225 research outputs found
A novel evolutionary formulation of the maximum independent set problem
We introduce a novel evolutionary formulation of the problem of finding a
maximum independent set of a graph. The new formulation is based on the
relationship that exists between a graph's independence number and its acyclic
orientations. It views such orientations as individuals and evolves them with
the aid of evolutionary operators that are very heavily based on the structure
of the graph and its acyclic orientations. The resulting heuristic has been
tested on some of the Second DIMACS Implementation Challenge benchmark graphs,
and has been found to be competitive when compared to several of the other
heuristics that have also been tested on those graphs
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
Consistency and Random Constraint Satisfaction Models
In this paper, we study the possibility of designing non-trivial random CSP
models by exploiting the intrinsic connection between structures and
typical-case hardness. We show that constraint consistency, a notion that has
been developed to improve the efficiency of CSP algorithms, is in fact the key
to the design of random CSP models that have interesting phase transition
behavior and guaranteed exponential resolution complexity without putting much
restriction on the parameter of constraint tightness or the domain size of the
problem. We propose a very flexible framework for constructing problem
instances withinteresting behavior and develop a variety of concrete methods to
construct specific random CSP models that enforce different levels of
constraint consistency. A series of experimental studies with interesting
observations are carried out to illustrate the effectiveness of introducing
structural elements in random instances, to verify the robustness of our
proposal, and to investigate features of some specific models based on our
framework that are highly related to the behavior of backtracking search
algorithms
On Maximum Weight Clique Algorithms, and How They Are Evaluated
Maximum weight clique and maximum weight independent set solvers are often benchmarked using maximum clique problem instances, with weights allocated to vertices by taking the vertex number mod 200 plus 1. For constraint programming approaches, this rule has clear implications, favouring weight-based rather than degree-based heuristics. We show that similar implications hold for dedicated algorithms, and that additionally, weight distributions affect whether certain inference rules are cost-effective. We look at other families of benchmark instances for the maximum weight clique problem, coming from winner determination problems, graph colouring, and error-correcting codes, and introduce two new families of instances, based upon kidney exchange and the Research Excellence Framework. In each case the weights carry much more interesting structure, and do not in any way resemble the 200 rule. We make these instances available in the hopes of improving the quality of future experiments
Depth-first simplicial partition for copositivity detection, with an application to MaxClique
Detection of copositivity plays an important role in combinatorial and quadratic optimization. Recently, an algorithm for copositivity detection by simplicial partition has been proposed. In this paper, we develop an improved depth-first simplicial partition algorithm which reduces memory requirements significantly and therefore enables copositivity checks of much larger matrices – of size up to a few thousands instead of a few hundreds. The algorithm has been investigated experimentally on a number of MaxClique problems as well as on generated random problems. We present numerical results showing that the algorithm is much faster than a recently published linear algebraic algorithm for copositivity detection based on traditional ideas – checking properties of principal sub-matrices. We also show that the algorithm works very well for solving MaxClique problems through copositivity checks
Dynamic Local Search for the Maximum Clique Problem
In this paper, we introduce DLS-MC, a new stochastic local search algorithm
for the maximum clique problem. DLS-MC alternates between phases of iterative
improvement, during which suitable vertices are added to the current clique,
and plateau search, during which vertices of the current clique are swapped
with vertices not contained in the current clique. The selection of vertices is
solely based on vertex penalties that are dynamically adjusted during the
search, and a perturbation mechanism is used to overcome search stagnation. The
behaviour of DLS-MC is controlled by a single parameter, penalty delay, which
controls the frequency at which vertex penalties are reduced. We show
empirically that DLS-MC achieves substantial performance improvements over
state-of-the-art algorithms for the maximum clique problem over a large range
of the commonly used DIMACS benchmark instances
A conceptual heuristic for solving the maximum clique problem
The maximum clique problem (MCP) is the problem of finding the clique with maximum cardinality in a graph. It has been intensively studied for years by computer scientists and mathematicians. It has many practical applications and it is usually the computational bottleneck. Due to the complexity of the problem, exact solutions can be very computationally expensive. In the scope of this thesis, a polynomial time heuristic that is based on Formal Concept Analysis has been developed. The developed approach has three variations that use different algorithm design approaches to solve the problem, a greedy algorithm, a backtracking algorithm and a branch and bound algorithm. The parameters of the branch and bound algorithm are tuned in a training phase and the tuned parameters are tested on the BHOSLIB benchmark graphs. The developed approach is tested on all the instances of the DIMACS benchmark graphs, and the results show that the maximum clique is obtained for 70% of the graph instances. The developed approach is compared to several of the most effective recent algorithms.NPRP grant #06-1220-1-233 from the Qatar National Research Fund (a member of Qatar Foundation)
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