Adversarially Robust Coloring for Graph Streams

Abstract

A streaming algorithm is considered to be adversarially robust if it provides correct outputs with high probability even when the stream updates are chosen by an adversary who may observe and react to the past outputs of the algorithm. We grow the burgeoning body of work on such algorithms in a new direction by studying robust algorithms for the problem of maintaining a valid vertex coloring of an n-vertex graph given as a stream of edges. Following standard practice, we focus on graphs with maximum degree at most ∆ and aim for colorings using a small number f (∆) of colors.A recent breakthrough (Assadi, Chen, and Khanna; SODA 2019) shows that in the standard, nonrobust, streaming setting, (∆ + 1)-colorings can be obtained while using only O(n) space. Here, we prove that an adversarially robust algorithm running under a similar space bound must spend almost Ω(∆ 2 ) colors and that robust O(∆)-coloring requires a linear amount of space, namely Ω(n∆). We in fact obtain a more general lower bound, trading off the space usage against the number of colors used. From a complexity-theoretic standpoint, these lower bounds provide (i) the first significant separation between adversarially robust algorithms and ordinary randomized algorithms for a natural problem on insertion-only streams and (ii) the first significant separation between randomized and deterministic coloring algorithms for graph streams, since deterministic streaming algorithms are automatically robust.We complement our lower bounds with a suite of positive results, giving adversarially robust coloring algorithms using sublinear space. In particular, we can maintain an O(∆ 2 )-coloring using O(n √ ∆) space and an O(∆ 3 )-coloring using O(n) space.

A Framework for Adversarially Robust Streaming Algorithms

“…A first result in this flavor has very recently been established by [15], who considered the problem of coloring a graph in the semi-streaming model. They proved that coloring with few colors requires substantially more space in the adversarial model compared to the static one; for example, O (Δ) colors require Ω(nΔ) space in the robust setting but only O (n) space in the static setting [6].…”

High-performance and balanced parallel graph coloring on multicore platforms

“…Instead, our work efficiently parallelizes the distance-1 graph coloring kernel on multicore platforms, in which any two adjacent vertices of the graph connected with a direct edge are assigned with different colors. Finally, prior works propose algorithms for edge coloring [67], dynamic or streaming coloring [68][69][70][71][72][73][74], k-distance coloring [75,76] and sequential exact coloring [77][78][79][80]. All these works are not closely related to our work, since we focus on designing high-performance parallel algorithms for the distance-1 vertex graph coloring kernel.…”