Edge conflicts do not determine geodesics in the associahedron

TL;DR

This paper demonstrates that minimizing conflicting edge pairs does not reliably lead to shortest paths in associahedra, challenging the effectiveness of greedy algorithms based on conflict reduction.

Contribution

It provides counterexamples showing conflicts do not always decrease along geodesics, revealing limitations of conflict-based greedy algorithms in associahedra.

Findings

Conflicting edge pairs do not always decrease along geodesics.

Greedy conflict-reduction algorithms may fail to find shortest paths.

Conflicts can increase arbitrarily along geodesics.

Abstract

There are no known efficient algorithms to calculate distance in the one-skeleta of associahedra, a problem that is equivalent to finding rotation distance between rooted binary trees or the flip distance between polygonal triangulations. One measure of the difference between trees is the number of conflicting edge pairs, and a natural way of trying to find short paths is to minimize successively this number of conflicting edge pairs using flip operations in the corresponding triangulations. We describe examples that show that the number of such conflicts does not always decrease along geodesics. Thus, a greedy algorithm that always chooses a transformation that reduces conflicts will not produce a geodesic in all cases. Further, for any specified amount, there are examples of pairs of all large sizes showing that the number of conflicts can increase by that amount along any geodesic between the pairs.