Fair Maximal Independent Sets

Abstract

Finding a maximal independent set (MIS) is a classic problem in graph theory that has been widely studied in the context of distributed algorithms. Standard distributed solutions to the MIS problem focus on time complexity. In this paper, we also consider fairness. For a given MIS algorithm A and graph G, we define the inequality factor for A on G to be the largest ratio between the probabilities of the nodes joining an MIS in the graph. We say an algorithm is fair with respect to a family of graphs if it achieves a constant inequality factor for all graphs in the family. In this paper, we seek efficient and fair algorithms for common graph families. We begin by describing an algorithm that is fair and runs in O(log* n)-time in rooted trees of size n. Moving to unrooted trees, we describe a fair algorithm that runs in O(log n) time. Generalizing further to bipartite graphs, we describe a third fair algorithm that requires O(log2 n) rounds. We also show a fair algorithm for planar graphs that runs in O(log2 n) rounds, and describe an algorithm that can be run in any graph, yielding good bounds on inequality in regions that can be efficiently colored with a small number of colors. We conclude our theoretical analysis with a lower bound that identifies a graph where all MIS algorithms achieve an inequality bound in Ω(n)-eliminating the possibility of an MIS algorithm that is fair in all graphs. Finally, to motivate the need for provable fairness guarantees, we simulate both our tree algorithm and Luby's MIS algorithm [13] in a variety of different tree topologies-some synthetic and some derived from real world data. Whereas our algorithm always yield an inequality factor ≤3.25 in these simulations, Luby's algorithms yields factors as large as 168.