9,311 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
On combinatorial optimisation in analysis of protein-protein interaction and protein folding networks
Abstract: Protein-protein interaction networks and protein folding networks represent prominent research topics at the intersection of bioinformatics and network science. In this paper, we present a study of these networks from combinatorial optimisation point of view. Using a combination of classical heuristics and stochastic optimisation techniques, we were able to identify several interesting combinatorial properties of biological networks of the COSIN project. We obtained optimal or near-optimal solutions to maximum clique and chromatic number problems for these networks. We also explore patterns of both non-overlapping and overlapping cliques in these networks. Optimal or near-optimal solutions to partitioning of these networks into non-overlapping cliques and to maximum independent set problem were discovered. Maximal cliques are explored by enumerative techniques. Domination in these networks is briefly studied, too. Applications and extensions of our findings are discussed
GraphCombEx: A Software Tool for Exploration of Combinatorial Optimisation Properties of Large Graphs
We present a prototype of a software tool for exploration of multiple
combinatorial optimisation problems in large real-world and synthetic complex
networks. Our tool, called GraphCombEx (an acronym of Graph Combinatorial
Explorer), provides a unified framework for scalable computation and
presentation of high-quality suboptimal solutions and bounds for a number of
widely studied combinatorial optimisation problems. Efficient representation
and applicability to large-scale graphs and complex networks are particularly
considered in its design. The problems currently supported include maximum
clique, graph colouring, maximum independent set, minimum vertex clique
covering, minimum dominating set, as well as the longest simple cycle problem.
Suboptimal solutions and intervals for optimal objective values are estimated
using scalable heuristics. The tool is designed with extensibility in mind,
with the view of further problems and both new fast and high-performance
heuristics to be added in the future. GraphCombEx has already been successfully
used as a support tool in a number of recent research studies using
combinatorial optimisation to analyse complex networks, indicating its promise
as a research software tool
On the Enumeration of Minimal Dominating Sets and Related Notions
A dominating set in a graph is a subset of its vertex set such that each
vertex is either in or has a neighbour in . In this paper, we are
interested in the enumeration of (inclusion-wise) minimal dominating sets in
graphs, called the Dom-Enum problem. It is well known that this problem can be
polynomially reduced to the Trans-Enum problem in hypergraphs, i.e., the
problem of enumerating all minimal transversals in a hypergraph. Firstly we
show that the Trans-Enum problem can be polynomially reduced to the Dom-Enum
problem. As a consequence there exists an output-polynomial time algorithm for
the Trans-Enum problem if and only if there exists one for the Dom-Enum
problem. Secondly, we study the Dom-Enum problem in some graph classes. We give
an output-polynomial time algorithm for the Dom-Enum problem in split graphs,
and introduce the completion of a graph to obtain an output-polynomial time
algorithm for the Dom-Enum problem in -free chordal graphs, a proper
superclass of split graphs. Finally, we investigate the complexity of the
enumeration of (inclusion-wise) minimal connected dominating sets and minimal
total dominating sets of graphs. We show that there exists an output-polynomial
time algorithm for the Dom-Enum problem (or equivalently Trans-Enum problem) if
and only if there exists one for the following enumeration problems: minimal
total dominating sets, minimal total dominating sets in split graphs, minimal
connected dominating sets in split graphs, minimal dominating sets in
co-bipartite graphs.Comment: 15 pages, 3 figures, In revisio
Polynomial Delay Algorithm for Listing Minimal Edge Dominating sets in Graphs
The Transversal problem, i.e, the enumeration of all the minimal transversals
of a hypergraph in output-polynomial time, i.e, in time polynomial in its size
and the cumulated size of all its minimal transversals, is a fifty years old
open problem, and up to now there are few examples of hypergraph classes where
the problem is solved. A minimal dominating set in a graph is a subset of its
vertex set that has a non empty intersection with the closed neighborhood of
every vertex. It is proved in [M. M. Kant\'e, V. Limouzy, A. Mary, L. Nourine,
On the Enumeration of Minimal Dominating Sets and Related Notions, In Revision
2014] that the enumeration of minimal dominating sets in graphs and the
enumeration of minimal transversals in hypergraphs are two equivalent problems.
Hoping this equivalence can help to get new insights in the Transversal
problem, it is natural to look inside graph classes. It is proved independently
and with different techniques in [Golovach et al. - ICALP 2013] and [Kant\'e et
al. - ISAAC 2012] that minimal edge dominating sets in graphs (i.e, minimal
dominating sets in line graphs) can be enumerated in incremental
output-polynomial time. We provide the first polynomial delay and polynomial
space algorithm that lists all the minimal edge dominating sets in graphs,
answering an open problem of [Golovach et al. - ICALP 2013]. Besides the
result, we hope the used techniques that are a mix of a modification of the
well-known Berge's algorithm and a strong use of the structure of line graphs,
are of great interest and could be used to get new output-polynomial time
algorithms.Comment: proofs simplified from previous version, 12 pages, 2 figure
An Efficient Streaming Algorithm for the Submodular Cover Problem
We initiate the study of the classical Submodular Cover (SC) problem in the
data streaming model which we refer to as the Streaming Submodular Cover (SSC).
We show that any single pass streaming algorithm using sublinear memory in the
size of the stream will fail to provide any non-trivial approximation
guarantees for SSC. Hence, we consider a relaxed version of SSC, where we only
seek to find a partial cover.
We design the first Efficient bicriteria Submodular Cover Streaming
(ESC-Streaming) algorithm for this problem, and provide theoretical guarantees
for its performance supported by numerical evidence. Our algorithm finds
solutions that are competitive with the near-optimal offline greedy algorithm
despite requiring only a single pass over the data stream. In our numerical
experiments, we evaluate the performance of ESC-Streaming on active set
selection and large-scale graph cover problems.Comment: To appear in NIPS'1
The Directed Dominating Set Problem: Generalized Leaf Removal and Belief Propagation
A minimum dominating set for a digraph (directed graph) is a smallest set of
vertices such that each vertex either belongs to this set or has at least one
parent vertex in this set. We solve this hard combinatorial optimization
problem approximately by a local algorithm of generalized leaf removal and by a
message-passing algorithm of belief propagation. These algorithms can construct
near-optimal dominating sets or even exact minimum dominating sets for random
digraphs and also for real-world digraph instances. We further develop a core
percolation theory and a replica-symmetric spin glass theory for this problem.
Our algorithmic and theoretical results may facilitate applications of
dominating sets to various network problems involving directed interactions.Comment: 11 pages, 3 figures in EPS forma
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