201 research outputs found
Quantum Algorithm to Solve Satisfiability Problems
A new quantum algorithm is proposed to solve Satisfiability(SAT) problems by
taking advantage of non-unitary transformation in ground state quantum
computer. The energy gap scale of the ground state quantum computer is analyzed
for 3-bit Exact Cover problems. The time cost of this algorithm on general SAT
problems is discussed.Comment: 5 pages, 3 figure
NP-hardness of the cluster minimization problem revisited
The computational complexity of the "cluster minimization problem" is
revisited [L. T. Wille and J. Vennik, J. Phys. A 18, L419 (1985)]. It is argued
that the original NP-hardness proof does not apply to pairwise potentials of
physical interest, such as those that depend on the geometric distance between
the particles. A geometric analog of the original problem is formulated, and a
new proof for such potentials is provided by polynomial time transformation
from the independent set problem for unit disk graphs. Limitations of this
formulation are pointed out, and new subproblems that bear more direct
consequences to the numerical study of clusters are suggested.Comment: 8 pages, 2 figures, accepted to J. Phys. A: Math. and Ge
Travelling Salesman Problem with a Center
We study a travelling salesman problem where the path is optimized with a
cost function that includes its length as well as a certain measure of
its distance from the geometrical center of the graph. Using simulated
annealing (SA) we show that such a problem has a transition point that
separates two phases differing in the scaling behaviour of and , in
efficiency of SA, and in the shape of minimal paths.Comment: 4 pages, minor changes, accepted in Phys.Rev.
Optimal combinations of imperfect objects
We address the question of how to make best use of imperfect objects, such as
defective analog and digital components. We show that perfect, or near-perfect,
devices can be constructed by taking combinations of such defects. Any
remaining objects can be recycled efficiently. In addition to its practical
applications, our `defect combination problem' provides a novel generalization
of classical optimization problems.Comment: 4 pages, 3 figures, minor change
Extremal Optimization at the Phase Transition of the 3-Coloring Problem
We investigate the phase transition of the 3-coloring problem on random
graphs, using the extremal optimization heuristic. 3-coloring is among the
hardest combinatorial optimization problems and is closely related to a 3-state
anti-ferromagnetic Potts model. Like many other such optimization problems, it
has been shown to exhibit a phase transition in its ground state behavior under
variation of a system parameter: the graph's mean vertex degree. This phase
transition is often associated with the instances of highest complexity. We use
extremal optimization to measure the ground state cost and the ``backbone'', an
order parameter related to ground state overlap, averaged over a large number
of instances near the transition for random graphs of size up to 512. For
graphs up to this size, benchmarks show that extremal optimization reaches
ground states and explores a sufficient number of them to give the correct
backbone value after about update steps. Finite size scaling gives
a critical mean degree value . Furthermore, the
exploration of the degenerate ground states indicates that the backbone order
parameter, measuring the constrainedness of the problem, exhibits a first-order
phase transition.Comment: RevTex4, 8 pages, 4 postscript figures, related information available
at http://www.physics.emory.edu/faculty/boettcher
Minimizing Unsatisfaction in Colourful Neighbourhoods
Colouring sparse graphs under various restrictions is a theoretical problem
of significant practical relevance. Here we consider the problem of maximizing
the number of different colours available at the nodes and their
neighbourhoods, given a predetermined number of colours. In the analytical
framework of a tree approximation, carried out at both zero and finite
temperatures, solutions obtained by population dynamics give rise to estimates
of the threshold connectivity for the incomplete to complete transition, which
are consistent with those of existing algorithms. The nature of the transition
as well as the validity of the tree approximation are investigated.Comment: 28 pages, 12 figures, substantially revised with additional
explanatio
Hybrid next-fit algorithm for the two-dimensional rectangle bin-packing problem
We present a new approximation algorithm for the two-dimensional bin-packing problem. The algorithm is based on two one-dimensional bin-packing algorithms. Since the algorithm is of next-fit type it can also be used for those cases where the output is required to be on-line (e. g. if we open an new bin we have no possibility to pack elements into the earlier opened bins). We give a tight bound for its worst-case and show that this bound is a parameter of the maximal sizes of the items to be packed. Moreover, we also present a probabilistic analysis of this algorithm
The Computational Complexity of the Lorentz Lattice Gas
The Lorentz lattice gas is studied from the perspective of computational
complexity theory. It is shown that using massive parallelism, particle
trajectories can be simulated in a time that scales logarithmically in the
length of the trajectory. This result characterizes the ``logical depth" of the
Lorentz lattice gas and allows us to compare it to other models in statistical
physics.Comment: 9 pages, LaTeX, to appear in J. Stat. Phy
Instance Space of the Number Partitioning Problem
Within the replica framework we study analytically the instance space of the
number partitioning problem. This classic integer programming problem consists
of partitioning a sequence of N positive real numbers \{a_1, a_2,..., a_N}
(the instance) into two sets such that the absolute value of the difference of
the sums of over the two sets is minimized. We show that there is an
upper bound to the number of perfect partitions (i.e. partitions
for which that difference is zero) and characterize the statistical properties
of the instances for which those partitions exist. In particular, in the case
that the two sets have the same cardinality (balanced partitions) we find
. Moreover, we show that the disordered model resulting from hte
instance space approach can be viewed as a model of replicators where the
random interactions are given by the Hebb rule.Comment: 7 page
Extremal Optimization of Graph Partitioning at the Percolation Threshold
The benefits of a recently proposed method to approximate hard optimization
problems are demonstrated on the graph partitioning problem. The performance of
this new method, called Extremal Optimization, is compared to Simulated
Annealing in extensive numerical simulations. While generally a complex
(NP-hard) problem, the optimization of the graph partitions is particularly
difficult for sparse graphs with average connectivities near the percolation
threshold. At this threshold, the relative error of Simulated Annealing for
large graphs is found to diverge relative to Extremal Optimization at equalized
runtime. On the other hand, Extremal Optimization, based on the extremal
dynamics of self-organized critical systems, reproduces known results about
optimal partitions at this critical point quite well.Comment: 7 pages, RevTex, 9 ps-figures included, as to appear in Journal of
Physics
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