A New Tree Encoding for the Degree-Constrained Spanning Tree Problem

The degree-constrained minimum spanning tree (dc- MST) problem is of high practical importance. Up to now there are few effective algorithms to solve this problem because of its NP-hard complexity. More recently, a ge- netic algorithm (GA) approach for this problem was tried by using Pr¨ufer number to encode a spanning tree. The Pr¨ufer number is a skillful encoding for tree but not efficient enough to deal with the dc-MST problem. In this paper, a new tree encoding is developed directly based on the de- gree constraint on each vertex and the connectivity among vertices. We denote it as degree-based permutation encod- ing and apply it to the dc-MST problem by using the GA approach. Compared with the numerical results and CPU runtimes between two encodings, the new degree-based per- mutation is effective to deal with the dc-MST problem and even more efficient than the Pr¨ufer number to evolve to the optimal or near-optimal solutions.