Graph Coloring and Semidefinite Rank
Renee Mirka, Devin Smedira, and David P. Williamson

TL;DR
This paper explores the relationship between semidefinite programming, matrix rank, and graph coloring, providing characterizations for specific graph classes and proposing a heuristic for coloring planar graphs.
Contribution
It characterizes graphs with high dual rank solutions, links dual rank to graph colorability, and introduces a cost-based heuristic for graph coloring.
Findings
High dual rank solutions exist for k-trees and certain planar graphs.
Non-uniquely colorable graphs lack high-rank dual solutions.
Constructed cost functions enable effective heuristics for planar graph coloring.
Abstract
This paper considers the interplay between semidefinite programming, matrix rank, and graph coloring. Karger, Motwani, and Sudan \cite{KMS98} give a vector program for which a coloring of the graph can be encoded as a semidefinite matrix of low rank. By complementary slackness conditions of semidefinite programming, if an optimal dual solution has sufficiently high rank, any optimal primal solution must have low rank. We attempt to characterize graphs for which we can show that the corresponding dual optimal solution must have sufficiently high rank. In the case of the original Karger, Motwani, and Sudan vector program, we show that any graph which is a -tree has sufficiently high dual rank, and we can extract the coloring from the corresponding low-rank primal solution. We can also show that if the graph is not uniquely colorable, then no sufficiently high rank dual optimal solution…
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Taxonomy
TopicsNuclear Receptors and Signaling · Advanced Graph Theory Research · Complexity and Algorithms in Graphs
