46 research outputs found
Solution-space structure of (some) optimization problems
We study numerically the cluster structure of random ensembles of two NP-hard
optimization problems originating in computational complexity, the vertex-cover
problem and the number partitioning problem. We use branch-and-bound type
algorithms to obtain exact solutions of these problems for moderate system
sizes. Using two methods, direct neighborhood-based clustering and hierarchical
clustering, we investigate the structure of the solution space. The main result
is that the correspondence between solution structure and the phase diagrams of
the problems is not unique. Namely, for vertex cover we observe a drastic
change of the solution space from large single clusters to multiple nested
levels of clusters. In contrast, for the number-partitioning problem, the phase
space looks always very simple, similar to a random distribution of the
lowest-energy configurations. This holds in the ``easy''/solvable phase as well
as in the ``hard''/unsolvable phase.Comment: 10 pages, 5 figures, Fig. 4 in reduced quality to reduce size,
Proceedings of the International Workshop on Statistical-Mechanical
Informatics 2007, Kyoto (Japan) September 16-19, 200