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 \(\tilde{\Theta }(\sqrt {n})\) or \(\tilde{\Theta }(n)\) , 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, as well as the application of practical exact solvers. We further provide efficient algorithms with much improved approximation guarantees for practical special cases where the cardinalities of the input sets or the number of input sets each items belongs to are not too large. Finally, we also generalize our results to DAG-based and non-hierarchical categorization.

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 Θ̃(√n) or Θ̃(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.

Category trees play a central role in many web applications, enabling browsing-style information access. Building trees that reflect users' dynamic interests is, however, a challenging task, carried out by taxonomists. This manual construction leads to outdated trees as it is hard to keep track of market trends. While taxonomists can identify candidate categories, i.e. sets of items with a shared label, most such categories cannot simultaneously exist in the tree, as platforms set a bound on the number of categories an item may belong to. To address this setting, we formalize the problem of constructing a tree where the categories are maximally similar to desirable candidate categories while satisfying combinatorial requirements and provide a model that captures practical considerations.

Category trees play a central role in e-commerce platforms, enabling browsing-style information access. Building category trees that reflect users' dynamic information needs is a challenging task, mostly carried out by in-house taxonomists. This manual construction often leads to trees that are lacking or outdated since it is hard to keep track of market trends, seasonal changes, holidays, and special events.To support a browsing experience that better matches the user information needs, and to considerably reduce the manual work performed by taxonomists, we propose CONCAT - a system that leverages the demand-based nature of the query paradigm to automatically build a category tree that is maximally similar to the result sets for search queries. We demonstrate the effectiveness of CONCAT on real-world data, taken from a large e-commerce platform, by interacting with the ICDE'21 participants who act both as the consumers and the taxonomists.