Epsilon-transformation: exploiting phase transitions to solve combinatorial optimization problems

Abstract

It has been shown that there exists a transition in the average-case complexity of tree search problems, from exponential to polynomial in the search depth. We develop a new method, called ε-transformation, which makes use of this complexity transition, to find a suboptimal solution. With a random tree model, we show that, as the tree depth approaches infinity, the expected number of nodes expanded by branch-and-bound (BnB) using ε-transformation is at most cubic in the search depth, and that the relative error of the solution cost found with respect to the optimal solution cost is bounded above by a small constant. We also present an iterative version of ε-transformation that can be used to find both suboptimal and optimal goal nodes. Depth-first BnB (DFBnB) using iterative ε-transformation significantly improves upon DFBnB on random trees with large branching factors and deep goal nodes, finding a better solution sooner. We then present experimental results for ε-transformation and iterative ε-transformation on the asymmetric traveling salesman problem (ATSP) and the maximum boolean satisfiability problem, and identify the conditions under which these two methods are effective. On the ATSP, DFBnB using ε-transformation outperforms a well-known local search method, and DFBnB using iterative ε-transformation improves upon DFBnB.