Sparse linear complementarity problems

Hanna Sumita, Naonori Kakimura, Kazuhisa Makino

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)

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(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.

Original languageEnglish
Title of host publicationAlgorithms and Complexity - 8th International Conference, CIAC 2013, Proceedings
Pages358-369
Number of pages12
Volume7878 LNCS
DOIs
Publication statusPublished - 2013
Externally publishedYes
Event8th International Conference on Algorithms and Complexity, CIAC 2013 - Barcelona, Spain
Duration: 2013 May 222013 May 24

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume7878 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other8th International Conference on Algorithms and Complexity, CIAC 2013
CountrySpain
CityBarcelona
Period13/5/2213/5/24

Fingerprint

Linear Complementarity Problem
NP-complete problem
Linear Time
Integer
Coefficient

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

Cite this

Sumita, H., Kakimura, N., & Makino, K. (2013). Sparse linear complementarity problems. In 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.

Algorithms and Complexity - 8th International Conference, CIAC 2013, Proceedings. Vol. 7878 LNCS 2013. p. 358-369 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7878 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Sumita, H, Kakimura, N & Makino, K 2013, Sparse linear complementarity problems. in 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
Sumita H, Kakimura N, Makino K. Sparse linear complementarity problems. In Algorithms and Complexity - 8th International Conference, CIAC 2013, Proceedings. Vol. 7878 LNCS. 2013. p. 358-369. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-38233-8_30
Sumita, Hanna ; Kakimura, Naonori ; Makino, Kazuhisa. / Sparse linear complementarity problems. Algorithms and Complexity - 8th International Conference, CIAC 2013, Proceedings. Vol. 7878 LNCS 2013. pp. 358-369 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
@inproceedings{77c35ff5070d42fca0d3a05ef0a237d7,
title = "Sparse linear complementarity problems",
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(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.",
author = "Hanna Sumita and Naonori Kakimura and Kazuhisa Makino",
year = "2013",
doi = "10.1007/978-3-642-38233-8_30",
language = "English",
isbn = "9783642382321",
volume = "7878 LNCS",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
pages = "358--369",
booktitle = "Algorithms and Complexity - 8th International Conference, CIAC 2013, Proceedings",

}

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 -