Stephan Mertens | Otto-von-Guericke-Universität Magdeburg (original) (raw)
Papers by Stephan Mertens
arXiv (Cornell University), Sep 15, 2000
The statistical physics approach to the number partioning problem, a classical NPhard problem, is... more The statistical physics approach to the number partioning problem, a classical NPhard problem, is both simple and rewarding. Very basic notions and methods from statistical mechanics are enough to obtain analytical results for the phase boundary that separates the "easy-to-solve" from the "hard-to-solve" phase of the NPP as well as for the probability distributions of the optimal and sub-optimal solutions. In addition, it can be shown that solving a number partioning problem of size N to some extent corresponds to locating the minimum in an unsorted list of O(2 N) numbers. Considering this correspondence it is not surprising that known heuristics for the partitioning problem are not significantly better than simple random search.
arXiv (Cornell University), Oct 14, 2003
Number partitioning is one of the classical NP-hard problems of combinatorial optimization. It ha... more Number partitioning is one of the classical NP-hard problems of combinatorial optimization. It has applications in areas like public key encryption and task scheduling. The random version of number partitioning has an "easy-hard" phase transition similar to the phase transitions observed in other combinatorial problems like k-SAT. In contrast to most other problems, number partitioning is simple enough to obtain detailled and rigorous results on the "hard" and "easy" phase and the transition that separates them. We review the known results on random integer partitioning, give a very simple derivation of the phase transition and discuss the algorithmic implications of both phases.
Random Structures and Algorithms, Dec 15, 2008
The number partitioning problem is a classic problem of combinatorial optimization in which a set... more The number partitioning problem is a classic problem of combinatorial optimization in which a set of n numbers is partitioned into two subsets such that the sum of the numbers in one subset is as close as possible to the sum of the numbers in the other set. When the n numbers are i.i.d. variables drawn from some distribution, the partitioning problem turns out to be equivalent to a mean-field antiferromagnetic Ising spin glass. In the spin glass representation, it is natural to define energies-corresponding to the costs of the partitions, and overlaps-corresponding to the correlations between partitions. Although the energy levels of this model are a priori highly correlated, a surprising recent conjecture asserts that the energy spectrum of number partitioning is locally that of a random energy model (REM): the spacings between nearby energy levels are uncorrelated. In other words, the properly scaled energies converge to a Poisson process. The conjecture also asserts that the corresponding spin configurations are uncorrelated, indicating vanishing overlaps in the spin glass representation. In this paper, we prove these two claims, collectively known as the local REM conjecture.
arXiv (Cornell University), Dec 12, 2009
Consider "lagged" Fibonacci sequences a(n) = a(n − 1) + a(⌊n/k⌋) for k > 1. We show that limn→∞ a... more Consider "lagged" Fibonacci sequences a(n) = a(n − 1) + a(⌊n/k⌋) for k > 1. We show that limn→∞ a(kn)/a(n) • ln n/n = k ln k and we demonstrate the slow numerical convergence to this limit and how to deal with this slow convergence. We also discuss the connection between two classical results of N.G. de Bruijn and K. Mahler on the asymptotics of a(n).
arXiv (Cornell University), Jan 31, 2005
The number partitioning problem is a classic problem of combinatorial optimization in which a set... more The number partitioning problem is a classic problem of combinatorial optimization in which a set of nnn numbers is partitioned into two subsets such that the sum of the numbers in one subset is as close as possible to the sum of the numbers in the other set. When the nnn numbers are i.i.d. variables drawn from some distribution, the partitioning problem turns out to be equivalent to a mean-field antiferromagnetic Ising spin glass. In the spin glass representation, it is natural to define energies -- corresponding to the costs of the partitions, and overlaps -- corresponding to the correlations between partitions. Although the energy levels of this model are {\em a priori} highly correlated, a surprising recent conjecture asserts that the energy spectrum of number partitioning is locally that of a random energy model (REM): the spacings between nearby energy levels are uncorrelated. In other words, the properly scaled energies converge to a Poisson process. The conjecture also asserts that the corresponding spin configurations are uncorrelated, indicating vanishing overlaps in the spin glass representation. In this paper, we prove these two claims, collectively known as the local REM conjecture.
arXiv (Cornell University), May 26, 2009
Monte Carlo simulations are an important tool in statistical physics, complex systems science, an... more Monte Carlo simulations are an important tool in statistical physics, complex systems science, and many other fields. An increasing number of these simulations is run on parallel systems ranging from multicore desktop computers to supercomputers with thousands of CPUs. This raises the issue of generating large amounts of random numbers in a parallel application. In this lecture we will learn just enough of the theory of pseudo random number generation to make wise decisions on how to choose and how to use random number generators when it comes to large scale, parallel simulations.
arXiv (Cornell University), Mar 11, 1999
Given a set of numbers, the balanced partioning problem is to divide them into two subsets, so th... more Given a set of numbers, the balanced partioning problem is to divide them into two subsets, so that the sum of the numbers in each subset are as nearly equal as possible, subject to the constraint that the cardinalities of the subsets be within one of each other. We combine the balanced largest differencing method (BLDM) and Korf's complete Karmarkar-Karp algorithm to get a new algorithm that optimally solves the balanced partitioning problem. For numbers with twelve significant digits or less, the algorithm can optimally solve balanced partioning problems of arbitrary size in practice. For numbers with greater precision, it first returns the BLDM solution, then continues to find better solutions as time allows.
Physical Review E, Aug 23, 2004
Energy spectra of disordered systems share a common feature: if the entropy of the quenched disor... more Energy spectra of disordered systems share a common feature: if the entropy of the quenched disorder is larger than the entropy of the dynamical variables, the spectrum is locally that of a random energy model and the correlation between energy and configuration is lost. We demonstrate this effect for the Edwards-Anderson model, but we also discuss its universality.
Physical Review Letters, Feb 7, 2000
Oxford University Press eBooks, Aug 11, 2011
Oxford University Press eBooks, Aug 11, 2011
Oxford University Press eBooks, Aug 11, 2011
Oxford University Press eBooks, Aug 11, 2011
Oxford University Press eBooks, Aug 11, 2011
Oxford University Press eBooks, Aug 11, 2011
Oxford University Press eBooks, Aug 11, 2011
Oxford University Press eBooks, Aug 11, 2011
Oxford University Press eBooks, Aug 11, 2011
Journal of Statistical Mechanics: Theory and Experiment, Jun 29, 2015
Let pn denote the probability that a random instance of the stable roommates problem of size n ad... more Let pn denote the probability that a random instance of the stable roommates problem of size n admits a solution. We derive an explicit formula for pn and compute exact values of pn for n ≤ 12.
Journal of Statistical Physics, Feb 1, 2004
Tossing a coin is the most elementary Monte Carlo experiment. In a computer the coin is replaced ... more Tossing a coin is the most elementary Monte Carlo experiment. In a computer the coin is replaced by a pseudo random number generator. It can be shown analytically and by exact enumerations that popular random number generators are not capable of imitating a fair coin: pseudo random coins show more "heads" than "tails". This bias explains the empirically observed failure of some random number generators in random walk experiments. It can be traced down to the special role of the value zero in the algebra of finite fields.
arXiv (Cornell University), Sep 15, 2000
The statistical physics approach to the number partioning problem, a classical NPhard problem, is... more The statistical physics approach to the number partioning problem, a classical NPhard problem, is both simple and rewarding. Very basic notions and methods from statistical mechanics are enough to obtain analytical results for the phase boundary that separates the "easy-to-solve" from the "hard-to-solve" phase of the NPP as well as for the probability distributions of the optimal and sub-optimal solutions. In addition, it can be shown that solving a number partioning problem of size N to some extent corresponds to locating the minimum in an unsorted list of O(2 N) numbers. Considering this correspondence it is not surprising that known heuristics for the partitioning problem are not significantly better than simple random search.
arXiv (Cornell University), Oct 14, 2003
Number partitioning is one of the classical NP-hard problems of combinatorial optimization. It ha... more Number partitioning is one of the classical NP-hard problems of combinatorial optimization. It has applications in areas like public key encryption and task scheduling. The random version of number partitioning has an "easy-hard" phase transition similar to the phase transitions observed in other combinatorial problems like k-SAT. In contrast to most other problems, number partitioning is simple enough to obtain detailled and rigorous results on the "hard" and "easy" phase and the transition that separates them. We review the known results on random integer partitioning, give a very simple derivation of the phase transition and discuss the algorithmic implications of both phases.
Random Structures and Algorithms, Dec 15, 2008
The number partitioning problem is a classic problem of combinatorial optimization in which a set... more The number partitioning problem is a classic problem of combinatorial optimization in which a set of n numbers is partitioned into two subsets such that the sum of the numbers in one subset is as close as possible to the sum of the numbers in the other set. When the n numbers are i.i.d. variables drawn from some distribution, the partitioning problem turns out to be equivalent to a mean-field antiferromagnetic Ising spin glass. In the spin glass representation, it is natural to define energies-corresponding to the costs of the partitions, and overlaps-corresponding to the correlations between partitions. Although the energy levels of this model are a priori highly correlated, a surprising recent conjecture asserts that the energy spectrum of number partitioning is locally that of a random energy model (REM): the spacings between nearby energy levels are uncorrelated. In other words, the properly scaled energies converge to a Poisson process. The conjecture also asserts that the corresponding spin configurations are uncorrelated, indicating vanishing overlaps in the spin glass representation. In this paper, we prove these two claims, collectively known as the local REM conjecture.
arXiv (Cornell University), Dec 12, 2009
Consider "lagged" Fibonacci sequences a(n) = a(n − 1) + a(⌊n/k⌋) for k > 1. We show that limn→∞ a... more Consider "lagged" Fibonacci sequences a(n) = a(n − 1) + a(⌊n/k⌋) for k > 1. We show that limn→∞ a(kn)/a(n) • ln n/n = k ln k and we demonstrate the slow numerical convergence to this limit and how to deal with this slow convergence. We also discuss the connection between two classical results of N.G. de Bruijn and K. Mahler on the asymptotics of a(n).
arXiv (Cornell University), Jan 31, 2005
The number partitioning problem is a classic problem of combinatorial optimization in which a set... more The number partitioning problem is a classic problem of combinatorial optimization in which a set of nnn numbers is partitioned into two subsets such that the sum of the numbers in one subset is as close as possible to the sum of the numbers in the other set. When the nnn numbers are i.i.d. variables drawn from some distribution, the partitioning problem turns out to be equivalent to a mean-field antiferromagnetic Ising spin glass. In the spin glass representation, it is natural to define energies -- corresponding to the costs of the partitions, and overlaps -- corresponding to the correlations between partitions. Although the energy levels of this model are {\em a priori} highly correlated, a surprising recent conjecture asserts that the energy spectrum of number partitioning is locally that of a random energy model (REM): the spacings between nearby energy levels are uncorrelated. In other words, the properly scaled energies converge to a Poisson process. The conjecture also asserts that the corresponding spin configurations are uncorrelated, indicating vanishing overlaps in the spin glass representation. In this paper, we prove these two claims, collectively known as the local REM conjecture.
arXiv (Cornell University), May 26, 2009
Monte Carlo simulations are an important tool in statistical physics, complex systems science, an... more Monte Carlo simulations are an important tool in statistical physics, complex systems science, and many other fields. An increasing number of these simulations is run on parallel systems ranging from multicore desktop computers to supercomputers with thousands of CPUs. This raises the issue of generating large amounts of random numbers in a parallel application. In this lecture we will learn just enough of the theory of pseudo random number generation to make wise decisions on how to choose and how to use random number generators when it comes to large scale, parallel simulations.
arXiv (Cornell University), Mar 11, 1999
Given a set of numbers, the balanced partioning problem is to divide them into two subsets, so th... more Given a set of numbers, the balanced partioning problem is to divide them into two subsets, so that the sum of the numbers in each subset are as nearly equal as possible, subject to the constraint that the cardinalities of the subsets be within one of each other. We combine the balanced largest differencing method (BLDM) and Korf's complete Karmarkar-Karp algorithm to get a new algorithm that optimally solves the balanced partitioning problem. For numbers with twelve significant digits or less, the algorithm can optimally solve balanced partioning problems of arbitrary size in practice. For numbers with greater precision, it first returns the BLDM solution, then continues to find better solutions as time allows.
Physical Review E, Aug 23, 2004
Energy spectra of disordered systems share a common feature: if the entropy of the quenched disor... more Energy spectra of disordered systems share a common feature: if the entropy of the quenched disorder is larger than the entropy of the dynamical variables, the spectrum is locally that of a random energy model and the correlation between energy and configuration is lost. We demonstrate this effect for the Edwards-Anderson model, but we also discuss its universality.
Physical Review Letters, Feb 7, 2000
Oxford University Press eBooks, Aug 11, 2011
Oxford University Press eBooks, Aug 11, 2011
Oxford University Press eBooks, Aug 11, 2011
Oxford University Press eBooks, Aug 11, 2011
Oxford University Press eBooks, Aug 11, 2011
Oxford University Press eBooks, Aug 11, 2011
Oxford University Press eBooks, Aug 11, 2011
Oxford University Press eBooks, Aug 11, 2011
Oxford University Press eBooks, Aug 11, 2011
Journal of Statistical Mechanics: Theory and Experiment, Jun 29, 2015
Let pn denote the probability that a random instance of the stable roommates problem of size n ad... more Let pn denote the probability that a random instance of the stable roommates problem of size n admits a solution. We derive an explicit formula for pn and compute exact values of pn for n ≤ 12.
Journal of Statistical Physics, Feb 1, 2004
Tossing a coin is the most elementary Monte Carlo experiment. In a computer the coin is replaced ... more Tossing a coin is the most elementary Monte Carlo experiment. In a computer the coin is replaced by a pseudo random number generator. It can be shown analytically and by exact enumerations that popular random number generators are not capable of imitating a fair coin: pseudo random coins show more "heads" than "tails". This bias explains the empirically observed failure of some random number generators in random walk experiments. It can be traced down to the special role of the value zero in the algebra of finite fields.