The parameterized complexity of finding minimum bounded chains

IF 0.4 4区计算机科学 Q4 MATHEMATICS Computational Geometry-Theory and Applications Pub Date : 2024-04-29 DOI:10.1016/j.comgeo.2024.102102

Nello Blaser , Morten Brun , Lars M. Salbu , Erlend Raa Vågset

{"title":"The parameterized complexity of finding minimum bounded chains","authors":"Nello Blaser , Morten Brun , Lars M. Salbu , Erlend Raa Vågset","doi":"10.1016/j.comgeo.2024.102102","DOIUrl":null,"url":null,"abstract":"<div>Finding the smallest d-chain with a specific <math><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo></math>-boundary in a simplicial complex is known as the Minimum Bounded Chain problem (MBCd). MBCd is NP-hard for all <math><mi>d</mi><mo>≥</mo><mn>2</mn></math>. In this paper, we prove that it is also W[1]-hard for all <math><mi>d</mi><mo>≥</mo><mn>2</mn></math>, if we parameterize the problem by solution size. We also give an algorithm solving MBC1 in polynomial time and introduce and implement two fixed parameter tractable (FPT) algorithms solving MBCd for all d. The first algorithm uses a shortest path approach and is parameterized by solution size and coface degree. The second algorithm is a dynamic programming approach based on treewidth, which has the same runtime as a lower bound we prove under the exponential time hypothesis.</div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0925772124000245/pdfft?md5=783e1fbffafc12d2132a61d1e8077846&pid=1-s2.0-S0925772124000245-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Geometry-Theory and Applications","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0925772124000245","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Finding the smallest d-chain with a specific $(d - 1)$ -boundary in a simplicial complex is known as the Minimum Bounded Chain problem (MBC_d). MBC_d is NP-hard for all $d \geq 2$ . In this paper, we prove that it is also W[1]-hard for all $d \geq 2$ , if we parameterize the problem by solution size. We also give an algorithm solving MBC₁ in polynomial time and introduce and implement two fixed parameter tractable (FPT) algorithms solving MBC_d for all d. The first algorithm uses a shortest path approach and is parameterized by solution size and coface degree. The second algorithm is a dynamic programming approach based on treewidth, which has the same runtime as a lower bound we prove under the exponential time hypothesis.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

寻找最小有界链的参数化复杂性

寻找简单复合物中具有特定（d-1）边界的最小 d 链被称为最小有界链问题（MBCd）。对于所有 d≥2 的情况，MBCd 都是 NP-困难的。在本文中，我们证明，如果以解的大小为参数，对于所有 d≥2 的问题，MBCd 也是 W[1]-hard 的。我们还给出了一种多项式时间内求解 MBC1 的算法，并引入和实现了两种固定参数可求解 MBCd 的算法（FPT）。第二种算法是基于树宽的动态编程方法，其运行时间与我们在指数时间假设下证明的下限相同。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Computational Geometry-Theory and Applications 数学-数学

CiteScore

1.60

自引率

16.70%

发文量

审稿时长

>12 weeks

期刊介绍： Computational Geometry is a forum for research in theoretical and applied aspects of computational geometry. The journal publishes fundamental research in all areas of the subject, as well as disseminating information on the applications, techniques, and use of computational geometry. Computational Geometry publishes articles on the design and analysis of geometric algorithms. All aspects of computational geometry are covered, including the numerical, graph theoretical and combinatorial aspects. Also welcomed are computational geometry solutions to fundamental problems arising in computer graphics, pattern recognition, robotics, image processing, CAD-CAM, VLSI design and geographical information systems. Computational Geometry features a special section containing open problems and concise reports on implementations of computational geometry tools.

期刊最新文献

Editorial Board Largest unit rectangles inscribed in a convex polygon Packing unequal disks in the Euclidean plane Editorial Board Improved approximation for two-dimensional vector multiple knapsack