28,808 research outputs found
Local Conflict Coloring
Locally finding a solution to symmetry-breaking tasks such as
vertex-coloring, edge-coloring, maximal matching, maximal independent set,
etc., is a long-standing challenge in distributed network computing. More
recently, it has also become a challenge in the framework of centralized local
computation. We introduce conflict coloring as a general symmetry-breaking task
that includes all the aforementioned tasks as specific instantiations ---
conflict coloring includes all locally checkable labeling tasks from
[Naor\&Stockmeyer, STOC 1993]. Conflict coloring is characterized by two
parameters and , where the former measures the amount of freedom given
to the nodes for selecting their colors, and the latter measures the number of
constraints which colors of adjacent nodes are subject to.We show that, in the
standard LOCAL model for distributed network computing, if l/d \textgreater{}
\Delta, then conflict coloring can be solved in rounds in -node graphs with maximum degree
, where ignores the polylog factors in . The
dependency in~ is optimal, as a consequence of the lower
bound by [Linial, SIAM J. Comp. 1992] for -coloring. An important
special case of our result is a significant improvement over the best known
algorithm for distributed -coloring due to [Barenboim, PODC 2015],
which required rounds. Improvements for other
variants of coloring, including -list-coloring,
-edge-coloring, -coloring, etc., also follow from our general
result on conflict coloring. Likewise, in the framework of centralized local
computation algorithms (LCAs), our general result yields an LCA which requires
a smaller number of probes than the previously best known algorithm for
vertex-coloring, and works for a wide range of coloring problems
Local Conflict Coloring Revisited: Linial for Lists
Linial's famous color reduction algorithm reduces a given -coloring of a
graph with maximum degree to a -coloring, in a
single round in the LOCAL model. We show a similar result when nodes are
restricted to choose their color from a list of allowed colors: given an
-coloring in a directed graph of maximum outdegree , if every node
has a list of size from a color space then they can select a color
in two rounds in the LOCAL model. Moreover, the communication of a node
essentially consists of sending its list to the neighbors. This is obtained as
part of a framework that also contains Linial's color reduction (with an
alternative proof) as a special case. Our result also leads to a defective list
coloring algorithm. As a corollary, we improve the state-of-the-art truly local
-list coloring algorithm from Barenboim et al. [PODC'18] by slightly
reducing the runtime to and significantly
reducing the message size (from huge to roughly ). Our techniques are
inspired by the local conflict coloring framework of Fraigniaud et al.
[FOCS'16].Comment: to appear at DISC 202
Proper Conflict-free Coloring of Graphs with Large Maximum Degree
A proper coloring of a graph is conflict-free if, for every non-isolated
vertex, some color is used exactly once on its neighborhood. Caro,
Petru\v{s}evski, and \v{S}krekovski proved that every graph has a proper
conflict-free coloring with at most colors and conjectured that
colors suffice for every connected graph with . Our first main result is that even for list-coloring, colors suffice for every graph
with ; we also prove slightly weaker bounds for all
graphs with . These results follow from our more general
framework on proper conflict-free list-coloring of a pair consisting of a graph
and a ``conflict'' hypergraph . As another corollary of our
results in this general framework, every graph has a proper
-list-coloring such that every bi-chromatic
component is a path on at most three vertices, where the number of colors is
optimal up to a constant factor. Our proof uses a fairly new type of recursive
counting argument called Rosenfeld counting, which is a variant of the
Lov\'{a}sz Local Lemma or entropy compression.
We also prove an asymptotically optimal result for a fractional analogue of
our general framework for proper conflict-free coloring for pairs of a graph
and a conflict hypergraph. A corollary states that every graph has a
fractional -coloring such that every fractionally
bi-chromatic component has at most two vertices. In particular, it implies that
the fractional analogue of the conjecture of Caro et al. holds asymptotically
in a strong sense
Fast Coloring Despite Congested Relays
We provide a -round randomized algorithm for distance-2
coloring in CONGEST with colors. For
, this improves exponentially on the
algorithm of [Halld\'orsson,
Kuhn, Maus, Nolin, DISC'20].
Our study is motivated by the ubiquity and hardness of local reductions in
CONGEST. For instance, algorithms for the Local Lov\'asz Lemma [Moser, Tardos,
JACM'10; Fischer, Ghaffari, DISC'17; Davies, SODA'23] usually assume
communication on the conflict graph, which can be simulated in LOCAL with only
constant overhead, while this may be prohibitively expensive in CONGEST. We
hope our techniques help tackle in CONGEST other coloring problems defined by
local relations.Comment: 37 pages. To appear in proceedings of DISC 202
On the Complexity of Local Distributed Graph Problems
This paper is centered on the complexity of graph problems in the
well-studied LOCAL model of distributed computing, introduced by Linial [FOCS
'87]. It is widely known that for many of the classic distributed graph
problems (including maximal independent set (MIS) and -vertex
coloring), the randomized complexity is at most polylogarithmic in the size
of the network, while the best deterministic complexity is typically
. Understanding and narrowing down this exponential gap
is considered to be one of the central long-standing open questions in the area
of distributed graph algorithms. We investigate the problem by introducing a
complexity-theoretic framework that allows us to shed some light on the role of
randomness in the LOCAL model. We define the SLOCAL model as a sequential
version of the LOCAL model. Our framework allows us to prove completeness
results with respect to the class of problems which can be solved efficiently
in the SLOCAL model, implying that if any of the complete problems can be
solved deterministically in rounds in the LOCAL model, we can
deterministically solve all efficient SLOCAL-problems (including MIS and
-coloring) in rounds in the LOCAL model. We show
that a rather rudimentary looking graph coloring problem is complete in the
above sense: Color the nodes of a graph with colors red and blue such that each
node of sufficiently large polylogarithmic degree has at least one neighbor of
each color. The problem admits a trivial zero-round randomized solution. The
result can be viewed as showing that the only obstacle to getting efficient
determinstic algorithms in the LOCAL model is an efficient algorithm to
approximately round fractional values into integer values
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