Search Trees on Trees via LP

TL;DR

This paper investigates the LP relaxation approach for computing optimal search trees on trees, disproves a conjecture about its optimality, and explores the limitations and future research directions.

Contribution

It demonstrates that Golinsky's LP does not always produce optimal solutions and introduces the normals method to analyze the LP's polytope.

Findings

Golinsky's LP has a non-zero integrality gap.

The LP does not always yield an optimal search tree.

The normals method helps analyze the LP's vertices and bounds.

Abstract

We consider the problem of computing optimal search trees on trees (STTs). STTs generalize binary search trees (BSTs) in which we search nodes in a path (linear order) to search trees that facilitate search over general tree topologies. Golinsky proposed a linear programming (LP) relaxation of the problem of computing an optimal static STT over a given tree topology. He used this LP formulation to compute an STT that is a $2$-approximation to an optimal STT, and conjectured that it is, in fact, an extended formulation of the convex-hull of all depths-vectors of STTs, and thus always gives an optimal solution. In this work we study this LP approach further. We show that the conjecture is false and that Golinsky's LP does not always give an optimal solution. To show this we use what we call the ``normals method''. We use this method to enumerate over vertices of Golinsky's polytope for all tree topologies of no more than 8 nodes. We give a lower bound on the integrality gap of the LP and on the approximation ratio of Golinsky's rounding method. We further enumerate several research directions that can lead to the resolution of the question whether one can compute an optimal STT in polynomial time.