75 research outputs found
Non-Local Probes Do Not Help with Graph Problems
This work bridges the gap between distributed and centralised models of
computing in the context of sublinear-time graph algorithms. A priori, typical
centralised models of computing (e.g., parallel decision trees or centralised
local algorithms) seem to be much more powerful than distributed
message-passing algorithms: centralised algorithms can directly probe any part
of the input, while in distributed algorithms nodes can only communicate with
their immediate neighbours. We show that for a large class of graph problems,
this extra freedom does not help centralised algorithms at all: for example,
efficient stateless deterministic centralised local algorithms can be simulated
with efficient distributed message-passing algorithms. In particular, this
enables us to transfer existing lower bound results from distributed algorithms
to centralised local algorithms
Approximating Subdense Instances of Covering Problems
We study approximability of subdense instances of various covering problems
on graphs, defined as instances in which the minimum or average degree is
Omega(n/psi(n)) for some function psi(n)=omega(1) of the instance size. We
design new approximation algorithms as well as new polynomial time
approximation schemes (PTASs) for those problems and establish first
approximation hardness results for them. Interestingly, in some cases we were
able to prove optimality of the underlying approximation ratios, under usual
complexity-theoretic assumptions. Our results for the Vertex Cover problem
depend on an improved recursive sampling method which could be of independent
interest
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
Monotone properties of random geometric graphs have sharp thresholds
Random geometric graphs result from taking uniformly distributed points
in the unit cube, , and connecting two points if their Euclidean
distance is at most , for some prescribed . We show that monotone
properties for this class of graphs have sharp thresholds by reducing the
problem to bounding the bottleneck matching on two sets of points
distributed uniformly in . We present upper bounds on the threshold
width, and show that our bound is sharp for and at most a sublogarithmic
factor away for . Interestingly, the threshold width is much sharper for
random geometric graphs than for Bernoulli random graphs. Further, a random
geometric graph is shown to be a subgraph, with high probability, of another
independently drawn random geometric graph with a slightly larger radius; this
property is shown to have no analogue for Bernoulli random graphs.Comment: Published at http://dx.doi.org/10.1214/105051605000000575 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings
We provide CONGEST model algorithms for approximating minimum weighted vertex
cover and the maximum weighted matching. For bipartite graphs, we show that a
-approximate weighted vertex cover can be computed
deterministically in polylogarithmic time. This generalizes a corresponding
result for the unweighted vertex cover problem shown in [Faour, Kuhn; OPODIS
'20]. Moreover, we show that in general weighted graph families that are closed
under taking subgraphs and in which we can compute an independent set of weight
at least a -fraction of the total weight, one can compute a
-approximate weighted vertex cover in
polylogarithmic time in the CONGEST model. Our result in particular implies
that in graphs of arboricity , one can compute a
-approximate weighted vertex cover.
For maximum weighted matchings, we show that a -approximate
solution can be computed deterministically in polylogarithmic CONGEST rounds
(for constant ). We also provide a more efficient randomized
algorithm. Our algorithm generalizes results of [Lotker, Patt-Shamir, Pettie;
SPAA '08] and [Bar-Yehuda, Hillel, Ghaffari, Schwartzman; PODC '17] for the
unweighted case.
Finally, we show that even in the LOCAL model and in bipartite graphs of
degree , if for some constant
, then computing a -approximation for the
unweighted minimum vertex cover problem requires rounds. This generalizes aresult of [G\"o\"os, Suomela;
DISC '12], who showed that computing a -approximation in
such graphs requires rounds
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
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