13 research outputs found
NP-hardness of the cluster minimization problem revisited
The computational complexity of the "cluster minimization problem" is
revisited [L. T. Wille and J. Vennik, J. Phys. A 18, L419 (1985)]. It is argued
that the original NP-hardness proof does not apply to pairwise potentials of
physical interest, such as those that depend on the geometric distance between
the particles. A geometric analog of the original problem is formulated, and a
new proof for such potentials is provided by polynomial time transformation
from the independent set problem for unit disk graphs. Limitations of this
formulation are pointed out, and new subproblems that bear more direct
consequences to the numerical study of clusters are suggested.Comment: 8 pages, 2 figures, accepted to J. Phys. A: Math. and Ge
Number partitioning as random energy model
Number partitioning is a classical problem from combinatorial optimisation.
In physical terms it corresponds to a long range anti-ferromagnetic Ising spin
glass. It has been rigorously proven that the low lying energies of number
partitioning behave like uncorrelated random variables. We claim that
neighbouring energy levels are uncorrelated almost everywhere on the energy
axis, and that energetically adjacent configurations are uncorrelated, too.
Apparently there is no relation between geometry (configuration) and energy
that could be exploited by an optimization algorithm. This ``local random
energy'' picture of number partitioning is corroborated by numerical
simulations and heuristic arguments.Comment: 8+2 pages, 9 figures, PDF onl
Polynomial-time approximation schemes for scheduling problems with time lags
We identify two classes of machine scheduling problems with time lags that possess Polynomial-Time Approximation Schemes (PTASs). These classes together, one for minimizing makespan and one for minimizing total completion time, include many well-studied time lag scheduling problems. The running times of these approximation schemes are polynomial in the number of jobs, but exponential in the number of machines and the ratio between the largest time lag and the smallest positive operation time. These classes constitute the first PTAS results for scheduling problems with time lags