The linear complementarity problems with a few variables per constraint

Hanna Sumita, Naonori Kakimura, Kazuhisa Makino

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


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 languageEnglish
Pages (from-to)1015-1026
Number of pages12
JournalMathematics of Operations Research
Issue number4
Publication statusPublished - 2015 Nov
Externally publishedYes


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

ASJC Scopus subject areas

  • General Mathematics
  • Computer Science Applications
  • Management Science and Operations Research


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