A New Look at the Easy-Hard-Easy Pattern of Combinatorial Search Difficulty

TL;DR

This paper investigates the easy-hard-easy pattern in combinatorial search difficulty, revealing that it can occur independently of solution count changes and may be due to variations in minimal unsolvable subproblems.

Contribution

It challenges existing explanations by showing the pattern persists even with fixed solution counts, highlighting the role of minimal unsolvable subproblems.

Findings

Easy-hard-easy pattern occurs even with fixed solution numbers

Pattern linked to changes in minimal unsolvable subproblems

Existing theory does not fully explain the difficulty peak

Abstract

The easy-hard-easy pattern in the difficulty of combinatorial search problems as constraints are added has been explained as due to a competition between the decrease in number of solutions and increased pruning. We test the generality of this explanation by examining one of its predictions: if the number of solutions is held fixed by the choice of problems, then increased pruning should lead to a monotonic decrease in search cost. Instead, we find the easy-hard-easy pattern in median search cost even when the number of solutions is held constant, for some search methods. This generalizes previous observations of this pattern and shows that the existing theory does not explain the full range of the peak in search cost. In these cases the pattern appears to be due to changes in the size of the minimal unsolvable subproblems, rather than changing numbers of solutions.