arXiv subjects

Stephan Mertens

Publications and source records attributed to Stephan Mertens.

40 records · Page 3Linked to original sources

arXiv1999

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.

cond-mat.dis-nn↗

arXiv1999

A complete anytime algorithm for balanced number partitioning

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.

cs.DS↗

arXiv1998

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.

cond-mat↗

arXiv1997

On the ground states of the Bernasconi model

The ground states of the Bernasconi model are binary +1/-1 sequences of length N with low autocorrelations. We introduce the notion of perfect sequences, binary sequences with one-valued off-peak correlations of minimum amount. If they exist, they are ground states. Using results from the mathematical theory of cyclic difference sets, we specify all values of N for which perfect sequences do exist and how to construct them. For other values of N, we investigate almost perfect sequences, i.e. sequences with two-valued off-peak correlations of minimum amount. Numerical and analytical results support the conjecture that almost perfect sequences do exist for all values of N, but that they are not always ground states. We present a construction for low-energy configurations that works if N is the product of two odd primes.

cond-mat.dis-nn↗