 2013 |

Externally published | Yes |

Event | 8th International Conference on Algorithms and Complexity, CIAC 2013 - Barcelona, Spain Duration: 2013 May 22 → 2013 May 24 |

### 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 | 7878 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 8th International Conference on Algorithms and Complexity, CIAC 2013 |
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Country | Spain |

City | Barcelona |

Period | 13/5/22 → 13/5/24 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Algorithms and Complexity - 8th International Conference, CIAC 2013, Proceedings*(Vol. 7878 LNCS, pp. 358-369). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7878 LNCS). https://doi.org/10.1007/978-3-642-38233-8_30

**Sparse linear complementarity problems.** / Sumita, Hanna; Kakimura, Naonori; Makino, Kazuhisa.

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

*Algorithms and Complexity - 8th International Conference, CIAC 2013, Proceedings.*vol. 7878 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7878 LNCS, pp. 358-369, 8th International Conference on Algorithms and Complexity, CIAC 2013, Barcelona, Spain, 13/5/22. https://doi.org/10.1007/978-3-642-38233-8_30

}

TY - GEN

T1 - Sparse linear complementarity problems

AU - Sumita, Hanna

AU - Kakimura, Naonori

AU - Makino, Kazuhisa

PY - 2013

Y1 - 2013

N2 - In this paper, we study the sparse linear complementarity problem, denoted by k-LCP: the coefficient matrix has at most k nonzero entries per row. It is known that 1-LCP is solvable in linear time, while 3-LCP is strongly NP-hard. We show that 2-LCP is strongly NP-hard, while it can be solved in O(n3 logn) time if it is sign-balanced, i.e., each row has at most one positive and one negative entries, where n is the number of constraints. Our second result matches with the currently best known complexity bound for the corresponding sparse linear feasibility problem. In addition, we show that an integer variant of sign-balanced 2-LCP is weakly NP-hard and pseudo-polynomially solvable, and the generalized 1-LCP is strongly NP-hard.

AB - In this paper, we study the sparse linear complementarity problem, denoted by k-LCP: the coefficient matrix has at most k nonzero entries per row. It is known that 1-LCP is solvable in linear time, while 3-LCP is strongly NP-hard. We show that 2-LCP is strongly NP-hard, while it can be solved in O(n3 logn) time if it is sign-balanced, i.e., each row has at most one positive and one negative entries, where n is the number of constraints. Our second result matches with the currently best known complexity bound for the corresponding sparse linear feasibility problem. In addition, we show that an integer variant of sign-balanced 2-LCP is weakly NP-hard and pseudo-polynomially solvable, and the generalized 1-LCP is strongly NP-hard.

UR - http://www.scopus.com/inward/record.url?scp=84883351739&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84883351739&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-38233-8_30

DO - 10.1007/978-3-642-38233-8_30

M3 - Conference contribution

AN - SCOPUS:84883351739

SN - 9783642382321

VL - 7878 LNCS

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

SP - 358

EP - 369

BT - Algorithms and Complexity - 8th International Conference, CIAC 2013, Proceedings

ER -