134 research outputs found
Brief Announcement: Streaming and Massively Parallel Algorithms for Edge Coloring
A valid edge-coloring of a graph is an assignment of "colors" to its edges such that no two incident edges receive the same color. The goal is to find a proper coloring that uses few colors. In this paper, we revisit this problem in two models of computation specific to massive graphs, the Massively Parallel Computations (MPC) model and the Graph Streaming model:
Massively Parallel Computation. We give a randomized MPC algorithm that w.h.p., returns a (1+o(1))Delta edge coloring in O(1) rounds using O~(n) space per machine and O(m) total space. The space per machine can also be further improved to n^{1-Omega(1)} if Delta = n^{Omega(1)}. This is, to our knowledge, the first constant round algorithm for a natural graph problem in the strongly sublinear regime of MPC. Our algorithm improves a previous result of Harvey et al. [SPAA 2018] which required n^{1+Omega(1)} space to achieve the same result.
Graph Streaming. Since the output of edge-coloring is as large as its input, we consider a standard variant of the streaming model where the output is also reported in a streaming fashion. The main challenge is that the algorithm cannot "remember" all the reported edge colors, yet has to output a proper edge coloring using few colors.
We give a one-pass O~(n)-space streaming algorithm that always returns a valid coloring and uses 5.44 Delta colors w.h.p., if the edges arrive in a random order. For adversarial order streams, we give another one-pass O~(n)-space algorithm that requires O(Delta^2) colors
Two lower bounds for -centered colorings
Given a graph and an integer , a coloring is
\emph{-centered} if for every connected subgraph of , either uses
more than colors on or there is a color that appears exactly once in
. The notion of -centered colorings plays a central role in the theory of
sparse graphs. In this note we show two lower bounds on the number of colors
required in a -centered coloring.
First, we consider monotone classes of graphs whose shallow minors have
average degree bounded polynomially in the radius, or equivalently (by a result
of Dvo\v{r}\'ak and Norin), admitting strongly sublinear separators. We
construct such a class such that -centered colorings require a number of
colors super-polynomial in . This is in contrast with a recent result of
Pilipczuk and Siebertz, who established a polynomial upper bound in the special
case of graphs excluding a fixed minor.
Second, we consider graphs of maximum degree . D\k{e}bski, Felsner,
Micek, and Schr\"{o}der recently proved that these graphs have -centered
colorings with colors. We show that there are graphs of
maximum degree that require colors in any -centered coloring, thus matching their
upper bound up to a logarithmic factor.Comment: v3: final version with journal layout v2: revised following referees'
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