On the Hardness of Category Tree Construction

Shay Gershtein, Uri Avron, Ido Guy, Tova Milo, Slava Novgorodov

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Category trees, or taxonomies, are rooted trees where each node, called a category, corresponds to a set of related items. The construction of taxonomies has been studied in various domains, including e-commerce, document management, and question answering. Multiple algorithms for automating construction have been proposed, employing a variety of clustering approaches and crowdsourcing. However, no formal model to capture such categorization problems has been devised, and their complexity has not been studied. To address this, we propose in this work a combinatorial model that captures many practical settings and show that the aforementioned empirical approach has been warranted, as we prove strong inapproximability bounds for various problem variants and special cases when the goal is to produce a categorization of the maximum utility. In our model, the input is a set of n weighted item sets that the tree would ideally contain as categories. Each category, rather than perfectly match the corresponding input set, is allowed to exceed a given threshold for a given similarity function. The goal is to produce a tree that maximizes the total weight of the sets for which it contains a matching category. A key parameter is an upper bound on the number of categories an item may belong to, which produces the hardness of the problem, as initially each item may be contained in an arbitrary number of input sets. For this model, we prove inapproximability bounds, of order Θ(√n) or Θ(n), for various problem variants and special cases, loosely justifying the aforementioned heuristic approach. Our work includes reductions based on parameterized randomized constructions that highlight how various problem parameters and properties of the input may affect the hardness. Moreover, for the special case where the category must be identical to the corresponding input set, we devise an algorithm whose approximation guarantee depends solely on a more granular parameter, allowing improved worst-case guarantees. Finally, we also generalize our results to DAG-based and non-hierarchical categorization.

Original languageEnglish
Title of host publication25th International Conference on Database Theory, ICDT 2022
EditorsDan Olteanu, Nils Vortmeier
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772235
DOIs
StatePublished - 1 Mar 2022
Event25th International Conference on Database Theory, ICDT 2022 - Virtual, Edinburgh, United Kingdom
Duration: 29 Mar 20221 Apr 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume220
ISSN (Print)1868-8969

Conference

Conference25th International Conference on Database Theory, ICDT 2022
Country/TerritoryUnited Kingdom
CityVirtual, Edinburgh
Period29/03/221/04/22

Keywords

  • approximation algorithms
  • approximation hardness bounds
  • category tree construction
  • maximum independent set
  • taxonomy construction

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