5,116 research outputs found
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
Best Response Games on Regular Graphs
With the growth of the internet it is becoming increasingly important to
understand how the behaviour of players is affected by the topology of the
network interconnecting them. Many models which involve networks of interacting
players have been proposed and best response games are amongst the simplest. In
best response games each vertex simultaneously updates to employ the best
response to their current surroundings. We concentrate upon trying to
understand the dynamics of best response games on regular graphs with many
strategies. When more than two strategies are present highly complex dynamics
can ensue. We focus upon trying to understand exactly how best response games
on regular graphs sample from the space of possible cellular automata. To
understand this issue we investigate convex divisions in high dimensional space
and we prove that almost every division of dimensional space into
convex regions includes a single point where all regions meet. We then find
connections between the convex geometry of best response games and the theory
of alternating circuits on graphs. Exploiting these unexpected connections
allows us to gain an interesting answer to our question of when cellular
automata are best response games
Using Differential Evolution for the Graph Coloring
Differential evolution was developed for reliable and versatile function
optimization. It has also become interesting for other domains because of its
ease to use. In this paper, we posed the question of whether differential
evolution can also be used by solving of the combinatorial optimization
problems, and in particular, for the graph coloring problem. Therefore, a
hybrid self-adaptive differential evolution algorithm for graph coloring was
proposed that is comparable with the best heuristics for graph coloring today,
i.e. Tabucol of Hertz and de Werra and the hybrid evolutionary algorithm of
Galinier and Hao. We have focused on the graph 3-coloring. Therefore, the
evolutionary algorithm with method SAW of Eiben et al., which achieved
excellent results for this kind of graphs, was also incorporated into this
study. The extensive experiments show that the differential evolution could
become a competitive tool for the solving of graph coloring problem in the
future
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
Decomposition, Reformulation, and Diving in University Course Timetabling
In many real-life optimisation problems, there are multiple interacting
components in a solution. For example, different components might specify
assignments to different kinds of resource. Often, each component is associated
with different sets of soft constraints, and so with different measures of soft
constraint violation. The goal is then to minimise a linear combination of such
measures. This paper studies an approach to such problems, which can be thought
of as multiphase exploitation of multiple objective-/value-restricted
submodels. In this approach, only one computationally difficult component of a
problem and the associated subset of objectives is considered at first. This
produces partial solutions, which define interesting neighbourhoods in the
search space of the complete problem. Often, it is possible to pick the initial
component so that variable aggregation can be performed at the first stage, and
the neighbourhoods to be explored next are guaranteed to contain feasible
solutions. Using integer programming, it is then easy to implement heuristics
producing solutions with bounds on their quality.
Our study is performed on a university course timetabling problem used in the
2007 International Timetabling Competition, also known as the Udine Course
Timetabling Problem. In the proposed heuristic, an objective-restricted
neighbourhood generator produces assignments of periods to events, with
decreasing numbers of violations of two period-related soft constraints. Those
are relaxed into assignments of events to days, which define neighbourhoods
that are easier to search with respect to all four soft constraints. Integer
programming formulations for all subproblems are given and evaluated using ILOG
CPLEX 11. The wider applicability of this approach is analysed and discussed.Comment: 45 pages, 7 figures. Improved typesetting of figures and table
Anagram-free Graph Colouring
An anagram is a word of the form where is a non-empty word and
is a permutation of . We study anagram-free graph colouring and give bounds
on the chromatic number. Alon et al. (2002) asked whether anagram-free
chromatic number is bounded by a function of the maximum degree. We answer this
question in the negative by constructing graphs with maximum degree 3 and
unbounded anagram-free chromatic number. We also prove upper and lower bounds
on the anagram-free chromatic number of trees in terms of their radius and
pathwidth. Finally, we explore extensions to edge colouring and
-anagram-free colouring.Comment: Version 2: Changed 'abelian square' to 'anagram' for consistency with
'Anagram-free colourings of graphs' by Kam\v{c}ev, {\L}uczak, and Sudakov.
Minor changes based on referee feedbac
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