The Replica Method in Optimization Problems.
Abstract
In this thesis I discuss the application of the replica method in combinatorial optimization problems. In particular, I study certain graphpartitioning problems. One problem that I consider is the following. We are given a set of vertices V = (V_1,V_2,ldots V_{N}), with N even, and a set of edges E = {(V_{i},V _{j})i not= j}. Let each edge be connected with probability P. The bipartitioning problem is to divide V into two parts of equal size, in such a way as to minimize the number of edges N _{c} connecting these two parts. We are interested in the behavior of N_{c }/N, averaged over all possible configurations of edges in the limit N > infty , as a function of the connectivity alpha = NP. When alpha is finite, the problem is shown to be similar, but not identical, to the mean field theory of a spin glass with finite connectivity. The replicasymmetric solution is derived. It is shown to be consistent with exact results for the infinite cluster obtained by P. Erdos.
 Publication:

Ph.D. Thesis
 Pub Date:
 1988
 Bibcode:
 1988PhDT.......104L
 Keywords:

 Physics: General