Computing inductive vertex orderings

Abstract

Inductive vertex orderings or edge orientations of graphs, such as inductive k-independence and k-simpliciality orderings, and k-perfect orientation, arise naturally in geometric intersection graphs and have been systematically studied in Ye and Borodin (2012) [4] and Kammer and Tholey (2014) [5]. In general, those orderings and orientations are hard to compute, even approximately. In this note, we show that an inductive k-independence ordering can be computed in k-simplicial graphs, and a k-approximation (that is, a solution with approximation ratio at most k) to the optimal inductive independence ordering can be computed in k-perfectly orientable graphs. As a result, we obtain a k-approximation (2k-approximation) algorithm for maximum weight independent sets in k-simplicial (k-perfectly orientable, respectively) graphs that runs in polynomial time independent of k without prior knowledge of the ordering.