5,811 research outputs found
Vertex elimination orderings for hereditary graph classes
We provide a general method to prove the existence and compute efficiently
elimination orderings in graphs. Our method relies on several tools that were
known before, but that were not put together so far: the algorithm LexBFS due
to Rose, Tarjan and Lueker, one of its properties discovered by Berry and
Bordat, and a local decomposition property of graphs discovered by Maffray,
Trotignon and Vu\vskovi\'c. We use this method to prove the existence of
elimination orderings in several classes of graphs, and to compute them in
linear time. Some of the classes have already been studied, namely
even-hole-free graphs, square-theta-free Berge graphs, universally signable
graphs and wheel-free graphs. Some other classes are new. It turns out that all
the classes that we study in this paper can be defined by excluding some of the
so-called Truemper configurations. For several classes of graphs, we obtain
directly bounds on the chromatic number, or fast algorithms for the maximum
clique problem or the coloring problem
Compact Labelings For Efficient First-Order Model-Checking
We consider graph properties that can be checked from labels, i.e., bit
sequences, of logarithmic length attached to vertices. We prove that there
exists such a labeling for checking a first-order formula with free set
variables in the graphs of every class that is \emph{nicely locally
cwd-decomposable}. This notion generalizes that of a \emph{nicely locally
tree-decomposable} class. The graphs of such classes can be covered by graphs
of bounded \emph{clique-width} with limited overlaps. We also consider such
labelings for \emph{bounded} first-order formulas on graph classes of
\emph{bounded expansion}. Some of these results are extended to counting
queries
On the Complexity of Distributed Splitting Problems
One of the fundamental open problems in the area of distributed graph
algorithms is the question of whether randomization is needed for efficient
symmetry breaking. While there are fast, -time randomized
distributed algorithms for all of the classic symmetry breaking problems, for
many of them, the best deterministic algorithms are almost exponentially
slower. The following basic local splitting problem, which is known as the
\emph{weak splitting} problem takes a central role in this context: Each node
of a graph has to be colored red or blue such that each node of
sufficiently large degree has at least one node of each color among its
neighbors. Ghaffari, Kuhn, and Maus [STOC '17] showed that this seemingly
simple problem is complete w.r.t. the above fundamental open question in the
following sense: If there is an efficient -time determinstic
distributed algorithm for weak splitting, then there is such an algorithm for
all locally checkable graph problems for which an efficient randomized
algorithm exists. In this paper, we investigate the distributed complexity of
weak splitting and some closely related problems. E.g., we obtain efficient
algorithms for special cases of weak splitting, where the graph is nearly
regular. In particular, we show that if and are the minimum
and maximum degrees of and if , weak splitting can
be solved deterministically in time
. Further, if and , there is a
randomized algorithm with time complexity
Fast Local Computation Algorithms
For input , let denote the set of outputs that are the "legal"
answers for a computational problem . Suppose and members of are
so large that there is not time to read them in their entirety. We propose a
model of {\em local computation algorithms} which for a given input ,
support queries by a user to values of specified locations in a legal
output . When more than one legal output exists for a given
, the local computation algorithm should output in a way that is consistent
with at least one such . Local computation algorithms are intended to
distill the common features of several concepts that have appeared in various
algorithmic subfields, including local distributed computation, local
algorithms, locally decodable codes, and local reconstruction.
We develop a technique, based on known constructions of small sample spaces
of -wise independent random variables and Beck's analysis in his algorithmic
approach to the Lov{\'{a}}sz Local Lemma, which under certain conditions can be
applied to construct local computation algorithms that run in {\em
polylogarithmic} time and space. We apply this technique to maximal independent
set computations, scheduling radio network broadcasts, hypergraph coloring and
satisfying -SAT formulas.Comment: A preliminary version of this paper appeared in ICS 2011, pp. 223-23
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
Fast Distributed Approximation for Max-Cut
Finding a maximum cut is a fundamental task in many computational settings.
Surprisingly, it has been insufficiently studied in the classic distributed
settings, where vertices communicate by synchronously sending messages to their
neighbors according to the underlying graph, known as the or
models. We amend this by obtaining almost optimal
algorithms for Max-Cut on a wide class of graphs in these models. In
particular, for any , we develop randomized approximation
algorithms achieving a ratio of to the optimum for Max-Cut on
bipartite graphs in the model, and on general graphs in the
model.
We further present efficient deterministic algorithms, including a
-approximation for Max-Dicut in our models, thus improving the best known
(randomized) ratio of . Our algorithms make non-trivial use of the greedy
approach of Buchbinder et al. (SIAM Journal on Computing, 2015) for maximizing
an unconstrained (non-monotone) submodular function, which may be of
independent interest
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