### 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 |

DOIs | |

Publication status | Published - 2013 Sep 9 |

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 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

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