### Abstract

In this paper, we introduce total dual integrality of the linear complementarity problem (LCP) by analogy with the linear programming problem. The main idea of defining the notion is to propose the LCP with orientation, a variant of the LCP whose feasible complementary cones are specified by an additional input vector. Then we naturally define the dual problem of the LCP with orientation and total dual integrality of the LCP. We show that if the LCP is totally dual integral, then all basic solutions are integral. If the input matrix is sufficient or rank-symmetric, and the LCP is totally dual integral, then our result implies that the LCP always has an integral solution whenever it has a solution. We also introduce a class of matrices such that any LCP instance having the matrix as a coefficient matrix is totally dual integral. We investigate relationships between matrix classes in the LCP literature such as principally unimodular matrices. Principally unimodular matrices are known that all basic solutions to the LCP are integral for any integral input vector. In addition, we show that it is coNP-hard to decide whether a given LCP instance is totally dual integral.

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

Pages (from-to) | 1-23 |

Number of pages | 23 |

Journal | Annals of Operations Research |

DOIs | |

Publication status | Accepted/In press - 2018 Jun 14 |

### Fingerprint

### Keywords

- Linear complementarity problem
- Principal unimodularity
- Total dual integrality

### ASJC Scopus subject areas

- Decision Sciences(all)
- Management Science and Operations Research

### Cite this

*Annals of Operations Research*, 1-23. https://doi.org/10.1007/s10479-018-2926-8

**Total dual integrality of the linear complementarity problem.** / Sumita, Hanna; Kakimura, Naonori; Makino, Kazuhisa.

Research output: Contribution to journal › Article

*Annals of Operations Research*, pp. 1-23. https://doi.org/10.1007/s10479-018-2926-8

}

TY - JOUR

T1 - Total dual integrality of the linear complementarity problem

AU - Sumita, Hanna

AU - Kakimura, Naonori

AU - Makino, Kazuhisa

PY - 2018/6/14

Y1 - 2018/6/14

N2 - In this paper, we introduce total dual integrality of the linear complementarity problem (LCP) by analogy with the linear programming problem. The main idea of defining the notion is to propose the LCP with orientation, a variant of the LCP whose feasible complementary cones are specified by an additional input vector. Then we naturally define the dual problem of the LCP with orientation and total dual integrality of the LCP. We show that if the LCP is totally dual integral, then all basic solutions are integral. If the input matrix is sufficient or rank-symmetric, and the LCP is totally dual integral, then our result implies that the LCP always has an integral solution whenever it has a solution. We also introduce a class of matrices such that any LCP instance having the matrix as a coefficient matrix is totally dual integral. We investigate relationships between matrix classes in the LCP literature such as principally unimodular matrices. Principally unimodular matrices are known that all basic solutions to the LCP are integral for any integral input vector. In addition, we show that it is coNP-hard to decide whether a given LCP instance is totally dual integral.

AB - In this paper, we introduce total dual integrality of the linear complementarity problem (LCP) by analogy with the linear programming problem. The main idea of defining the notion is to propose the LCP with orientation, a variant of the LCP whose feasible complementary cones are specified by an additional input vector. Then we naturally define the dual problem of the LCP with orientation and total dual integrality of the LCP. We show that if the LCP is totally dual integral, then all basic solutions are integral. If the input matrix is sufficient or rank-symmetric, and the LCP is totally dual integral, then our result implies that the LCP always has an integral solution whenever it has a solution. We also introduce a class of matrices such that any LCP instance having the matrix as a coefficient matrix is totally dual integral. We investigate relationships between matrix classes in the LCP literature such as principally unimodular matrices. Principally unimodular matrices are known that all basic solutions to the LCP are integral for any integral input vector. In addition, we show that it is coNP-hard to decide whether a given LCP instance is totally dual integral.

KW - Linear complementarity problem

KW - Principal unimodularity

KW - Total dual integrality

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U2 - 10.1007/s10479-018-2926-8

DO - 10.1007/s10479-018-2926-8

M3 - Article

SP - 1

EP - 23

JO - Annals of Operations Research

JF - Annals of Operations Research

SN - 0254-5330

ER -