878 research outputs found
Multi-layer local optima networks for the analysis of advanced local search-based algorithms
A Local Optima Network (LON) is a graph model that compresses the fitness
landscape of a particular combinatorial optimization problem based on a
specific neighborhood operator and a local search algorithm. Determining which
and how landscape features affect the effectiveness of search algorithms is
relevant for both predicting their performance and improving the design
process. This paper proposes the concept of multi-layer LONs as well as a
methodology to explore these models aiming at extracting metrics for fitness
landscape analysis. Constructing such models, extracting and analyzing their
metrics are the preliminary steps into the direction of extending the study on
single neighborhood operator heuristics to more sophisticated ones that use
multiple operators. Therefore, in the present paper we investigate a twolayer
LON obtained from instances of a combinatorial problem using bitflip and swap
operators. First, we enumerate instances of NK-landscape model and use the hill
climbing heuristic to build the corresponding LONs. Then, using LON metrics, we
analyze how efficiently the search might be when combining both strategies. The
experiments show promising results and demonstrate the ability of multi-layer
LONs to provide useful information that could be used for in metaheuristics
based on multiple operators such as Variable Neighborhood Search.Comment: Accepted in GECCO202
Clustering of Local Optima in Combinatorial Fitness Landscapes
Using the recently proposed model of combinatorial landscapes: local optima
networks, we study the distribution of local optima in two classes of instances
of the quadratic assignment problem. Our results indicate that the two problem
instance classes give rise to very different configuration spaces. For the
so-called real-like class, the optima networks possess a clear modular
structure, while the networks belonging to the class of random uniform
instances are less well partitionable into clusters. We briefly discuss the
consequences of the findings for heuristically searching the corresponding
problem spaces.Comment: Learning and Intelligent OptimizatioN Conference (LION 5), Rome :
Italy (2011
Analysis of combinatorial search spaces for a class of NP-hard problems, An
2011 Spring.Includes bibliographical references.Given a finite but very large set of states X and a real-valued objective function ƒ defined on X, combinatorial optimization refers to the problem of finding elements of X that maximize (or minimize) ƒ. Many combinatorial search algorithms employ some perturbation operator to hill-climb in the search space. Such perturbative local search algorithms are state of the art for many classes of NP-hard combinatorial optimization problems such as maximum k-satisfiability, scheduling, and problems of graph theory. In this thesis we analyze combinatorial search spaces by expanding the objective function into a (sparse) series of basis functions. While most analyses of the distribution of function values in the search space must rely on empirical sampling, the basis function expansion allows us to directly study the distribution of function values across regions of states for combinatorial problems without the need for sampling. We concentrate on objective functions that can be expressed as bounded pseudo-Boolean functions which are NP-hard to solve in general. We use the basis expansion to construct a polynomial-time algorithm for exactly computing constant-degree moments of the objective function ƒ over arbitrarily large regions of the search space. On functions with restricted codomains, these moments are related to the true distribution by a system of linear equations. Given low moments supplied by our algorithm, we construct bounds of the true distribution of ƒ over regions of the space using a linear programming approach. A straightforward relaxation allows us to efficiently approximate the distribution and hence quickly estimate the count of states in a given region that have certain values under the objective function. The analysis is also useful for characterizing properties of specific combinatorial problems. For instance, by connecting search space analysis to the theory of inapproximability, we prove that the bound specified by Grover's maximum principle for the Max-Ek-Lin-2 problem is sharp. Moreover, we use the framework to prove certain configurations are forbidden in regions of the Max-3-Sat search space, supplying the first theoretical confirmation of empirical results by others. Finally, we show that theoretical results can be used to drive the design of algorithms in a principled manner by using the search space analysis developed in this thesis in algorithmic applications. First, information obtained from our moment retrieving algorithm can be used to direct a hill-climbing search across plateaus in the Max-k-Sat search space. Second, the analysis can be used to control the mutation rate on a (1+1) evolutionary algorithm on bounded pseudo-Boolean functions so that the offspring of each search point is maximized in expectation. For these applications, knowledge of the search space structure supplied by the analysis translates to significant gains in the performance of search
Local Optima Networks of NK Landscapes with Neutrality
In previous work we have introduced a network-based model that abstracts many
details of the underlying landscape and compresses the landscape information
into a weighted, oriented graph which we call the local optima network. The
vertices of this graph are the local optima of the given fitness landscape,
while the arcs are transition probabilities between local optima basins. Here
we extend this formalism to neutral fitness landscapes, which are common in
difficult combinatorial search spaces. By using two known neutral variants of
the NK family (i.e. NKp and NKq) in which the amount of neutrality can be tuned
by a parameter, we show that our new definitions of the optima networks and the
associated basins are consistent with the previous definitions for the
non-neutral case. Moreover, our empirical study and statistical analysis show
that the features of neutral landscapes interpolate smoothly between landscapes
with maximum neutrality and non-neutral ones. We found some unknown structural
differences between the two studied families of neutral landscapes. But
overall, the network features studied confirmed that neutrality, in landscapes
with percolating neutral networks, may enhance heuristic search. Our current
methodology requires the exhaustive enumeration of the underlying search space.
Therefore, sampling techniques should be developed before this analysis can
have practical implications. We argue, however, that the proposed model offers
a new perspective into the problem difficulty of combinatorial optimization
problems and may inspire the design of more effective search heuristics.Comment: IEEE Transactions on Evolutionary Computation volume 14, 6 (2010) to
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