2,513 research outputs found
A Discrete State Transition Algorithm for Generalized Traveling Salesman Problem
Generalized traveling salesman problem (GTSP) is an extension of classical
traveling salesman problem (TSP), which is a combinatorial optimization problem
and an NP-hard problem. In this paper, an efficient discrete state transition
algorithm (DSTA) for GTSP is proposed, where a new local search operator named
\textit{K-circle}, directed by neighborhood information in space, has been
introduced to DSTA to shrink search space and strengthen search ability. A
novel robust update mechanism, restore in probability and risk in probability
(Double R-Probability), is used in our work to escape from local minima. The
proposed algorithm is tested on a set of GTSP instances. Compared with other
heuristics, experimental results have demonstrated the effectiveness and strong
adaptability of DSTA and also show that DSTA has better search ability than its
competitors.Comment: 8 pages, 1 figur
A Method to Change Phase Transition Nature -- Toward Annealing Method --
In this paper, we review a way to change nature of phase transition with
annealing methods in mind. Annealing methods are regarded as a general
technique to solve optimization problems efficiently. In annealing methods, we
introduce a controllable parameter which represents a kind of fluctuation and
decrease the parameter gradually. Annealing methods face with a difficulty when
a phase transition point exists during the protocol. Then, it is important to
develop a method to avoid the phase transition by introducing a new type of
fluctuation. By taking the Potts model for instance, we review a way to change
the phase transition nature. Although the method described in this paper does
not succeed to avoid the phase transition, we believe that the concept of the
method will be useful for optimization problems.Comment: 27 pages, 3 figures, revised version will appear in proceedings of
Kinki University Quantum Computing Series Vo.
Phase Transitions and Backbones of the Asymmetric Traveling Salesman Problem
In recent years, there has been much interest in phase transitions of
combinatorial problems. Phase transitions have been successfully used to
analyze combinatorial optimization problems, characterize their typical-case
features and locate the hardest problem instances. In this paper, we study
phase transitions of the asymmetric Traveling Salesman Problem (ATSP), an
NP-hard combinatorial optimization problem that has many real-world
applications. Using random instances of up to 1,500 cities in which intercity
distances are uniformly distributed, we empirically show that many properties
of the problem, including the optimal tour cost and backbone size, experience
sharp transitions as the precision of intercity distances increases across a
critical value. Our experimental results on the costs of the ATSP tours and
assignment problem agree with the theoretical result that the asymptotic cost
of assignment problem is pi ^2 /6 the number of cities goes to infinity. In
addition, we show that the average computational cost of the well-known
branch-and-bound subtour elimination algorithm for the problem also exhibits a
thrashing behavior, transitioning from easy to difficult as the distance
precision increases. These results answer positively an open question regarding
the existence of phase transitions in the ATSP, and provide guidance on how
difficult ATSP problem instances should be generated
Quantum Annealing and Analog Quantum Computation
We review here the recent success in quantum annealing, i.e., optimization of
the cost or energy functions of complex systems utilizing quantum fluctuations.
The concept is introduced in successive steps through the studies of mapping of
such computationally hard problems to the classical spin glass problems. The
quantum spin glass problems arise with the introduction of quantum
fluctuations, and the annealing behavior of the systems as these fluctuations
are reduced slowly to zero. This provides a general framework for realizing
analog quantum computation.Comment: 22 pages, 7 figs (color online); new References Added. Reviews of
Modern Physics (in press
Traveling Salesman Problem
The idea behind TSP was conceived by Austrian mathematician Karl Menger in mid 1930s who invited the research community to consider a problem from the everyday life from a mathematical point of view. A traveling salesman has to visit exactly once each one of a list of m cities and then return to the home city. He knows the cost of traveling from any city i to any other city j. Thus, which is the tour of least possible cost the salesman can take? In this book the problem of finding algorithmic technique leading to good/optimal solutions for TSP (or for some other strictly related problems) is considered. TSP is a very attractive problem for the research community because it arises as a natural subproblem in many applications concerning the every day life. Indeed, each application, in which an optimal ordering of a number of items has to be chosen in a way that the total cost of a solution is determined by adding up the costs arising from two successively items, can be modelled as a TSP instance. Thus, studying TSP can never be considered as an abstract research with no real importance
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