TY - JOUR

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

AU - Sumita, Hanna

AU - Kakimura, Naonori

AU - Makino, Kazuhisa

N1 - Publisher Copyright:
© 2015 INFORMS.

PY - 2015/11

Y1 - 2015/11

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

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U2 - 10.1287/moor.2014.0708

DO - 10.1287/moor.2014.0708

M3 - Article

AN - SCOPUS:84947050901

VL - 40

SP - 1015

EP - 1026

JO - Mathematics of Operations Research

JF - Mathematics of Operations Research

SN - 0364-765X

IS - 4

ER -