Approximation Bounds For Minimum Degree Matching

TL;DR

This paper analyzes the approximation performance of the MINGREEDY strategy for maximum matching, establishing tight bounds for graphs with bounded degree and demonstrating optimality among adaptive priority algorithms.

Contribution

It provides the first tight worst-case approximation bounds for MINGREEDY on bounded degree graphs and proves its optimality among adaptive priority algorithms.

Findings

MINGREEDY achieves an approximation ratio of at least (Δ-1)/(2Δ-3).

The bounds are tight and optimal among adaptive priority algorithms.

No better worst-case bounds are known for randomized greedy strategies on small degree graphs.

Abstract

We consider the MINGREEDY strategy for Maximum Cardinality Matching. MINGREEDY repeatedly selects an edge incident with a node of minimum degree. For graphs of degree at most $\Delta$ we show that MINGREEDY achieves approximation ratio at least $ \frac{\Delta-1}{2\Delta-3} $ in the worst case and that this performance is optimal among adaptive priority algorithms in the vertex model, which include many prominent greedy matching heuristics. Even when considering expected approximation ratios of randomized greedy strategies, no better worst case bounds are known for graphs of small degrees.