High-contrast algorithm behavior: observation, hypothesis, and experimental design

After extensive experiments with two algorithms, CPLEX and our implementation of all-integer dual simplex, we observed extreme differences between the two on a set of design automation benchmarks. In many cases one of the two would find an optimal solution within seconds while the other timed out at one hour. We conjecture that this contrast is accounted for by the extent to which the constraint matrix can be made block diagonal via row/column permutations. The actual structure of the matrix without the permutations is not important. We used crossing minimization to discover the right permutations that make the underlying structure, whether block diagonal or random, visible. Our conjecture leads to a testable hypothesis for a limited domain. The hypothesis is based on earlier observations reported in an earlier version of this paper and subsequent exploratory experiments. We present a sample of the results validating the hypothesis and make additional observations outside its scope. As far as we are aware the approach taken here is unique and, we hope, will inspire other research of its kind.