Popular Roommates in Simply Exponential Time

IF 0.7 4区计算机科学 Q4 COMPUTER SCIENCE, SOFTWARE ENGINEERING Algorithmica Pub Date : 2024-12-10 DOI:10.1007/s00453-024-01287-4

Telikepalli Kavitha

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Abstract

We consider the popular matching problem in a roommates instance G on n vertices, i.e., G is a graph where each vertex has a strict preference order over its neighbors. A matching M is popular if there is no matching N such that the vertices that prefer N to M outnumber those that prefer M to N. It is known that it is NP-hard to decide if G admits a popular matching or not. There is no better algorithm known for this problem than the brute force algorithm that enumerates all matchings and tests each for popularity—this could take n! time. Here we show an \(O^*(k^n)\) time algorithm for this problem, where \(k < 7.32\). We use the recent breakthrough result on the maximum number of stable matchings possible in a roommates instance to analyze our algorithm for the popular matching problem. We identify a natural (also, hard) subclass of popular matchings called truly popular matchings that are “popular fractional” and show an \(O^*(2^n)\) time algorithm for the truly popular matching problem in G. We also identify a subclass of max-size popular matchings called super-dominant matchings and show a linear time algorithm for the super-dominant roommates problem.

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受欢迎的室友在指数时间

我们考虑在n个顶点上的室友实例G中流行的匹配问题，即，G是一个图，其中每个顶点对其邻居具有严格的优先顺序。如果不存在匹配N，使得倾向于N而不是M的顶点多于倾向于M而不是N的顶点，那么匹配M就是流行的。众所周知，决定G是否允许流行匹配是np困难的。对于这个问题，没有比蛮力算法更好的算法了，蛮力算法枚举所有匹配并测试每个匹配的受欢迎程度——这可能需要n个时间！时间。这里我们展示了用于此问题的\(O^*(k^n)\)时间算法，其中\(k < 7.32\)。我们使用最近关于室友实例中可能的最大稳定匹配数的突破性结果来分析我们对流行匹配问题的算法。我们确定了流行匹配的一个自然（也是困难的）子类，称为真正流行的匹配，它是“流行分数”，并展示了g中真正流行的匹配问题的\(O^*(2^n)\)时间算法。我们还确定了最大规模的流行匹配的一个子类，称为超优势匹配，并展示了超优势室友问题的线性时间算法。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Algorithmica 工程技术-计算机：软件工程

CiteScore

2.80

自引率

9.10%

发文量

158

审稿时长

12 months

期刊介绍： Algorithmica is an international journal which publishes theoretical papers on algorithms that address problems arising in practical areas, and experimental papers of general appeal for practical importance or techniques. The development of algorithms is an integral part of computer science. The increasing complexity and scope of computer applications makes the design of efficient algorithms essential. Algorithmica covers algorithms in applied areas such as: VLSI, distributed computing, parallel processing, automated design, robotics, graphics, data base design, software tools, as well as algorithms in fundamental areas such as sorting, searching, data structures, computational geometry, and linear programming. In addition, the journal features two special sections: Application Experience, presenting findings obtained from applications of theoretical results to practical situations, and Problems, offering short papers presenting problems on selected topics of computer science.

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