113 research outputs found
The -strong induced arboricity of a graph
The induced arboricity of a graph is the smallest number of induced
forests covering the edges of . This is a well-defined parameter bounded
from above by the number of edges of when each forest in a cover consists
of exactly one edge. Not all edges of a graph necessarily belong to induced
forests with larger components. For , we call an edge -valid if it
is contained in an induced tree on edges. The -strong induced arboricity
of , denoted by , is the smallest number of induced forests with
components of sizes at least that cover all -valid edges in . This
parameter is highly non-monotone.
However, we prove that for any proper minor-closed graph class ,
and more generally for any class of bounded expansion, and any , the
maximum value of for is bounded from above by a
constant depending only on and . This implies that the
adjacent closed vertex-distinguishing number of graphs from a class of bounded
expansion is bounded by a constant depending only on the class. We further
prove that for any graph of tree-width~
and that for any graph of tree-depth . In addition, we
prove that when is planar.Comment: 24 pages, 11 figure
Acyclic homomorphisms to stars of graph Cartesian products and chordal bipartite graphs
AbstractHomomorphisms to a given graph H (H-colourings) are considered in the literature among other graph colouring concepts. We restrict our attention to a special class of H-colourings, namely H is assumed to be a star. Our additional requirement is that the set of vertices of a graph G mapped into the central vertex of the star and any other colour class induce in G an acyclic subgraph. We investigate the existence of such a homomorphism to a star of given order. The complexity of this problem is studied. Moreover, the smallest order of a star for which a homomorphism of a given graph G with desired features exists is considered. Some exact values and many bounds of this number for chordal bipartite graphs, cylinders, grids, in particular hypercubes, are given. As an application of these results, we obtain some bounds on the cardinality of the minimum feedback vertex set for specified graph classes
Higher Lower Bounds from the 3SUM Conjecture
The 3SUM conjecture has proven to be a valuable tool for proving conditional
lower bounds on dynamic data structures and graph problems. This line of work
was initiated by P\v{a}tra\c{s}cu (STOC 2010) who reduced 3SUM to an offline
SetDisjointness problem. However, the reduction introduced by P\v{a}tra\c{s}cu
suffers from several inefficiencies, making it difficult to obtain tight
conditional lower bounds from the 3SUM conjecture.
In this paper we address many of the deficiencies of P\v{a}tra\c{s}cu's
framework. We give new and efficient reductions from 3SUM to offline
SetDisjointness and offline SetIntersection (the reporting version of
SetDisjointness) which leads to polynomially higher lower bounds on several
problems. Using our reductions, we are able to show the essential optimality of
several algorithms, assuming the 3SUM conjecture.
- Chiba and Nishizeki's -time algorithm (SICOMP 1985) for
enumerating all triangles in a graph with arboricity/degeneracy is
essentially optimal, for any .
- Bj{\o}rklund, Pagh, Williams, and Zwick's algorithm (ICALP 2014) for
listing triangles is essentially optimal (assuming the matrix
multiplication exponent is ).
- Any static data structure for SetDisjointness that answers queries in
constant time must spend time in preprocessing, where
is the size of the set system.
These statements were unattainable via P\v{a}tra\c{s}cu's reductions.
We also introduce several new reductions from 3SUM to pattern matching
problems and dynamic graph problems. Of particular interest are new conditional
lower bounds for dynamic versions of Maximum Cardinality Matching, which
introduce a new technique for obtaining amortized lower bounds.Comment: Full version of SODA 2016 pape
Distributed Dominating Set Approximations beyond Planar Graphs
The Minimum Dominating Set (MDS) problem is one of the most fundamental and
challenging problems in distributed computing. While it is well-known that
minimum dominating sets cannot be approximated locally on general graphs, over
the last years, there has been much progress on computing local approximations
on sparse graphs, and in particular planar graphs.
In this paper we study distributed and deterministic MDS approximation
algorithms for graph classes beyond planar graphs. In particular, we show that
existing approximation bounds for planar graphs can be lifted to bounded genus
graphs, and present (1) a local constant-time, constant-factor MDS
approximation algorithm and (2) a local -time
approximation scheme. Our main technical contribution is a new analysis of a
slightly modified variant of an existing algorithm by Lenzen et al.
Interestingly, unlike existing proofs for planar graphs, our analysis does not
rely on direct topological arguments.Comment: arXiv admin note: substantial text overlap with arXiv:1602.0299
Cover and variable degeneracy
Let be a nonnegative integer valued function on the vertex set of a
graph. A graph is {\bf strictly -degenerate} if each nonempty subgraph
has a vertex such that . In this
paper, we define a new concept, strictly -degenerate transversal, which
generalizes list coloring, signed coloring, DP-coloring, -forested-coloring,
and -partition. A {\bf cover} of a graph is a
graph with vertex set , where ; the edge set , where is a matching between
and . A vertex set is a {\bf transversal} of if
for each . A transversal is a {\bf
strictly -degenerate transversal} if is strictly -degenerate. The
main result of this paper is a degree type result, which generalizes Brooks'
theorem, Gallai's theorem, degree-choosable result, signed degree-colorable
result, and DP-degree-colorable result. Similar to Borodin, Kostochka and
Toft's variable degeneracy, this degree type result is also self-strengthening.
We also give some structural results on critical graphs with respect to
strictly -degenerate transversal. Using these results, we can uniformly
prove many new and known results. In the final section, we pose some open
problems
Distributed Deterministic Edge Coloring using Bounded Neighborhood Independence
We study the {edge-coloring} problem in the message-passing model of
distributed computing. This is one of the most fundamental and well-studied
problems in this area. Currently, the best-known deterministic algorithms for
(2Delta -1)-edge-coloring requires O(Delta) + log-star n time \cite{PR01},
where Delta is the maximum degree of the input graph. Also, recent results of
\cite{BE10} for vertex-coloring imply that one can get an
O(Delta)-edge-coloring in O(Delta^{epsilon} \cdot \log n) time, and an
O(Delta^{1 + epsilon})-edge-coloring in O(log Delta log n) time, for an
arbitrarily small constant epsilon > 0.
In this paper we devise a drastically faster deterministic edge-coloring
algorithm. Specifically, our algorithm computes an O(Delta)-edge-coloring in
O(Delta^{epsilon}) + log-star n time, and an O(Delta^{1 +
epsilon})-edge-coloring in O(log Delta) + log-star n time. This result improves
the previous state-of-the-art {exponentially} in a wide range of Delta,
specifically, for 2^{Omega(\log-star n)} \leq Delta \leq polylog(n). In
addition, for small values of Delta our deterministic algorithm outperforms all
the existing {randomized} algorithms for this problem.
On our way to these results we study the {vertex-coloring} problem on the
family of graphs with bounded {neighborhood independence}. This is a large
family, which strictly includes line graphs of r-hypergraphs for any r = O(1),
and graphs of bounded growth. We devise a very fast deterministic algorithm for
vertex-coloring graphs with bounded neighborhood independence. This algorithm
directly gives rise to our edge-coloring algorithms, which apply to {general}
graphs.
Our main technical contribution is a subroutine that computes an
O(Delta/p)-defective p-vertex coloring of graphs with bounded neighborhood
independence in O(p^2) + \log-star n time, for a parameter p, 1 \leq p \leq
Delta
The Three Tree Theorem
We prove that every 2-sphere graph different from a prism can be vertex
4-colored in such a way that all Kempe chains are forests. This implies the
following three tree theorem: the arboricity of a discrete 2-sphere is 3.
Moreover, the three trees can be chosen so that each hits every triangle. A
consequence is a result of an exercise in the book of Bondy and Murty based on
work of A. Frank, A. Gyarfas and C. Nash-Williams: the arboricity of a planar
graph is less or equal than 3.Comment: 17 pages, 3 figure
Fully Dynamic Matching in Bipartite Graphs
Maximum cardinality matching in bipartite graphs is an important and
well-studied problem. The fully dynamic version, in which edges are inserted
and deleted over time has also been the subject of much attention. Existing
algorithms for dynamic matching (in general graphs) seem to fall into two
groups: there are fast (mostly randomized) algorithms that do not achieve a
better than 2-approximation, and there slow algorithms with \O(\sqrt{m})
update time that achieve a better-than-2 approximation. Thus the obvious
question is whether we can design an algorithm -- deterministic or randomized
-- that achieves a tradeoff between these two: a approximation
and a better-than-2 approximation simultaneously. We answer this question in
the affirmative for bipartite graphs.
Our main result is a fully dynamic algorithm that maintains a 3/2 + \eps
approximation in worst-case update time O(m^{1/4}\eps^{/2.5}). We also give
stronger results for graphs whose arboricity is at most \al, achieving a (1+
\eps) approximation in worst-case time O(\al (\al + \log n)) for constant
\eps. When the arboricity is constant, this bound is and when the
arboricity is polylogarithmic the update time is also polylogarithmic.
The most important technical developement is the use of an intermediate graph
we call an edge degree constrained subgraph (EDCS). This graph places
constraints on the sum of the degrees of the endpoints of each edge: upper
bounds for matched edges and lower bounds for unmatched edges. The main
technical content of our paper involves showing both how to maintain an EDCS
dynamically and that and EDCS always contains a sufficiently large matching. We
also make use of graph orientations to help bound the amount of work done
during each update.Comment: Longer version of paper that appears in ICALP 201
Massively Parallel Symmetry Breaking on Sparse Graphs: MIS and Maximal Matching
The success of modern parallel paradigms such as MapReduce, Hadoop, or Spark,
has attracted a significant attention to the Massively Parallel Computation
(MPC) model over the past few years, especially on graph problems. In this
work, we consider symmetry breaking problems of maximal independent set (MIS)
and maximal matching (MM), which are among the most intensively studied
problems in distributed/parallel computing, in MPC.
These problems are known to admit efficient MPC algorithms if the space per
machine is near-linear in , the number of vertices in the graph. This space
requirement however, as observed in the literature, is often significantly
larger than we can afford; especially when the input graph is sparse. In a
sharp contrast, in the truly sublinear regime of space per
machine, all the known algorithms take rounds which is
considered inefficient.
Motivated by this shortcoming, we parametrize our algorithms by the
arboricity of the input graph, which is a well-received measure of its
sparsity. We show that both MIS and MM admit round algorithms using space per
machine for any constant and using
total space. Therefore, for the wide range of sparse graphs with small
arboricity---such as minor-free graphs, bounded-genus graphs or bounded
treewidth graphs---we get an round algorithm which
exponentially improves prior algorithms.
By known reductions, our results also imply a -approximation of
maximum cardinality matching, a -approximation of maximum
weighted matching, and a 2-approximation of minimum vertex cover with
essentially the same round complexity and memory requirements.Comment: A merger of this paper and the independent and concurrent paper
[arxiv:1807.05374] appeared at PODC 201
Orienting edges to fight fire in graphs
We investigate a new oriented variant of the Firefighter Problem. In the
traditional Firefighter Problem, a fire breaks out at a given vertex of a
graph, and at each time interval spreads to neighbouring vertices that have not
been protected, while a constant number of vertices are protected at each time
interval. In the version of the problem considered here, the firefighters are
able to orient the edges of the graph before the fire breaks out, but the fire
could start at any vertex. We consider this problem when played on a graph in
one of several graph classes, and give upper and lower bounds on the number of
vertices that can be saved. In particular, when one firefighter is available at
each time interval, and the given graph is a complete graph, or a complete
bipartite graph, we present firefighting strategies that are provably optimal.
We also provide lower bounds on the number of vertices that can be saved as a
function of the chromatic number, of the maximum degree, and of the treewidth
of a graph. For a subcubic graph, we show that the firefighters can save all
but two vertices, and this is best possible
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