1,821 research outputs found
On Computing Maximal Independent Sets of Hypergraphs in Parallel
Whether or not the problem of finding maximal independent sets (MIS) in
hypergraphs is in (R)NC is one of the fundamental problems in the theory of
parallel computing. Unlike the well-understood case of MIS in graphs, for the
hypergraph problem, our knowledge is quite limited despite considerable work.
It is known that the problem is in \emph{RNC} when the edges of the hypergraph
have constant size. For general hypergraphs with vertices and edges,
the fastest previously known algorithm works in time with
processors. In this paper we give an EREW PRAM algorithm
that works in time with processors on general
hypergraphs satisfying , where
and . Our algorithm is
based on a sampling idea that reduces the dimension of the hypergraph and
employs the algorithm for constant dimension hypergraphs as a subroutine
Distributed Symmetry Breaking in Hypergraphs
Fundamental local symmetry breaking problems such as Maximal Independent Set
(MIS) and coloring have been recognized as important by the community, and
studied extensively in (standard) graphs. In particular, fast (i.e.,
logarithmic run time) randomized algorithms are well-established for MIS and
-coloring in both the LOCAL and CONGEST distributed computing
models. On the other hand, comparatively much less is known on the complexity
of distributed symmetry breaking in {\em hypergraphs}. In particular, a key
question is whether a fast (randomized) algorithm for MIS exists for
hypergraphs.
In this paper, we study the distributed complexity of symmetry breaking in
hypergraphs by presenting distributed randomized algorithms for a variety of
fundamental problems under a natural distributed computing model for
hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can
be solved in rounds ( is the number of nodes of the
hypergraph) in the LOCAL model. We then present a key result of this paper ---
an -round hypergraph MIS algorithm in
the CONGEST model where is the maximum node degree of the hypergraph
and is any arbitrarily small constant.
To demonstrate the usefulness of hypergraph MIS, we present applications of
our hypergraph algorithm to solving problems in (standard) graphs. In
particular, the hypergraph MIS yields fast distributed algorithms for the {\em
balanced minimal dominating set} problem (left open in Harris et al. [ICALP
2013]) and the {\em minimal connected dominating set problem}. We also present
distributed algorithms for coloring, maximal matching, and maximal clique in
hypergraphs.Comment: Changes from the previous version: More references adde
Distributed local approximation algorithms for maximum matching in graphs and hypergraphs
We describe approximation algorithms in Linial's classic LOCAL model of
distributed computing to find maximum-weight matchings in a hypergraph of rank
. Our main result is a deterministic algorithm to generate a matching which
is an -approximation to the maximum weight matching, running in rounds. (Here, the
notations hides and factors).
This is based on a number of new derandomization techniques extending methods
of Ghaffari, Harris & Kuhn (2017).
As a main application, we obtain nearly-optimal algorithms for the
long-studied problem of maximum-weight graph matching. Specifically, we get a
approximation algorithm using randomized time and deterministic time.
The second application is a faster algorithm for hypergraph maximal matching,
a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of
local graph algorithms. This gives an algorithm for -edge-list
coloring in rounds deterministically or
rounds randomly. Another consequence (with
additional optimizations) is an algorithm which generates an edge-orientation
with out-degree at most for a graph of
arboricity ; for fixed this runs in
rounds deterministically or rounds randomly
Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching
We present a deterministic distributed algorithm that computes a
-edge-coloring, or even list-edge-coloring, in any -node graph
with maximum degree , in rounds. This answers
one of the long-standing open questions of \emph{distributed graph algorithms}
from the late 1980s, which asked for a polylogarithmic-time algorithm. See,
e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and
Elkin. The previous best round complexities were by
Panconesi and Srinivasan [STOC'92] and
by Fraigniaud, Heinrich, and Kosowski [FOCS'16]. A corollary of our
deterministic list-edge-coloring also improves the randomized complexity of
-edge-coloring to poly rounds.
The key technical ingredient is a deterministic distributed algorithm for
\emph{hypergraph maximal matching}, which we believe will be of interest beyond
this result. In any hypergraph of rank --- where each hyperedge has at most
vertices --- with nodes and maximum degree , this algorithm
computes a maximal matching in rounds.
This hypergraph matching algorithm and its extensions lead to a number of
other results. In particular, a polylogarithmic-time deterministic distributed
maximal independent set algorithm for graphs with bounded neighborhood
independence, hence answering Open Problem 5 of Barenboim and Elkin's book, a
-round deterministic
algorithm for -approximation of maximum matching, and a
quasi-polylogarithmic-time deterministic distributed algorithm for orienting
-arboricity graphs with out-degree at most ,
for any constant , hence partially answering Open Problem 10 of
Barenboim and Elkin's book
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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