## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Thursday, March 19, 2015, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Hanna Sumita (University of Tokyo)

## The Linear Complementarity Problem: Sparsity and Integrality

The linear complementarity problem (LCP) is one of the most fundamental problems in mathematical programming from both theoretical and practical points of view. In this talk, we discuss sparsity and integrality of the LCP.

As for the sparsity, it is known that 3-LCP (i.e., each row of the input matrix contains at most three nonzero entries) is NP-hard and 1-LCP is linearly solvable. We show that 2-LCP is NP-hard, and sign-balanced 2-LCP (i.e., each row of the input matrix contains at most one positive entry and at most one negative entry) can be solved in polynomial time. The latter result matches with the currently best known complexity bound for the corresponding sparse linear feasibility problem. We also discuss fixed parameter tractability of sparse LCP.

As for the integral LCP, we discuss principal unimodularity and total dual integrality of the LCP. It is known that If the input matrix is principally unimodular, then all basic solutions to the LCP are integral. We introduce total dual integrality of the 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. This allows us to define naturally its dual problem and the total dual integrality of the LCP. Then 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, then this implies that the LCP always has an integral solution whenever it has a solution.

This is a joint work with N. Kakimura and K. Makino.

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