{"title":"A PTAS for the horizontal rectangle stabbing problem","authors":"Arindam Khan, Aditya Subramanian, Andreas Wiese","doi":"10.1007/s10107-024-02106-y","DOIUrl":null,"url":null,"abstract":"<p>We study rectangle stabbing problems in which we are given <i>n</i> axis-aligned rectangles in the plane that we want to <i>stab</i>, that is, we want to select line segments such that for each given rectangle there is a line segment that intersects two opposite edges of it. In the <i>horizontal rectangle stabbing problem</i> (<span>Stabbing</span>), the goal is to find a set of horizontal line segments of minimum total length such that all rectangles are stabbed. In the <i>horizontal–vertical stabbing problem</i> (<span>HV-Stabbing</span>), the goal is to find a set of rectilinear (that is, either vertical or horizontal) line segments of minimum total length such that all rectangles are stabbed. Both variants are NP-hard. Chan et al. (ISAAC, 2018) initiated the study of these problems by providing constant approximation algorithms. Recently, Eisenbrand et al. (A QPTAS for stabbing rectangles, 2021) have presented a QPTAS and a polynomial-time 8-approximation algorithm for <span>Stabbing</span>, but it was open whether the problem admits a PTAS. In this paper, we obtain a PTAS for <span>Stabbing</span>, settling this question. For <span>HV-Stabbing</span>, we obtain a <span>\\((2+\\varepsilon )\\)</span>-approximation. We also obtain PTASs for special cases of <span>HV-Stabbing</span>: (i) when all rectangles are squares, (ii) when each rectangle’s width is at most its height, and (iii) when all rectangles are <span>\\(\\delta \\)</span>-large, that is, have at least one edge whose length is at least <span>\\(\\delta \\)</span>, while all edge lengths are at most 1. Our result also implies improved approximations for other problems such as <i>generalized minimum Manhattan network</i>.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"46 1","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Programming","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10107-024-02106-y","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
引用次数: 0
Abstract
We study rectangle stabbing problems in which we are given n axis-aligned rectangles in the plane that we want to stab, that is, we want to select line segments such that for each given rectangle there is a line segment that intersects two opposite edges of it. In the horizontal rectangle stabbing problem (Stabbing), the goal is to find a set of horizontal line segments of minimum total length such that all rectangles are stabbed. In the horizontal–vertical stabbing problem (HV-Stabbing), the goal is to find a set of rectilinear (that is, either vertical or horizontal) line segments of minimum total length such that all rectangles are stabbed. Both variants are NP-hard. Chan et al. (ISAAC, 2018) initiated the study of these problems by providing constant approximation algorithms. Recently, Eisenbrand et al. (A QPTAS for stabbing rectangles, 2021) have presented a QPTAS and a polynomial-time 8-approximation algorithm for Stabbing, but it was open whether the problem admits a PTAS. In this paper, we obtain a PTAS for Stabbing, settling this question. For HV-Stabbing, we obtain a \((2+\varepsilon )\)-approximation. We also obtain PTASs for special cases of HV-Stabbing: (i) when all rectangles are squares, (ii) when each rectangle’s width is at most its height, and (iii) when all rectangles are \(\delta \)-large, that is, have at least one edge whose length is at least \(\delta \), while all edge lengths are at most 1. Our result also implies improved approximations for other problems such as generalized minimum Manhattan network.
期刊介绍:
Mathematical Programming publishes original articles dealing with every aspect of mathematical optimization; that is, everything of direct or indirect use concerning the problem of optimizing a function of many variables, often subject to a set of constraints. This involves theoretical and computational issues as well as application studies. Included, along with the standard topics of linear, nonlinear, integer, conic, stochastic and combinatorial optimization, are techniques for formulating and applying mathematical programming models, convex, nonsmooth and variational analysis, the theory of polyhedra, variational inequalities, and control and game theory viewed from the perspective of mathematical programming.