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Searching in trees with $k$-up-modular cost functions

Michał Szyfelbein

TL;DR

The paper tackles tree search with nonuniform vertex-query costs, formalizing the Tree Search Problem on a tree $T$ with costs $c(v)$. It introduces the notion of $k$-up-modularity to capture cost irregularity and develops a parametrized $O( ext{loglog } n)$-approximation algorithm with running time $k^{O( ext{log } k)}\cdot\text{poly}(n)$. The approach combines cost-interval decomposition, heavy-module analysis, an auxiliary tree $\mathcal{T}_{\mathcal{Z}}$ solved via a QPTAS, and a vertex-ranking-based construction to assemble a full decision tree with provable approximation guarantees. This yields a significant improvement over general-cost results and provides a structured path toward PTAS or stronger inapproximability results for this class of tree-search problems. The work also highlights practical considerations for irregular cost models in distributed testing, DB operations, and hierarchical clustering, where cost irregularities can be controlled by preprocessing to reduce $k$.

Abstract

Consider the following generalization of the classic binary search problem: a searcher is required to find a hidden vertex $x$ in a tree $T$. To do so, they iteratively perform queries to an oracle, each about a chosen vertex $v$. After each such call, the oracle responds whether the target was found and if not, the searcher receives as a reply the connected component of $T-v$ which contains $x$. Additionally, each vertex $v$ may have a different query cost $c(v)$. The goal is to find the optimal querying strategy which minimizes the worst case cost required to find $x$. The problem is known to be NP-hard even in restricted classes of trees such as bounded diameter spiders [Cicalese et al. 2016], and no constant factor approximation algorithm is known for general trees. Following the recent studies of [Dereniowski et al. 2022, Dereniowski et al. 2024], instead of restricted classes of trees, we explore restrictions on the cost function. We generalize the notion of up-monotonic functions and introduce the concept of \textit{$k$-up-modularity}. We show that an $O(\log\log n)$-approximate solution can be found within $k^{O(\log k)}\cdot\text{poly}(n)$ time.

Searching in trees with $k$-up-modular cost functions

TL;DR

The paper tackles tree search with nonuniform vertex-query costs, formalizing the Tree Search Problem on a tree with costs . It introduces the notion of -up-modularity to capture cost irregularity and develops a parametrized -approximation algorithm with running time . The approach combines cost-interval decomposition, heavy-module analysis, an auxiliary tree solved via a QPTAS, and a vertex-ranking-based construction to assemble a full decision tree with provable approximation guarantees. This yields a significant improvement over general-cost results and provides a structured path toward PTAS or stronger inapproximability results for this class of tree-search problems. The work also highlights practical considerations for irregular cost models in distributed testing, DB operations, and hierarchical clustering, where cost irregularities can be controlled by preprocessing to reduce .

Abstract

Consider the following generalization of the classic binary search problem: a searcher is required to find a hidden vertex in a tree . To do so, they iteratively perform queries to an oracle, each about a chosen vertex . After each such call, the oracle responds whether the target was found and if not, the searcher receives as a reply the connected component of which contains . Additionally, each vertex may have a different query cost . The goal is to find the optimal querying strategy which minimizes the worst case cost required to find . The problem is known to be NP-hard even in restricted classes of trees such as bounded diameter spiders [Cicalese et al. 2016], and no constant factor approximation algorithm is known for general trees. Following the recent studies of [Dereniowski et al. 2022, Dereniowski et al. 2024], instead of restricted classes of trees, we explore restrictions on the cost function. We generalize the notion of up-monotonic functions and introduce the concept of \textit{-up-modularity}. We show that an -approximate solution can be found within time.

Paper Structure

This paper contains 18 sections, 9 theorems, 5 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem description
  3. Motivation and applications
  4. State-of-the-art and related results
  5. Organization of the paper
  6. Preliminaries
  7. Notation and query model
  8. Definition of a decision tree
  9. $k$-up-modularity
  10. The parametrized $O\mathopen{}\left( \log\log n \right)$-approx. solution
  11. Cost levels
  12. Vertex ranking
  13. The main recursive procedure
  14. Base of the recursion
  15. Construction of the Auxiliary Tree
  16. ...and 3 more sections

Key Result

Corollary 2

There exists an $O\mathopen{}\left( n \right)$ time procedure RankingBasedDT that finds the optimal decision tree for the Tree Search Problem when all costs are uniform. Moreover, the depth of such a decision tree, i.e., the worst-case number of queries, is at most $\mathopen{}\left\lfloor \log n \r

Figures (6)

  • Figure 1: Example input tree $T$ and cost function $c$.
  • Figure 2: Example decision tree $D$ for $T$ of cost $11/5$.
  • Figure 3: A visual depiction of a tree $T$ with a $3$-up-modular cost function $c$. Each vertex of a tree is mapped onto some value of $c$. The yellow plane represents some threshold value $t\in \mathbb{R}_{\geq 0}$ (in this particular example $k\mathopen{}\left( T,c,t \right)=2$). The (two) blue subtrees represent members of $\mathcal{H}_{T,c}\mathopen{}\left( t \right)$.
  • Figure 4: Example tree $\mathcal{T}$. Dark grey circles represent heavy modules. Light grey regions represent light subtrees. Black vertices represent $\mathcal{X}$. Gray and black vertices represent $\mathcal{Y}$. White, gray and black vertices represent $\mathcal{Z}$. Lines represent paths of vertices between vertices of $\mathcal{Z}$.
  • Figure 5: Auxiliary tree $\mathcal{T}_{\mathcal{Z}}$ built from vertices of set $\mathcal{Z}$.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Corollary 2
  • Lemma 3
  • Theorem 4
  • Lemma 9
  • Lemma 11
  • Lemma 12
  • Lemma 13
  • Lemma 14
  • Theorem 15