Improving Envy Freeness up to Any Good Guarantees Through Rainbow Cycle Number

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-11-22 DOI:10.1287/moor.2021.0252

Bhaskar Ray Chaudhury, Jugal Garg, Kurt Mehlhorn, Ruta Mehta, Pranabendu Misra

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引用次数: 0

Abstract

We study the problem of fairly allocating a set of indivisible goods among n agents with additive valuations. Envy freeness up to any good (EFX) is arguably the most compelling fairness notion in this context. However, the existence of an EFX allocation has not been settled and is one of the most important problems in fair division. Toward resolving this question, many impressive results show the existence of its relaxations. In particular, it is known that 0.618-EFX allocations exist and that EFX allocation exists if we do not allocate at most (n-1) goods. Reducing the number of unallocated goods has emerged as a systematic way to tackle the main question. For example, follow-up works on three- and four-agents cases, respectively, allocated two more unallocated goods through an involved procedure. In this paper, we study the general case and achieve sublinear numbers of unallocated goods. Through a new approach, we show that for every [Formula: see text], there always exists a [Formula: see text]-EFX allocation with sublinear number of unallocated goods and high Nash welfare. For this, we reduce the EFX problem to a novel problem in extremal graph theory. We define the notion of rainbow cycle number [Formula: see text] in directed graphs. For all [Formula: see text] is the largest k such that there exists a k-partite graph [Formula: see text], in which each part has at most d vertices (i.e., [Formula: see text] for all [Formula: see text]); for any two parts Vi and Vj, each vertex in Vi has an incoming edge from some vertex in Vj and vice versa; and there exists no cycle in G that contains at most one vertex from each part. We show that any upper bound on [Formula: see text] directly translates to a sublinear bound on the number of unallocated goods. We establish a polynomial upper bound on [Formula: see text], yielding our main result. Furthermore, our approach is constructive, which also gives a polynomial-time algorithm for finding such an allocation.Funding: J. Garg was supported by the Directorate for Computer and Information Science and Engineering [Grant CCF-1942321]. R. Mehta was supported by the Directorate for Computer and Information Science and Engineering [Grant CCF-1750436].

查看原文本刊更多论文

通过彩虹循环数提高嫉妒自由度

研究了具有可加性定价的n个agent之间的一组不可分商品的公平分配问题。在这种情况下，嫉妒自由(EFX)可以说是最令人信服的公平概念。但是，EFX分配的存在性一直没有得到解决，这是公平分配中最重要的问题之一。为了解决这个问题，许多令人印象深刻的结果表明它的松弛存在。特别是，已知存在0.618-EFX分配，并且如果我们不分配最多(n-1)个货物，则存在EFX分配。减少未分配货物的数量已成为解决主要问题的一种系统方法。例如，分别针对三个和四个代理案件的后续工作，通过涉及的程序分配了两个未分配的货物。在本文中，我们研究了一般情况，并实现了未分配商品的次线性数量。通过一种新的方法，我们证明了对于每一个[公式:见文]，总是存在一个[公式:见文]-EFX分配，其未分配商品数量是次线性的，且纳什福利很高。为此，我们将EFX问题简化为极值图论中的一个新问题。我们在有向图中定义了彩虹圈数的概念[公式:见文]。对于所有[公式:见文]是最大的k，使得存在一个k部图[公式:见文]，其中每个部分最多有d个顶点(即[公式:见文]);对于任意两个部分Vi和Vj, Vi中的每个顶点都有一条从Vj的某个顶点来的边，反之亦然;并且在G中不存在每个部分最多包含一个顶点的循环。我们证明了[公式:见正文]的任何上界直接转化为未分配货物数量的次线性边界。我们在[公式:见文本]上建立了一个多项式上界，得到了我们的主要结果。此外，我们的方法是建设性的，它也给出了一个多项式时间算法来寻找这样的分配。资助:J. Garg由计算机和信息科学与工程理事会资助[Grant CCF-1942321]。R. Mehta由计算机与信息科学与工程理事会[Grant CCF-1750436]资助。

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

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.