Sublinear Algorithms for (Δ + 1) Vertex Coloring

Abstract

Any graph with maximum degree ∆ admits a proper vertex coloring with ∆ + 1 colors that can be found via a simple sequential greedy algorithm in linear time and space. But can one find such a coloring via a sublinear algorithm?We answer this fundamental question in the affirmative for several canonical classes of sublinear algorithms including graph streaming, sublinear time, and massively parallel computation (MPC) algorithms. In particular, we design:

Polynomial pass lower bounds for graph streaming algorithms

“…Up until this point, there is no contradiction as the answer to inputs (1, * ),(1, * ) to Alice and Bob is always 1 and hence there is no problem with the corresponding transcripts in Π [1 * , 1 * ] to be similar (similarly for Π [ * 1, * 1] separately). However, we combine this with the cut-and-paste property of randomized protocols based on Hellinger distance (see Fact B.14) to argue that in fact the distribution of Π [10,10] and Π [01, 01] are also similar. This then implies that Π [1 * , 1 * ] essentially has the same distribution as Π [ * 1, * 1] ; but then this is a contradiction as the answer to the protocol (which is only a function of the transcript) needs to be different between these two types of inputs.…”

“…Using a reduction from hidden-pointer chasing, we prove that any algorithm for submodular function minimization needs to make n 2−o(1) value queries to the function unless it has a polynomial degree of adaptivity.A vast body of work in graph streaming lower bounds concerns algorithms that make only one or a few passes over the stream. Examples of single-pass lower bounds include the ones for diameter [60], approximate matchings [13,14,63,84], exact minimum/maximum cuts [119], and maximal independent sets [10,46]. Examples of multi-pass lower bounds include the ones for BFS trees [60], perfect matchings [67], shortest path [67], and minimum vertex cover and dominating set [71].…”

“…Any p-pass streaming algorithm that with constant probability finds a lexicographically first maximal independent set of in a graph requires Ω(n 2 /p 5 ) space.The lexicographically-first MIS has a rich history in computer science and in particular parallel algorithms [5,29,44,97]. However, even though multiple variants of the independent set problem have been studied in the streaming model [10,45,46,62,[68][69][70], we are not aware of any work on this particular problem (we remark that standard MIS problem admits an O(n) space O(log log n) pass algorithm [62]). Besides being a fundamental problem in its own right, what makes this problem appealing for us is that it nicely illustrates the power of our techniques compared to previous approaches.…”

“…Recall that Π [x 1 x 2 , y 1 y 2 ] denotes the transcript of the protocol conditioned on the input (x 1 , x 2 ) to Alice, and (y 1 , y 2 ) to Bob. (1) h 2 (Π [11,10] , Π [10,10]…”

“…Define Π [1 * , 10] as the distribution of Π conditioned on the given value for x 1 , y 1 , y 2 (leaving out the assignment for x 2 ). We have, h 2 (Π [10,10] , Π [11,10] ).…”