1,072 research outputs found
A Weakly-Robust PTAS for Minimum Clique Partition in Unit Disk Graphs
We consider the problem of partitioning the set of vertices of a given unit
disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and
various constant factor approximations are known, with the current best ratio
of 3. Our main result is a {\em weakly robust} polynomial time approximation
scheme (PTAS) for UDGs expressed with edge-lengths, it either (i) computes a
clique partition or (ii) gives a certificate that the graph is not a UDG; for
the case (i) that it computes a clique partition, we show that it is guaranteed
to be within (1+\eps) ratio of the optimum if the input is UDG; however if
the input is not a UDG it either computes a clique partition as in case (i)
with no guarantee on the quality of the clique partition or detects that it is
not a UDG. Noting that recognition of UDG's is NP-hard even if we are given
edge lengths, our PTAS is a weakly-robust algorithm. Our algorithm can be
transformed into an O(\frac{\log^* n}{\eps^{O(1)}}) time distributed PTAS.
We consider a weighted version of the clique partition problem on vertex
weighted UDGs that generalizes the problem. We note some key distinctions with
the unweighted version, where ideas useful in obtaining a PTAS breakdown. Yet,
surprisingly, it admits a (2+\eps)-approximation algorithm for the weighted
case where the graph is expressed, say, as an adjacency matrix. This improves
on the best known 8-approximation for the {\em unweighted} case for UDGs
expressed in standard form.Comment: 21 pages, 9 figure
Optimization in Geometric Graphs: Complexity and Approximation
We consider several related problems arising in geometric graphs. In particular,
we investigate the computational complexity and approximability properties of several optimization problems in unit ball graphs and develop algorithms to find exact
and approximate solutions. In addition, we establish complexity-based theoretical
justifications for several greedy heuristics.
Unit ball graphs, which are defined in the three dimensional Euclidian space, have
several application areas such as computational geometry, facility location and, particularly, wireless communication networks. Efficient operation of wireless networks
involves several decision problems that can be reduced to well known optimization
problems in graph theory. For instance, the notion of a \virtual backbone" in a wire-
less network is strongly related to a minimum connected dominating set in its graph
theoretic representation.
Motivated by the vastness of application areas, we study several problems including maximum independent set, minimum vertex coloring, minimum clique partition,
max-cut and min-bisection. Although these problems have been widely studied in
the context of unit disk graphs, which are the two dimensional version of unit ball
graphs, there is no established result on the complexity and approximation status
for some of them in unit ball graphs. Furthermore, unit ball graphs can provide a
better representation of real networks since the nodes are deployed in the three dimensional space. We prove complexity results and propose solution procedures for
several problems using geometrical properties of these graphs.
We outline a matching-based branch and bound solution procedure for the maximum k-clique problem in unit disk graphs and demonstrate its effectiveness through
computational tests. We propose using minimum bottleneck connected dominating
set problem in order to determine the optimal transmission range of a wireless network that will ensure a certain size of "virtual backbone". We prove that this problem
is NP-hard in general graphs but solvable in polynomial time in unit disk and unit
ball graphs.
We also demonstrate work on theoretical foundations for simple greedy heuristics.
Particularly, similar to the notion of "best" approximation algorithms with respect to
their approximation ratios, we prove that several simple greedy heuristics are "best"
in the sense that it is NP-hard to recognize the gap between the greedy solution
and the optimal solution. We show results for several well known problems such as
maximum clique, maximum independent set, minimum vertex coloring and discuss
extensions of these results to a more general class of problems.
In addition, we propose a "worst-out" heuristic based on edge contractions for
the max-cut problem and provide analytical and experimental comparisons with a
well known "best-in" approach and its modified versions
Bidimensionality and Geometric Graphs
In this paper we use several of the key ideas from Bidimensionality to give a
new generic approach to design EPTASs and subexponential time parameterized
algorithms for problems on classes of graphs which are not minor closed, but
instead exhibit a geometric structure. In particular we present EPTASs and
subexponential time parameterized algorithms for Feedback Vertex Set, Vertex
Cover, Connected Vertex Cover, Diamond Hitting Set, on map graphs and unit disk
graphs, and for Cycle Packing and Minimum-Vertex Feedback Edge Set on unit disk
graphs. Our results are based on the recent decomposition theorems proved by
Fomin et al [SODA 2011], and our algorithms work directly on the input graph.
Thus it is not necessary to compute the geometric representations of the input
graph. To the best of our knowledge, these results are previously unknown, with
the exception of the EPTAS and a subexponential time parameterized algorithm on
unit disk graphs for Vertex Cover, which were obtained by Marx [ESA 2005] and
Alber and Fiala [J. Algorithms 2004], respectively.
We proceed to show that our approach can not be extended in its full
generality to more general classes of geometric graphs, such as intersection
graphs of unit balls in R^d, d >= 3. Specifically we prove that Feedback Vertex
Set on unit-ball graphs in R^3 neither admits PTASs unless P=NP, nor
subexponential time algorithms unless the Exponential Time Hypothesis fails.
Additionally, we show that the decomposition theorems which our approach is
based on fail for disk graphs and that therefore any extension of our results
to disk graphs would require new algorithmic ideas. On the other hand, we prove
that our EPTASs and subexponential time algorithms for Vertex Cover and
Connected Vertex Cover carry over both to disk graphs and to unit-ball graphs
in R^d for every fixed d
Hyperbolic intersection graphs and (quasi)-polynomial time
We study unit ball graphs (and, more generally, so-called noisy uniform ball
graphs) in -dimensional hyperbolic space, which we denote by .
Using a new separator theorem, we show that unit ball graphs in
enjoy similar properties as their Euclidean counterparts, but in one dimension
lower: many standard graph problems, such as Independent Set, Dominating Set,
Steiner Tree, and Hamiltonian Cycle can be solved in
time for any fixed , while the same problems need
time in . We also show that these algorithms in
are optimal up to constant factors in the exponent under ETH.
This drop in dimension has the largest impact in , where we
introduce a new technique to bound the treewidth of noisy uniform disk graphs.
The bounds yield quasi-polynomial () algorithms for all of the
studied problems, while in the case of Hamiltonian Cycle and -Coloring we
even get polynomial time algorithms. Furthermore, if the underlying noisy disks
in have constant maximum degree, then all studied problems can
be solved in polynomial time. This contrasts with the fact that these problems
require time under ETH in constant maximum degree
Euclidean unit disk graphs.
Finally, we complement our quasi-polynomial algorithm for Independent Set in
noisy uniform disk graphs with a matching lower bound
under ETH. This shows that the hyperbolic plane is a potential source of
NP-intermediate problems.Comment: Short version appears in SODA 202
QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs
A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time 2^{O~(n^{2/3})} for Maximum Clique on disk graphs. In stark contrast, Maximum Clique on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant ratio of approximation which cannot be attained even in time 2^{n^{1-epsilon}}, unless the Exponential Time Hypothesis fails
Self-stabilizing algorithms for Connected Vertex Cover and Clique decomposition problems
In many wireless networks, there is no fixed physical backbone nor
centralized network management. The nodes of such a network have to
self-organize in order to maintain a virtual backbone used to route messages.
Moreover, any node of the network can be a priori at the origin of a malicious
attack. Thus, in one hand the backbone must be fault-tolerant and in other hand
it can be useful to monitor all network communications to identify an attack as
soon as possible. We are interested in the minimum \emph{Connected Vertex
Cover} problem, a generalization of the classical minimum Vertex Cover problem,
which allows to obtain a connected backbone. Recently, Delbot et
al.~\cite{DelbotLP13} proposed a new centralized algorithm with a constant
approximation ratio of for this problem. In this paper, we propose a
distributed and self-stabilizing version of their algorithm with the same
approximation guarantee. To the best knowledge of the authors, it is the first
distributed and fault-tolerant algorithm for this problem. The approach
followed to solve the considered problem is based on the construction of a
connected minimal clique partition. Therefore, we also design the first
distributed self-stabilizing algorithm for this problem, which is of
independent interest
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