A linear complementarity based characterization of the weighted independence number and the independent domination number in graphs
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 Lovász 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2016
- DOI:
- 10.48550/arXiv.1603.05075
- arXiv:
- arXiv:1603.05075
- Bibcode:
- 2016arXiv160305075P
- Keywords:
-
- Computer Science - Discrete Mathematics;
- Computer Science - Computational Complexity;
- Mathematics - Combinatorics;
- Mathematics - Optimization and Control;
- 05C69;
- 68R10;
- 90C33;
- 90C27;
- 90C26
- E-Print:
- 16 pages