The Unexpected Virtue of Problem Reductions or How to Solve Problems Being Lazy but Wise

Abstract

The generalization of one problem to another is a useful technique in theoretical computer science; reductions among problems are a well established mathematical approach to demonstrate the structural relationships between problems. However, most of the reductions used to obtain theoretical results are relatively coarse-grained and chosen for their amenability in supporting mathematical proof, and represent a selection amongst many possible reduction schemas. We propose reexamining reductions as a practical tool, since choosing one reduction scheme over another may be decisive in solving a given instance in practical settings. In this work, we examine the impact of several new reduction schema. A total of 100 experiments were conducted using challenging Hamiltonian Cycle Problem instances using Concorde, a well known and effective TSP solver, and example of a complete memetic algorithm (MA). Benefits of using MA are that it uses multi-parent recombination, local search and also provides an optimality guarantee through its implicit enumeration complete search. We show that the choice of reduction scheme can result in dramatic speed-ups in practice, suggesting that when using general solvers, it pays “to be wise” and to explore alternative representations of instances.