4,452 research outputs found
An Order-based Algorithm for Minimum Dominating Set with Application in Graph Mining
Dominating set is a set of vertices of a graph such that all other vertices
have a neighbour in the dominating set. We propose a new order-based randomised
local search (RLS) algorithm to solve minimum dominating set problem in
large graphs. Experimental evaluation is presented for multiple types of
problem instances. These instances include unit disk graphs, which represent a
model of wireless networks, random scale-free networks, as well as samples from
two social networks and real-world graphs studied in network science. Our
experiments indicate that RLS performs better than both a classical greedy
approximation algorithm and two metaheuristic algorithms based on ant colony
optimisation and local search. The order-based algorithm is able to find small
dominating sets for graphs with tens of thousands of vertices. In addition, we
propose a multi-start variant of RLS that is suitable for solving the
minimum weight dominating set problem. The application of RLS in graph
mining is also briefly demonstrated
Local Approximation Schemes for Ad Hoc and Sensor Networks
We present two local approaches that yield polynomial-time approximation schemes (PTAS) for the Maximum Independent Set and Minimum Dominating Set problem in unit disk graphs. The algorithms run locally in each node and compute a (1+ε)-approximation to the problems at hand for any given ε > 0. The time complexity of both algorithms is O(TMIS + log*! n/εO(1)), where TMIS is the time required to compute a maximal independent set in the graph, and n denotes the number of nodes. We then extend these results to a more general class of graphs in which the maximum number of pair-wise independent nodes in every r-neighborhood is at most polynomial in r. Such graphs of polynomially bounded growth are introduced as a more realistic model for wireless networks and they generalize existing models, such as unit disk graphs or coverage area graphs
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
On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint
In the problem of minimum connected dominating set with routing cost
constraint, we are given a graph , and the goal is to find the
smallest connected dominating set of such that, for any two
non-adjacent vertices and in , the number of internal nodes on the
shortest path between and in the subgraph of induced by is at most times that in . For general graphs, the only
known previous approximability result is an -approximation algorithm
() for by Ding et al. For any constant , we
give an -approximation
algorithm. When , we give an -approximation
algorithm. Finally, we prove that, when , unless , for any constant , the problem admits no
polynomial-time -approximation algorithm, improving
upon the bound by Du et al. (albeit under a stronger hardness
assumption)
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
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
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