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 Δ and aim for colorings using a small number f(Δ) of colors.
A recent breakthrough (Assadi, Chen, and Khanna; SODA 2019) shows that in the standard, non-robust, streaming setting, (Δ+1)-colorings can be obtained while using only Õ(n) space. Here, we prove that an adversarially robust algorithm running under a similar space bound must spend almost Ω(Δ²) colors and that robust O(Δ)-coloring requires a linear amount of space, namely Ω(nΔ). 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(Δ²)-coloring using Õ(n √Δ) space and an O(Δ³)-coloring using Õ(n) space.
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Graphs with Tunable Chromatic Numbers for Parallel Coloring
We consider how to generate graphs of arbitrary size whose chromatic numbers can be chosen (or are well-bounded) for testing graph coloring algorithms on parallel computers. For the distance-1 graph coloring problem, we identify three classes of graphs with this property. The first is the Erdős-Rényi random graph with prescribed expected degree, where the chromatic number is known with high probability. It is also known that the Greedy algorithm colors this graph using at most twice the number of colors as the chromatic number. The second is a random geometric graph embedded in hyperbolic space where the size of the maximum clique provides a tight lower bound on the chromatic number. The third is a deterministic graph described by Mycielski, where the graph is recursively constructed such that its chromatic number is known and increases with graph size, although the size of the maximum clique remains two. For Jacobian estimation, we bound the distance-2 chromatic number of random bipartite graphs by considering its equivalence to distance-1 coloring of an intersection graph. We use a “balls and bins” probabilistic analysis to establish a lower bound and an upper bound on the distance-2 chromatic number. The regimes of graph sizes and probabilities that we consider are chosen to suit the Jacobian estimation problem, where the number of columns and rows are asymptotically nearly equal, and have number of nonzeros linearly related to the number of columns. Computationally we verify the theoretical predictions and show that the graphs are often be colored optimally by the serial and parallel Greedy algorithms.
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- Award ID(s):
- 1637534
- NSF-PAR ID:
- 10181552
- Date Published:
- Journal Name:
- SIAM 2020 Workshop on Combinatorial Scientific Computing
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1137/1.9781611976229.6
