Linear-in-Δ lower bounds in the LOCAL model

By prior work, there is a distributed graph algorithm that finds a maximal fractional matching (maximal edge packing) in O ( Δ ) rounds, independently of n ; here Δ is the maximum degree of the graph and n is the number of nodes in the graph. We show that this is optimal: there is no distributed algorithm that finds a maximal fractional matching in o ( Δ ) rounds, independently of n . Our work gives the first linear-in- Δ lower bound for a natural graph problem in the standard LOCAL model of distributed computing—prior lower bounds for a wide range of graph problems have been at best logarithmic in Δ .