16,638 research outputs found
Dynamics of bootstrap percolation
Bootstrap percolation transition may be first order or second order, or it
may have a mixed character where a first order drop in the order parameter is
preceded by critical fluctuations. Recent studies have indicated that the mixed
transition is characterized by power law avalanches, while the continuous
transition is characterized by truncated avalanches in a related sequential
bootstrap process. We explain this behavior on the basis of a through
analytical and numerical study of the avalanche distributions on a Bethe
lattice.Comment: Proceedings of the International Workshop and Conference on
Statistical Physics Approaches to Multidisciplinary Problems, IIT Guwahati,
India, 7-13 January 200
Ising formulations of many NP problems
We provide Ising formulations for many NP-complete and NP-hard problems,
including all of Karp's 21 NP-complete problems. This collects and extends
mappings to the Ising model from partitioning, covering and satisfiability. In
each case, the required number of spins is at most cubic in the size of the
problem. This work may be useful in designing adiabatic quantum optimization
algorithms.Comment: 27 pages; v2: substantial revision to intro/conclusion, many more
references; v3: substantial revision and extension, to-be-published versio
The linearization problem of a binary quadratic problem and its applications
We provide several applications of the linearization problem of a binary
quadratic problem. We propose a new lower bounding strategy, called the
linearization-based scheme, that is based on a simple certificate for a
quadratic function to be non-negative on the feasible set. Each
linearization-based bound requires a set of linearizable matrices as an input.
We prove that the Generalized Gilmore-Lawler bounding scheme for binary
quadratic problems provides linearization-based bounds. Moreover, we show that
the bound obtained from the first level reformulation linearization technique
is also a type of linearization-based bound, which enables us to provide a
comparison among mentioned bounds. However, the strongest linearization-based
bound is the one that uses the full characterization of the set of linearizable
matrices. Finally, we present a polynomial-time algorithm for the linearization
problem of the quadratic shortest path problem on directed acyclic graphs. Our
algorithm gives a complete characterization of the set of linearizable matrices
for the quadratic shortest path problem
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
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