### Abstract

In this paper, we consider the sparse linear complementarity problem, denoted by k-LCP: the coefficient matrices are restricted to have at most k nonzero entries per row. It is known that the 1-LCP is solvable in linear time, and the 3-LCP is strongly NP-hard. We show that the 2-LCP is strongly NP-hard, and it can be solved in polynomial time if it is sign-balanced, i.e., each row of the matrix has at most one positive and one negative entry. Our second result matches the currently best-known complexity bound for the corresponding sparse linear feasibility problem. In addition, we show that an integer variant of the sign-balanced 2-LCP is weakly NP-hard and pseudo-polynomially solvable, and the generalized 1-LCP is strongly NP-hard.

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

Pages (from-to) | 1015-1026 |

Number of pages | 12 |

Journal | Mathematics of Operations Research |

Volume | 40 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2015 Nov 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Combinatorial algorithm
- Linear complementarity problem
- NP-hardness
- Polynomial solvability
- Two-variable constraints

### ASJC Scopus subject areas

- Mathematics(all)
- Computer Science Applications
- Management Science and Operations Research

### Cite this

*Mathematics of Operations Research*,

*40*(4), 1015-1026. https://doi.org/10.1287/moor.2014.0708

**The linear complementarity problems with a few variables per constraint.** / Sumita, Hanna; Kakimura, Naonori; Makino, Kazuhisa.

Research output: Contribution to journal › Article

*Mathematics of Operations Research*, vol. 40, no. 4, pp. 1015-1026. https://doi.org/10.1287/moor.2014.0708

}

TY - JOUR

T1 - The linear complementarity problems with a few variables per constraint

AU - Sumita, Hanna

AU - Kakimura, Naonori

AU - Makino, Kazuhisa

PY - 2015/11/1

Y1 - 2015/11/1

N2 - In this paper, we consider the sparse linear complementarity problem, denoted by k-LCP: the coefficient matrices are restricted to have at most k nonzero entries per row. It is known that the 1-LCP is solvable in linear time, and the 3-LCP is strongly NP-hard. We show that the 2-LCP is strongly NP-hard, and it can be solved in polynomial time if it is sign-balanced, i.e., each row of the matrix has at most one positive and one negative entry. Our second result matches the currently best-known complexity bound for the corresponding sparse linear feasibility problem. In addition, we show that an integer variant of the 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 consider the sparse linear complementarity problem, denoted by k-LCP: the coefficient matrices are restricted to have at most k nonzero entries per row. It is known that the 1-LCP is solvable in linear time, and the 3-LCP is strongly NP-hard. We show that the 2-LCP is strongly NP-hard, and it can be solved in polynomial time if it is sign-balanced, i.e., each row of the matrix has at most one positive and one negative entry. Our second result matches the currently best-known complexity bound for the corresponding sparse linear feasibility problem. In addition, we show that an integer variant of the sign-balanced 2-LCP is weakly NP-hard and pseudo-polynomially solvable, and the generalized 1-LCP is strongly NP-hard.

KW - Combinatorial algorithm

KW - Linear complementarity problem

KW - NP-hardness

KW - Polynomial solvability

KW - Two-variable constraints

UR - http://www.scopus.com/inward/record.url?scp=84947050901&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84947050901&partnerID=8YFLogxK

U2 - 10.1287/moor.2014.0708

DO - 10.1287/moor.2014.0708

M3 - Article

VL - 40

SP - 1015

EP - 1026

JO - Mathematics of Operations Research

JF - Mathematics of Operations Research

SN - 0364-765X

IS - 4

ER -