33,164 research outputs found
Solving satisfiability problems by fluctuations: The dynamics of stochastic local search algorithms
Stochastic local search algorithms are frequently used to numerically solve
hard combinatorial optimization or decision problems. We give numerical and
approximate analytical descriptions of the dynamics of such algorithms applied
to random satisfiability problems. We find two different dynamical regimes,
depending on the number of constraints per variable: For low constraintness,
the problems are solved efficiently, i.e. in linear time. For higher
constraintness, the solution times become exponential. We observe that the
dynamical behavior is characterized by a fast equilibration and fluctuations
around this equilibrium. If the algorithm runs long enough, an exponentially
rare fluctuation towards a solution appears.Comment: 21 pages, 18 figures, revised version, to app. in PRE (2003
On the Effect of Connectedness for Biobjective Multiple and Long Path Problems
Recently, the property of connectedness has been claimed to give a strong
motivation on the design of local search techniques for multiobjective
combinatorial optimization (MOCO). Indeed, when connectedness holds, a basic
Pareto local search, initialized with at least one non-dominated solution,
allows to identify the efficient set exhaustively. However, this becomes
quickly infeasible in practice as the number of efficient solutions typically
grows exponentially with the instance size. As a consequence, we generally have
to deal with a limited-size approximation, where a good sample set has to be
found. In this paper, we propose the biobjective multiple and long path
problems to show experimentally that, on the first problems, even if the
efficient set is connected, a local search may be outperformed by a simple
evolutionary algorithm in the sampling of the efficient set. At the opposite,
on the second problems, a local search algorithm may successfully approximate a
disconnected efficient set. Then, we argue that connectedness is not the single
property to study for the design of local search heuristics for MOCO. This work
opens new discussions on a proper definition of the multiobjective fitness
landscape.Comment: Learning and Intelligent OptimizatioN Conference (LION 5), Rome :
Italy (2011
Quantum and Classical Multilevel Algorithms for (Hyper)Graphs
Combinatorial optimization problems on (hyper)graphs are ubiquitous in science and industry. Because many of these problems are NP-hard, development of sophisticated heuristics is of utmost importance for practical problems. In recent years, the emergence of Noisy Intermediate-Scale Quantum (NISQ) computers has opened up the opportunity to dramaticaly speedup combinatorial optimization. However, the adoption of NISQ devices is impeded by their severe limitations, both in terms of the number of qubits, as well as in their quality. NISQ devices are widely expected to have no more than hundreds to thousands of qubits with very limited error-correction, imposing a strict limit on the size and the structure of the problems that can be tackled directly. A natural solution to this issue is hybrid quantum-classical algorithms that combine a NISQ device with a classical machine with the goal of capturing “the best of both worlds”.
Being motivated by lack of high quality optimization solvers for hypergraph partitioning, in this thesis, we begin by discussing classical multilevel approaches for this problem. We present a novel relaxation-based vertex similarity measure termed algebraic distance for hypergraphs and the coarsening schemes based on it. Extending the multilevel method to include quantum optimization routines, we present Quantum Local Search (QLS) – a hybrid iterative improvement approach that is inspired by the classical local search approaches. Next, we introduce the Multilevel Quantum Local Search (ML-QLS) that incorporates the quantum-enhanced iterative improvement scheme introduced in QLS within the multilevel framework, as well as several techniques to further understand and improve the effectiveness of Quantum Approximate Optimization Algorithm used throughout our work
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Combinatorial optimization and metaheuristics
Today, combinatorial optimization is one of the youngest and most active areas of discrete mathematics. It is a branch of optimization in applied mathematics and computer science, related to operational research, algorithm theory and computational complexity theory. It sits at the intersection of several fields, including artificial intelligence, mathematics and software engineering. Its increasing interest arises for the fact that a large number of scientific and industrial problems can be formulated as abstract combinatorial optimization problems, through graphs and/or (integer) linear programs. Some of these problems have polynomial-time (“efficient”) algorithms, while most of them are NP-hard, i.e. it is not proved that they can be solved in polynomial-time. Mainly, it means that it is not possible to guarantee that an exact solution to the problem can be found and one has to settle for an approximate solution with known performance guarantees. Indeed, the goal of approximate methods is to find “quickly” (reasonable run-times), with “high” probability, provable “good” solutions (low error from the real optimal solution). In the last 20 years, a new kind of algorithm commonly called metaheuristics have emerged in this class, which basically try to combine heuristics in high level frameworks aimed at efficiently and effectively exploring the search space. This report briefly outlines the components, concepts, advantages and disadvantages of different metaheuristic approaches from a conceptual point of view, in order to analyze their similarities and differences. The two very significant forces of intensification and diversification, that mainly determine the behavior of a metaheuristic, will be pointed out. The report concludes by exploring the importance of hybridization and integration methods
Natural evolution strategies and variational Monte Carlo
A notion of quantum natural evolution strategies is introduced, which
provides a geometric synthesis of a number of known quantum/classical
algorithms for performing classical black-box optimization. Recent work of
Gomes et al. [2019] on heuristic combinatorial optimization using neural
quantum states is pedagogically reviewed in this context, emphasizing the
connection with natural evolution strategies. The algorithmic framework is
illustrated for approximate combinatorial optimization problems, and a
systematic strategy is found for improving the approximation ratios. In
particular it is found that natural evolution strategies can achieve
approximation ratios competitive with widely used heuristic algorithms for
Max-Cut, at the expense of increased computation time
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