Random Trees and the Analysis of Branch and Bound Procedures

Abstract

Branch and bound procedures are the most efficient known means for solving many NP-hard problems A special class of branch and bound procedures called relaxation-gutded procedures ig presented. While for some branch and bound procedures a worst-case complexity bound is known, the average case complexity is usually unknown, despite the fact that it may give more useful information about the performance of the algorithm. A random process which generates labeled trees is introduced as a model of the kind of trees that a relaxatlon-gutded procedure generates over random instances of a problem Results concerning the expected time and space complexity of searching these random trees are derived with respect to several search strategies. The best-bound search strategy is shown to be optimal in both time and space. These results are illustrated by data from random traveling salesman instances Evidence is presented that the asymmetric traveling salesman problem can be solved exactly in time O(n31n(n)) on the average. © 1984, ACM. All rights reserved.