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Graph coloring is a fundamental problem with wide reaching applications in various areas including ata mining and databases, e.g., in parallel query optimization. In recent years, there has been a growing interest in solving various graph coloring problems in the streaming model. The initial algorithms in this line of work are all crucially randomized, raising natural questions about how important a role randomization plays in streaming graph coloring. A couple of very recent works prove that deterministic or even adversarially robust coloring algorithms (that work on streams whose updates may depend on the algorithm's past outputs) are considerably weaker than standard randomized ones. However, there is still a significant gap between the upper and lower bounds for the number of colors needed (as a function of the maximum degree Δ) for robust coloring and multipass deterministic coloring. We contribute to this line of work by proving the following results. In the deterministic semi-streaming (i.e., O(n · polylog n) space) regime, we present an algorithm that achieves a combinatorially optimal (Δ+1)-coloring using O(logΔ log logΔ) passes. This improves upon the prior O(Δ)-coloring algorithm of Assadi, Chen, and Sun (STOC 2022) at the cost of only an O(log logΔ) factor in the number of passes. In the adversarially robust semi-streaming regime, we design an O(Δ5/2)-coloring algorithm that improves upon the previously best O(Δ3)-coloring algorithm of Chakrabarti, Ghosh, and Stoeckl (ITCS 2022). Further, we obtain a smooth colors/space tradeoff that improves upon another algorithm of the said work: whereas their algorithm uses O(Δ2) colors and O(nΔ1/2) space, ours, in particular, achieves (i)~O(Δ2) colors in O(nΔ1/3) space, and (ii)~O(Δ7/4) colors in O(nΔ1/2) space.
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Brooks' Theorem in Graph Streams: A Single-Pass Semi-Streaming Algorithm for $\Delta$-Coloring
Every graph with maximum degree $$\Delta$$ can be colored with $$(\Delta+1)$$colors using a simple greedy algorithm. Remarkably, recent work has shown thatone can find such a coloring even in the semi-streaming model. But, in reality,one almost never needs $$(\Delta+1)$$ colors to properly color a graph. Indeed,the celebrated \Brooks' theorem states that every (connected) graph besidecliques and odd cycles can be colored with $$\Delta$$ colors. Can we find a$$\Delta$$-coloring in the semi-streaming model as well? We settle this key question in the affirmative by designing a randomizedsemi-streaming algorithm that given any graph, with high probability, eithercorrectly declares that the graph is not $$\Delta$$-colorable or outputs a$$\Delta$$-coloring of the graph. The proof of this result starts with a detour. We first (provably) identifythe extent to which the previous approaches for streaming coloring fail for$$\Delta$$-coloring: for instance, all these approaches can handle streams withrepeated edges and they can run in $o(n^2)$ time -- we prove that neither ofthese tasks is possible for $$\Delta$$-coloring. These impossibility resultshowever pinpoint exactly what is missing from prior approaches when it comes to$$\Delta$$-coloring. We then build on these insights to design a semi-streaming algorithm thatuses $(i)$ a novel sparse-recovery approach based on sparse-densedecompositions to (partially) recover the problematic subgraphs of the input-- the ones that form the basis of our impossibility results -- and $(ii)$ anew coloring approach for these subgraphs that allows for recoloring of othervertices in a controlled way without relying on local explorations or findingaugmenting paths that are generally impossible for semi-streaming algorithms.We believe both these techniques can be of independent interest.
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- Award ID(s):
- 2047061
- PAR ID:
- 10488808
- Publisher / Repository:
- TheoretiCS
- Date Published:
- Journal Name:
- TheoretiCS
- Volume:
- Phase 2
- ISSN:
- 2751-4838
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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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.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.46298/THEORETICS.23.9
