2,021 research outputs found
Random Costs in Combinatorial Optimization
The random cost problem is the problem of finding the minimum in an
exponentially long list of random numbers. By definition, this problem cannot
be solved faster than by exhaustive search. It is shown that a classical
NP-hard optimization problem, number partitioning, is essentially equivalent to
the random cost problem. This explains the bad performance of heuristic
approaches to the number partitioning problem and allows us to calculate the
probability distributions of the optimum and sub-optimum costs.Comment: 4 pages, Revtex, 2 figures (eps), submitted to PR
Phase Transition in the Number Partitioning Problem
Number partitioning is an NP-complete problem of combinatorial optimization.
A statistical mechanics analysis reveals the existence of a phase transition
that separates the easy from the hard to solve instances and that reflects the
pseudo-polynomiality of number partitioning. The phase diagram and the value of
the typical ground state energy are calculated.Comment: minor changes (references, typos and discussion of results
Phase transition for cutting-plane approach to vertex-cover problem
We study the vertex-cover problem which is an NP-hard optimization problem
and a prototypical model exhibiting phase transitions on random graphs, e.g.,
Erdoes-Renyi (ER) random graphs. These phase transitions coincide with changes
of the solution space structure, e.g, for the ER ensemble at connectivity
c=e=2.7183 from replica symmetric to replica-symmetry broken. For the
vertex-cover problem, also the typical complexity of exact branch-and-bound
algorithms, which proceed by exploring the landscape of feasible
configurations, change close to this phase transition from "easy" to "hard". In
this work, we consider an algorithm which has a completely different strategy:
The problem is mapped onto a linear programming problem augmented by a
cutting-plane approach, hence the algorithm operates in a space OUTSIDE the
space of feasible configurations until the final step, where a solution is
found. Here we show that this type of algorithm also exhibits an "easy-hard"
transition around c=e, which strongly indicates that the typical hardness of a
problem is fundamental to the problem and not due to a specific representation
of the problem.Comment: 4 pages, 3 figure
Using small MUSes to explain how to solve pen and paper puzzles
Pen and paper puzzles like Sudoku, Futoshiki and Skyscrapers are hugely popular. Solving such puzzles can be a trivial task for modern AI systems. However, most AI systems solve problems using a form of backtracking, while people try to avoid backtracking as much as possible. This means that existing AI systems do not output explanations about their reasoning that are meaningful to people. We present Demystify, a tool which allows puzzles to be expressed in a high-level constraint programming language and uses MUSes to allow us to produce descriptions of steps in the puzzle solving. We give several improvements to the existing techniques for solving puzzles with MUSes, which allow us to solve a range of significantly more complex puzzles and give higher quality explanations. We demonstrate the effectiveness and generality of Demystify by comparing its results to documented strategies for solving a range of pen and paper puzzles by hand, showing that our technique can find many of the same explanations.Publisher PD
Observational constraints on the origin of the elements. V. Non-LTE abundance ratios of [Ni/Fe] in Galactic stars and enrichment by sub-Chandrasekhar mass SNe
We constrain the role of different SN Ia channels in the chemical enrichment
of the Galaxy by studying the abundances of nickel in Galactic stars. We
investigate four different SN Ia sub-classes, including the classical
single-degenerate near-Chandrasekhar mass SN Ia, the fainter SN Iax systems
associated with He accretion from the companion, as well as two sub-Ch mass SN
Ia channels. The latter include the double-detonation of a white dwarf
accreting helium-rich matter and violent white dwarf mergers. NLTE models of Fe
and Ni are used in the abundance analysis. In the GCE models, we include new
delay time distributions arising from the different SN Ia channels, as well as
recent yields for core-collapse supernovae and AGB stars. The data-model
comparison is performed using a Markov chain Monte Carlo framework that allows
us to explore the entire parameter space allowed by the diversity of explosion
mechanisms and the Galactic SN Ia rate, taking into account the uncertainties
of the observed data. We show that NLTE effects have a non-negligible impact on
the observed [Ni/Fe] ratios in the Galactic stars. The NLTE corrections to Ni
abundances are not large, but strictly positive, lifting the [Ni/Fe] ratios by
~+0.15 dex at [Fe/H] =-2. We find that that the distributions of [Ni/Fe] in LTE
and in NLTE are very tight, with a scatter of < 0.1 dex at all metallicities,
supporting earlier work. In LTE, most stars have scaled-solar Ni abundances,
[Ni/Fe] = 0, with a slight tendency for sub-solar [Ni/Fe] ratios at lower
[Fe/H]. In NLTE, however, we find a mild anti-correlation between [Ni/Fe] and
metallicity, and a slightly elevated [Ni/Fe] ratios at [Fe/H] < -1.0. The NLTE
data can be explained by the GCE models calculated with a substantial, ~ 75%,
fraction of sub-Ch SN Ia.Comment: accepted for publication in Astronomy & Astrophysics, abridged
version of the abstrac
CONJURE: automatic generation of constraint models from problem specifications
Funding: Engineering and Physical Sciences Research Council (EP/V027182/1, EP/P015638/1), Royal Society (URF/R/180015).When solving a combinatorial problem, the formulation or model of the problem is critical tothe efficiency of the solver. Automating the modelling process has long been of interest because of the expertise and time required to produce an effective model of a given problem. We describe a method to automatically produce constraint models from a problem specification written in the abstract constraint specification language Essence. Our approach is to incrementally refine the specification into a concrete model by applying a chosen refinement rule at each step. Any nontrivial specification may be refined in multiple ways, creating a space of models to choose from. The handling of symmetries is a particularly important aspect of automated modelling. Many combinatorial optimisation problems contain symmetry, which can lead to redundant search. If a partial assignment is shown to be invalid, we are wasting time if we ever consider a symmetric equivalent of it. A particularly important class of symmetries are those introduced by the constraint modelling process: modelling symmetries. We show how modelling symmetries may be broken automatically as they enter a model during refinement, obviating the need for an expensive symmetry detection step following model formulation. Our approach is implemented in a system called Conjure. We compare the models producedby Conjure to constraint models from the literature that are known to be effective. Our empirical results confirm that Conjure can reproduce successfully the kernels of the constraint models of 42 benchmark problems found in the literature.Publisher PDFPeer reviewe
Towards generic explanations for pen and paper puzzles with MUSes
This research was supported by the Royal Society URF\R\180015 .Pen and paper puzzles like Sudoku, Futoshiki and Star Battle are hugely popular. Solving such puzzles can be a trivial task for modern AI systems. However, most AI systems solve problems using a form of backtracking, while people try to avoid backtracking as much as possible. This means that existing AI systems do not output explanations about their reasoning that are meaningful to people. We present Demystify, a tool which allows puzzles to be expressed in a high-level constraint programming language and uses MUSes to allow us to produce descriptions of steps in the puzzle solving. We give several improvements to the existing techniques for solving puzzles with MUSes, which allow us to solve a range of significantly more complex puzzles and give higher quality explanations. We demonstrate the effectiveness and generality of Demystify by comparing its results to documented strategies for solving a range of pen and paper puzzles by hand, showing that our technique can find many of the same explanations.Publisher PD
New scaling for the alpha effect in slowly rotating turbulence
Using simulations of slowly rotating stratified turbulence, we show that the
alpha effect responsible for the generation of astrophysical magnetic fields is
proportional to the logarithmic gradient of kinetic energy density rather than
that of momentum, as was previously thought. This result is in agreement with a
new analytic theory developed in this paper for large Reynolds numbers. Thus,
the contribution of density stratification is less important than that of
turbulent velocity. The alpha effect and other turbulent transport coefficients
are determined by means of the test-field method. In addition to forced
turbulence, we also investigate supernova-driven turbulence and stellar
convection. In some cases (intermediate rotation rate for forced turbulence,
convection with intermediate temperature stratification, and supernova-driven
turbulence) we find that the contribution of density stratification might be
even less important than suggested by the analytic theory.Comment: 10 pages, 9 figures, revised version, Astrophys. J., in pres
Using Small MUSes to Explain How to Solve Pen and Paper Puzzles
In this paper, we present Demystify, a general tool for creating
human-interpretable step-by-step explanations of how to solve a wide range of
pen and paper puzzles from a high-level logical description. Demystify is based
on Minimal Unsatisfiable Subsets (MUSes), which allow Demystify to solve
puzzles as a series of logical deductions by identifying which parts of the
puzzle are required to progress. This paper makes three contributions over
previous work. First, we provide a generic input language, based on the Essence
constraint language, which allows us to easily use MUSes to solve a much wider
range of pen and paper puzzles. Second, we demonstrate that the explanations
that Demystify produces match those provided by humans by comparing our results
with those provided independently by puzzle experts on a range of puzzles. We
compare Demystify to published guides for solving a range of different pen and
paper puzzles and show that by using MUSes, Demystify produces solving
strategies which closely match human-produced guides to solving those same
puzzles (on average 89% of the time). Finally, we introduce a new randomised
algorithm to find MUSes for more difficult puzzles. This algorithm is focused
on optimised search for individual small MUSes
- …