Distributed Algorithms for Near-Equitable Coloring

For an $n$-vertex graph of maximum degree $\Delta$ and diameter $D$, an equitable $(\Delta+1)$-coloring is a vertex coloring where the frequency of each color (namely, the number of vertices it colors) are all equal to $\sigma=n/(\Delta+1)$ (up to rounding). The Hajnal-Szemer\'edi Theorem guarantees the existence of such a coloring for every graph, and an $O(n^2\Delta)$ time sequential algorithm is known for computing such a coloring. Here, we study near-equitable graph coloring in distributed networks. The main question of interest is how close one can remain to the desired palette size of $\Delta+1$ while computing, in few distributed rounds, a coloring whose frequencies are close to $\sigma$. It appears that these two conflicting parameters exhibit a tradeoff, which we attempt to explore. We present a suite of fast randomized distributed algorithms representing varying points on this tradeoff, analyze their properties, and study their time complexity in the sequential, CONGEST and Congested Clique (CC) models.