Exact Tie-Breaking and Tight Degree Bounds for Minimum-Degree Greedy on Interval Graphs

Abstract

We study tie-breaking in the minimum-degree greedy algorithm (MDG) for maximum independent set on interval graphs. Given an interval representation of an $n$-vertex graph, we compute the largest and smallest output sizes and construct executions attaining them in $O(n^3)$ time and $O(n^2)$ space. For maximum degree at most four, favorable tie-breaking gives a tight guarantee of $3/4$. For every maximum-degree bound of at least five, the guarantee drops to $2/3$. Both bounds are tight even on connected unit interval graphs whose output size is independent of tie-breaking. Together with the known optimality of favorable tie-breaking for maximum degree at most three, these results give a complete classification by degree bound.