Search Trees on Trees via LP

2025-01-29

AI Summary

This paper studies the Optimal Search Tree on Trees (STT) problem—constructing a search tree of minimum expected search cost over an arbitrary underlying tree topology, generalizing the classical Binary Search Tree model. Addressing Golinsky’s conjecture that a natural linear programming (LP) relaxation always yields an integral optimal solution, we provide the first counterexample: the LP exhibits an integrality gap, thus failing to guarantee optimality. We introduce the *normal-vector method* to efficiently enumerate vertices of the LP feasible region and rigorously derive lower bounds on both the integrality gap and the approximation ratio. A complete enumeration over all trees with up to eight nodes confirms these bounds. Our results show that the LP achieves only a nontrivial constant-factor approximation in the worst case, and computing an exact optimal STT in polynomial time remains an open problem. This work establishes fundamental hardness barriers for STT optimization, providing critical theoretical benchmarks for algorithm design and complexity analysis.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Social Networks and Social Media: Social mining and social search on the Web

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.

Problem

• optimizing search trees on trees
• evaluating LP relaxation accuracy
• determining polynomial-time optimal STT computation

Innovation

• LP relaxation for STTs
• Normals method for enumeration
• Integrality gap lower bound