2,063 research outputs found
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
Minor Excluded Network Families Admit Fast Distributed Algorithms
Distributed network optimization algorithms, such as minimum spanning tree,
minimum cut, and shortest path, are an active research area in distributed
computing. This paper presents a fast distributed algorithm for such problems
in the CONGEST model, on networks that exclude a fixed minor.
On general graphs, many optimization problems, including the ones mentioned
above, require rounds of communication in the CONGEST
model, even if the network graph has a much smaller diameter. Naturally, the
next step in algorithm design is to design efficient algorithms which bypass
this lower bound on a restricted class of graphs. Currently, the only known
method of doing so uses the low-congestion shortcut framework of Ghaffari and
Haeupler [SODA'16]. Building off of their work, this paper proves that excluded
minor graphs admit high-quality shortcuts, leading to an round
algorithm for the aforementioned problems, where is the diameter of the
network graph. To work with excluded minor graph families, we utilize the Graph
Structure Theorem of Robertson and Seymour. To the best of our knowledge, this
is the first time the Graph Structure Theorem has been used for an algorithmic
result in the distributed setting.
Even though the proof is involved, merely showing the existence of good
shortcuts is sufficient to obtain simple, efficient distributed algorithms. In
particular, the shortcut framework can efficiently construct near-optimal
shortcuts and then use them to solve the optimization problems. This, combined
with the very general family of excluded minor graphs, which includes most
other important graph classes, makes this result of significant interest
Odd-Minors I: Excluding small parity breaks
Given a graph class~, the -blind-treewidth of a
graph~ is the smallest integer~ such that~ has a tree-decomposition
where every bag whose torso does not belong to~ has size at
most~. In this paper we focus on the class~ of bipartite graphs
and the class~ of planar graphs together with the odd-minor
relation. For each of the two parameters, -blind-treewidth and
-blind-treewidth, we prove an analogue of the
celebrated Grid Theorem under the odd-minor relation. As a consequence we
obtain FPT-approximation algorithms for both parameters. We then provide
FPT-algorithms for \textsc{Maximum Independent Set} on graphs of bounded
-blind-treewidth and \textsc{Maximum Cut} on graphs of bounded
-blind-treewidth
Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Edge Crossings
We provide linear-time algorithms for geometric graphs with sublinearly many
crossings. That is, we provide algorithms running in O(n) time on connected
geometric graphs having n vertices and k crossings, where k is smaller than n
by an iterated logarithmic factor. Specific problems we study include Voronoi
diagrams and single-source shortest paths. Our algorithms all run in linear
time in the standard comparison-based computational model; hence, we make no
assumptions about the distribution or bit complexities of edge weights, nor do
we utilize unusual bit-level operations on memory words. Instead, our
algorithms are based on a planarization method that "zeroes in" on edge
crossings, together with methods for extending planar separator decompositions
to geometric graphs with sublinearly many crossings. Incidentally, our
planarization algorithm also solves an open computational geometry problem of
Chazelle for triangulating a self-intersecting polygonal chain having n
segments and k crossings in linear time, for the case when k is sublinear in n
by an iterated logarithmic factor.Comment: Expanded version of a paper appearing at the 20th ACM-SIAM Symposium
on Discrete Algorithms (SODA09
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