[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85526-en":3,"doc-seo-85526-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85526,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","Precedence-Constrained Decision Trees and Coverings","This work studies optimization problems under precedence constraints, focusing on Optimal Decision Trees and Set Cover. When one test (or set) X precedes Y, any feasible strategy must execute X before Y, enforcing precedence-closed decision trees and cover families. The paper considers worst-case and average identification time for decision trees, and cover size and average cover time for set cover. It develops approximation-preserving reductions, includes a Maximum Density Precedence-Closed Subfamily, and provides O*(√m)-approximation algorithms alongside hardness results for tightness, plus polylogarithmic guarantees for outforests and inforests.","arXiv :2602 .2 13 12v4 [ cs .DS] 12 Jul 2026  \nPrecedence-Constrained Decision Trees and Coverings  \nMichał Szyfelbein  \nGdańsk University of Technology, 80-233 Gdańsk, Poland [michal. szyfelbein@pg. edu. pl](michal. szyfelbein@pg. edu. pl)  \nDariusz Dereniowski  \nGdańsk University of Technology, 80-233 Gdańsk, Poland [deren@eti. pg. edu. pl](deren@eti. pg. edu. pl)  \nAbstract  \nThis work considers a number of optimization problems and reductive relations between them. The two main problems we are interested in are the Optimal Decision Tree and Set Cover. We study these two fundamental tasks under precedence constraints, that is, if a test (or set) X is a predecessor of Y , then in any feasible decision tree X needs to be an ancestor of Y (or respectively, if Y is added to set cover, then so must be X ) . For the Optimal Decision Tree we consider two optimization criteria: worst case identification time (height of the tree)  \nor the average identification time. Similarly, for the Set Cover we study two cost measures: the size of the cover or the average cover time.  \nOur approach is to develop a number of algorithmic reductions, where an approximation algorithm for one problem provides an approximation for another via a black-box usage of a procedure for the former. En route we introduce other optimization problems either to complete the ‘reduction landscape’ or because they hold the essence of combinatorial structure of our problems. The latter is brought by a problem of finding a Maximum Density PrecedenceClosed Subfamily, where the density is defined as the ratio of the number of items the family covers to its size. We provide O ∗ ( √m)-approximation polynomial-time algorithms for all aforementioned problems. The picture is complemented by a number of hardness reductions that provide O (m1/12−ϵ )-inapproximability results for the decision tree and covering problems. Besides giving a complete set of results for general precedence constraints, we also provide polylogarithmic approximation guarantees for two most typically studied and applicable graph types, outforests and inforests. By providing corresponding hardness results, we show most of these results to be tight.  \nKeywords: Optimal decision trees, Set Cover, Precedence Constraints, Approximation Algorithms 1 Introduction  \nConsider a set H of n hypotheses, a set T of m tests and an unknown target hypothesis h ∗ ∈ H that needs to be discovered through testing. Each test t ∈ T is a partition of H, that is, t consists of subsets of H such that for any X, Y ∈ t, X ∩ Y = ∅ and SX∈t X = H. As a result of executing a test t ∈ T, the questioner receives a reply that reveals H ∈ t such that h ∗ ∈ H. Upon receiving this response, the questioner adaptively selects the next test from T to perform, until the target hypothesis is identified. The goal is to conceive a strategy of testing, often modeled as a decision tree, which either minimizes the worst-case or the average-case number of tests performed until h ∗  \nis uncovered. We refer to the former problem as the Worst Case Decision Tree and to the latter as the Average Case Decision Tree. If the cost criterion is not mentioned explicitly, we refer to the problem as the Optimal Decision Tree (DT) .  \nIn this work we generalize the above setup by considering instances which are subject to arbitrary precedence constraints between tests given as a partial order (T, ⪯) on the test set T. It is required that in any valid strategy of testing, a given test t ∈ T can be performed only after all its predecessors have been performed previously (to which we refer as precedence-closed solution) . This setup is a natural generalization of the standard Optimal Decision Tree, and is partially motivated by the fact that the decision tree can be interpreted as a schedule of tests. Since task scheduling often involves precedence constraints, it is natural to ask how such constraints affect the complexity of the Optimal Decision Tree. On the applic","cbCaifpwkNQYEguE","https://ap.wps.com/l/cbCaifpwkNQYEguE","pdf",790942,2,1,34,"English","en",105,"# Abstract\n# Introduction\n## Problem setup: hypotheses and adaptive testing\n## Precedence constraints as precedence-closed strategies\n## Motivation and application examples\n## Algorithmic approach via reductions\n# Precedence-constrained Set Cover formulation\n# Keywords","[{\"question\":\"What does a precedence constraint mean in the decision tree setting?\",\"answer\":\"If a test X is a predecessor of Y, then any feasible strategy must place X as an ancestor of Y in the decision tree. This enforces that predecessors are performed before successors.\"},{\"question\":\"Which optimization criteria are studied for Optimal Decision Trees?\",\"answer\":\"The paper studies worst-case identification time, measured by the tree height, and average identification time over the adaptive testing process.\"},{\"question\":\"What cost measures are considered for Set Cover under precedence constraints?\",\"answer\":\"For set cover, it considers cover size and average cover time. The chosen subfamily must be precedence-closed with respect to the partial order on subsets.\"}]",1784204179,86,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"precedence-constrained-decision-trees-and-coverings","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/precedence-constrained-decision-trees-and-coverings/85526/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does a precedence constraint mean in the decision tree setting?","Question",{"text":75,"@type":76},"If a test X is a predecessor of Y, then any feasible strategy must place X as an ancestor of Y in the decision tree. This enforces that predecessors are performed before successors.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which optimization criteria are studied for Optimal Decision Trees?",{"text":80,"@type":76},"The paper studies worst-case identification time, measured by the tree height, and average identification time over the adaptive testing process.",{"name":82,"@type":73,"acceptedAnswer":83},"What cost measures are considered for Set Cover under precedence constraints?",{"text":84,"@type":76},"For set cover, it considers cover size and average cover time. 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