From Approximate to Optimal Solutions: Constructing Pruning and Propagation Rules

Abs t rac t At the heart of many optimizat ion procedures are powerful pruning and propagation rules. This paper presents a case study in the construction of such rules. We develop a new algor i thm, Complete Decreasing Best F i t , that finds the opt imal packing of objects into bins. The algor i thm use a branching rule based on the well known Decreasing Best Fi t approximat ion algor i thm. In addit ion, it includes a powerful pruning rule derived from a bound on the solution to the remaining subproblem. The bound is constructed by using modular ari thmetic to decompose the numerical constraints. We show that the pruning rule adds essentially a constant factor overhead to runtime, whilst reducing search significantly. On the hardest problems, runt ime can be reduced by an order of magnitude. Final ly we demonstrate how propagation rules can be buil t by adding lookahead to pruning rules. This general approach opt imizat ion procedures bui l t f rom branching rules based on good approximation algorithms, and pruning and propagation rules derived from bounds on the remaining subproblem may be effective on other NP-complete problems.