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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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From Cliques to Colorings and Back Again
We present an exact algorithm for graph coloring and maximum clique problems based on SAT technology. It relies on four sub-algorithms that alternatingly compute cliques of larger size and colorings with fewer colors. We show how these techniques can mutually help each other: larger cliques facilitate finding smaller colorings, which in turn can boost finding larger cliques. We evaluate our approach on the DIMACS graph coloring suite. For finding maximum cliques, we show that our algorithm can improve the state-of-the-art MaxSAT-based solver IncMaxCLQ, and for the graph coloring problem, we close two open instances, decrease two upper bounds, and increase one lower bound.
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- PAR ID:
- 10357179
- Date Published:
- Journal Name:
- 28th International Conference on Principles and Practice of Constraint Programming (CP 2022)
- 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/10.4230/LIPIcs.CP.2022.26
