On independent cliques and linear complementarity 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. Thus the maximum \( \ell _1 \) norm has an important economic interpretation as the maximum total investment over all equilibria of this game. Motivated by this, we consider a perturbation of this LCP that corresponds to a public goods game with imperfect substitutability. We identify the combinatorial structures on the graph that correspond to the maximum \(\ell _1\) norm of solutions (equivalently, investment maximizing equilibria) of the new LCP. We show that these solutions correspond to “independent cliques" – collections of cliques such that no two vertices from two distinct cliques are adjacent. When the perturbation becomes null, these cliques collapse to singletons and we recover the earlier relation to maximum independent sets. Independent cliques have been studied before as a generalization of independent sets. Our work establishes an intricate connection between independent cliques, LCPs and public goods games.