3,390 research outputs found
Symmetry within Solutions
We define the concept of an internal symmetry. This is a symmety within a
solution of a constraint satisfaction problem. We compare this to solution
symmetry, which is a mapping between different solutions of the same problem.
We argue that we may be able to exploit both types of symmetry when finding
solutions. We illustrate the potential of exploiting internal symmetries on two
benchmark domains: Van der Waerden numbers and graceful graphs. By identifying
internal symmetries we are able to extend the state of the art in both cases.Comment: AAAI 2010, Proceedings of Twenty-Fourth AAAI Conference on Artificial
Intelligenc
Quantum Kaleidoscopes and Bell's theorem
A quantum kaleidoscope is defined as a set of observables, or states,
consisting of many different subsets that provide closely related proofs of the
Bell-Kochen-Specker (BKS) and Bell nonlocality theorems. The kaleidoscopes
prove the BKS theorem through a simple parity argument, which also doubles as a
proof of Bell's nonlocality theorem if use is made of the right sort of
entanglement. Three closely related kaleidoscopes are introduced and discussed
in this paper: a 15-observable kaleidoscope, a 24-state kaleidoscope and a
60-state kaleidoscope. The close relationship of these kaleidoscopes to a
configuration of 12 points and 16 lines known as Reye's configuration is
pointed out. The "rotations" needed to make each kaleidoscope yield all its
apparitions are laid out. The 60-state kaleidoscope, whose underlying
geometrical structure is that of ten interlinked Reye's configurations
(together with their duals), possesses a total of 1120 apparitions that provide
proofs of the two Bell theorems. Some applications of these kaleidoscopes to
problems in quantum tomography and quantum state estimation are discussed.Comment: Two new references (No. 21 and 22) to related work have been adde
Markov Chain Selection Hyper-heuristic for the Optimisation of Constrained Magic Squares
UKCI 2015: UK Workshop on Computational Intelligence, University of Exeter, UK, 7-9 September 2015A square matrix of size n × n, containing each of the numbers (1, . . . , n2) in which every row, column and both diagonals has the same total is referred to as a magic square. The problem can be formulated as an optimisation problem where the task is to minimise the deviation from the magic square constraints and is tackled here by using hyper-heuristics. Hyper-heuristics have recently attracted the attention of the artificial intelligence, operations research, engineering and computer science communities where the aim is to design and develop high level strategies as general solvers which are applicable to a range of different problem domains. There are two main types of hyper-heuristics in the literature: methodologies to select and to generate heuristics and both types of approaches search the space of heuristics rather than solutions. In this study, we describe a Markov chain selection hyper-heuristic as an effective solution methodology for optimising constrained magic squares. The empirical results show that the proposed hyper-heuristic is able to outperform the current state-of-the-art method
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
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