{"title":"谢林隔离中福利保障的参数化复杂性","authors":"","doi":"10.1016/j.tcs.2024.114783","DOIUrl":null,"url":null,"abstract":"<div><p>Schelling's model considers <em>k</em> types of agents each of whom needs to select a vertex on an undirected graph and prefers neighboring agents of the same type. We are motivated by a recent line of work that studies solutions which are optimal with respect to notions related to the welfare of the agents. We explore the parameterized complexity of computing such solutions. We focus on the well-studied notions of social welfare (WO) and Pareto optimality (PO), alongside the recently proposed notions of group-welfare optimality (GWO) and utility-vector optimality (UVO), both of which lie between WO and PO. Firstly, we focus on the fundamental case where <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span> with <em>r</em> red agents and <em>b</em> blue agents. We show that all solution-notions we consider are intractable even when <span><math><mi>b</mi><mo>=</mo><mn>1</mn></math></span> and that they do not admit any <span>FPT</span> algorithm when parameterized by <em>r</em> and <em>b</em>, unless <span><math><mtext>FPT</mtext><mo>=</mo><mtext>W</mtext><mo>[</mo><mn>1</mn><mo>]</mo></math></span>. In addition, we show that WO and GWO remain intractable even on cubic graphs. We complement these negative results with an <span>FPT</span> algorithm parameterized by <span><math><mi>r</mi><mo>,</mo><mi>b</mi></math></span> and the maximum degree of the graph. For the general case with <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> types of agents, we prove that for any of the notions we consider the problem remains hard when parameterized by <em>k</em> for a large family of graphs that includes trees. We accompany these negative results with an <span>XP</span> algorithm parameterized by <em>k</em> and the treewidth of the graph.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0304397524004006/pdfft?md5=7297943aa04e1b17616df20ad757dfa3&pid=1-s2.0-S0304397524004006-main.pdf","citationCount":"0","resultStr":"{\"title\":\"The parameterized complexity of welfare guarantees in Schelling segregation\",\"authors\":\"\",\"doi\":\"10.1016/j.tcs.2024.114783\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Schelling's model considers <em>k</em> types of agents each of whom needs to select a vertex on an undirected graph and prefers neighboring agents of the same type. We are motivated by a recent line of work that studies solutions which are optimal with respect to notions related to the welfare of the agents. We explore the parameterized complexity of computing such solutions. We focus on the well-studied notions of social welfare (WO) and Pareto optimality (PO), alongside the recently proposed notions of group-welfare optimality (GWO) and utility-vector optimality (UVO), both of which lie between WO and PO. Firstly, we focus on the fundamental case where <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span> with <em>r</em> red agents and <em>b</em> blue agents. We show that all solution-notions we consider are intractable even when <span><math><mi>b</mi><mo>=</mo><mn>1</mn></math></span> and that they do not admit any <span>FPT</span> algorithm when parameterized by <em>r</em> and <em>b</em>, unless <span><math><mtext>FPT</mtext><mo>=</mo><mtext>W</mtext><mo>[</mo><mn>1</mn><mo>]</mo></math></span>. In addition, we show that WO and GWO remain intractable even on cubic graphs. We complement these negative results with an <span>FPT</span> algorithm parameterized by <span><math><mi>r</mi><mo>,</mo><mi>b</mi></math></span> and the maximum degree of the graph. For the general case with <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> types of agents, we prove that for any of the notions we consider the problem remains hard when parameterized by <em>k</em> for a large family of graphs that includes trees. We accompany these negative results with an <span>XP</span> algorithm parameterized by <em>k</em> and the treewidth of the graph.</p></div>\",\"PeriodicalId\":49438,\"journal\":{\"name\":\"Theoretical Computer Science\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-08-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0304397524004006/pdfft?md5=7297943aa04e1b17616df20ad757dfa3&pid=1-s2.0-S0304397524004006-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Theoretical Computer Science\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0304397524004006\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theoretical Computer Science","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304397524004006","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
摘要
谢林模型考虑了 k 种代理,每种代理都需要在无向图上选择一个顶点,并优先选择相邻的同类型代理。我们的灵感来自于最近的一项工作,即研究与代理人福利相关的最优解。我们探讨了计算此类解决方案的参数化复杂性。我们将重点放在已被广泛研究的社会福利(WO)和帕累托最优(PO)这两个概念上,以及最近提出的群体福利最优(GWO)和效用向量最优(UVO)这两个介于 WO 和 PO 之间的概念上。首先,我们重点讨论 k=2 的基本情况,即 r 个红色代理和 b 个蓝色代理。我们证明,即使当 b=1 时,我们所考虑的所有解-符号都是难以处理的,而且当以 r 和 b 为参数时,除非 FPT=W[1],否则它们不接受任何 FPT 算法。此外,我们还证明,即使在立方图上,WO 和 GWO 也仍然难以解决。我们用一种以 r、b 和图的最大度为参数的 FPT 算法来补充这些负面结果。对于有 k≥2 种代理的一般情况,我们证明,对于我们考虑的任何概念,当以 k 为参数时,对于包括树在内的一大类图,问题仍然很难解决。我们还提出了一种以 k 和图的树宽为参数的 XP 算法,与这些否定结果相辅相成。
The parameterized complexity of welfare guarantees in Schelling segregation
Schelling's model considers k types of agents each of whom needs to select a vertex on an undirected graph and prefers neighboring agents of the same type. We are motivated by a recent line of work that studies solutions which are optimal with respect to notions related to the welfare of the agents. We explore the parameterized complexity of computing such solutions. We focus on the well-studied notions of social welfare (WO) and Pareto optimality (PO), alongside the recently proposed notions of group-welfare optimality (GWO) and utility-vector optimality (UVO), both of which lie between WO and PO. Firstly, we focus on the fundamental case where with r red agents and b blue agents. We show that all solution-notions we consider are intractable even when and that they do not admit any FPT algorithm when parameterized by r and b, unless . In addition, we show that WO and GWO remain intractable even on cubic graphs. We complement these negative results with an FPT algorithm parameterized by and the maximum degree of the graph. For the general case with types of agents, we prove that for any of the notions we consider the problem remains hard when parameterized by k for a large family of graphs that includes trees. We accompany these negative results with an XP algorithm parameterized by k and the treewidth of the graph.
期刊介绍:
Theoretical Computer Science is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. Its aim is to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. All papers introducing or studying mathematical, logic and formal concepts and methods are welcome, provided that their motivation is clearly drawn from the field of computing.