A linear complementarity based characterization of the weighted independence number and the independent domination number in graphs

TL;DR

This paper introduces a novel continuous optimization approach using linear complementarity problems to characterize the weighted independence number and independent domination number in graphs, providing new theoretical insights and bounds.

Contribution

It presents a continuous optimization formulation for graph invariants via LCPs, and derives new graph-theoretic results including a stronger Lovász theta and conditions for well-covered graphs.

Findings

Weighted independence number characterized as max weighted ℓ₁ norm over LCP solutions

Lower bound on independent domination number via minimum ℓ₁ norm of LCP solutions

LPCCs are hard to approximate, indicating computational complexity

Abstract

The linear complementarity problem is a continuous optimization problem that generalizes convex quadratic programming, Nash equilibria of bimatrix games and several such problems. This paper presents a continuous optimization formulation for the weighted independence number of a graph by characterizing it as the maximum weighted $\ell_1$ norm over the solution set of a linear complementarity problem (LCP). The minimum $\ell_1$ norm of solutions of this LCP is a lower bound on the independent domination number of the graph. Unlike the case of the maximum $\ell_1$ norm, this lower bound is in general weak, but we show it to be tight if the graph is a forest. Using methods from the theory of LCPs, we obtain a few graph theoretic results. In particular, we provide a stronger variant of the Lov\'{a}sz theta of a graph. We then provide sufficient conditions for a graph to be well-covered, i.e., for all maximal independent sets to also be maximum. This condition is also shown to be necessary for well-coveredness if the graph is a forest. Finally, the reduction of the maximum independent set problem to a linear program with (linear) complementarity constraints (LPCC) shows that LPCCs are hard to approximate.