Abstract
In this paper, we study the parameterized complexity of the linear complementarity problem (LCP), which is one of the most fundamental mathematical optimization problems. The parameters we focus on are the sparsities of the input and the output of the LCP: the maximum numbers of nonzero entries per row/column in the coefficient matrix and the number of nonzero entries in a solution. Our main result is to present a fixed-parameter algorithm for the LCP with the combined parameter. We also show that if we drop any of the three parameters, then the LCP is NP-hard or W[1]-hard. In addition, we show the nonexistence of a polynomial kernel for the LCP unless coNP $$\subseteq $$ NP/poly.
Highlights
Given a square matrix M ∈ Rn×n and a vector q ∈ Rn, the linear complementarity problem (LCP) is to find a vector z ∈ Rn such thatM z + q ≥ 0, z ≥ 0, z (M z + q) = 0. (1)We denote a problem instance of the LCP with M and q by LCP (M, q)
We analyze the parameterized complexity of the LCP
We focus on the sparsities of the input and the output of the LCP
Summary
Given a square matrix M ∈ Rn×n and a vector q ∈ Rn, the linear complementarity problem (LCP) is to find a vector z ∈ Rn such that. The -sparse bimatrix game has payoff matrices each of whose rows and columns has at most nonzero entries, which can be described as the ( , )-sparse LCP when a mixed Nash equilibrium has a positive expected payoff. It is known that the mean payoff game falls into the (3, c)-sparse LCP [2], where c is the maximum indegree of a given graph plus two Another parameter discussed in this paper is the size of the support of a solution, that is, the number of nonzero entries of an output solution. We remark that the s-support LCP with only parameter s is W [2]-hard by reducing the problem of finding a Nash equilibrium with support size at most s of the bimatrix game [16].
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