On Independent Cliques and Linear Complementarity Problems

TL;DR

This paper explores the relationship between independent cliques in graphs and solutions to a perturbed Linear Complementarity Problem, identifying structures that maximize the solution norm and extending previous characterizations.

Contribution

It introduces independent clique solutions and analyzes their role in maximizing the LCP solutions' norm under perturbations, linking graph structures to LCP solutions.

Findings

Independent clique solutions support maximum LCP solutions for small perturbations.

The maximum $oldsymbol{ ext{l}_1}$ norm solutions correspond to independent clique structures.

Extension of previous graph-LCP characterizations to perturbed problems.

Abstract

In recent work (Pandit and Kulkarni [Discrete Applied Mathematics, 244 (2018), pp. 155--169]), the independence number of a graph was characterized as the maximum of the $\ell_1$ norm of solutions of a Linear Complementarity Problem (\LCP) defined suitably using parameters of the graph. Solutions of this LCP have another relation, namely, that they corresponded to Nash equilibria of a public goods game. Motivated by this, we consider a perturbation of this LCP and identify the combinatorial structures on the graph that correspond to the maximum $\ell_1$ norm of solutions of the new LCP. We introduce a new concept called independent clique solutions which are solutions of the LCP that are supported on independent cliques and show that for small perturbations, such solutions attain the maximum $\ell_1$ norm amongst all solutions of the new LCP.