{"title":"遗忘可以让你更快:一个O*(8.097k)时间的加权3集k-包装算法","authors":"M. Zehavi","doi":"10.1145/3599722","DOIUrl":null,"url":null,"abstract":"In this paper, we study the classic Weighted 3-Set k-Packing problem: given a universe U, a family \\(\\mathcal {S} \\) of subsets of size 3 of U, a weight function \\(w : {\\mathcal {S}} \\rightarrow \\mathbb {R} \\) , \\(W \\in \\mathbb {R} \\) and a parameter \\(k \\in \\mathbb {N} \\) , the objective is to decide if there is a subfamily \\({\\mathcal {S}}^{\\prime } \\subseteq {\\mathcal {S}} \\) of k disjoint sets and total weight at least W. We present a deterministic parameterized algorithm for this problem that runs in time O*(8.097k), where O* hides factors polynomial in the input size. This substantially improves upon the previously best deterministic algorithm for Weighted 3-Set k-Packing, which runs in time O*(12.155k) [SIDMA 2015], and was also the best deterministic algorithm for the unweighted version of this problem. Our algorithm is based on a novel application of the method of representative sets that might be of independent interest.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Forgetfulness Can Make You Faster: An O*(8.097k)-Time Algorithm for Weighted 3-Set k-Packing\",\"authors\":\"M. Zehavi\",\"doi\":\"10.1145/3599722\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we study the classic Weighted 3-Set k-Packing problem: given a universe U, a family \\\\(\\\\mathcal {S} \\\\) of subsets of size 3 of U, a weight function \\\\(w : {\\\\mathcal {S}} \\\\rightarrow \\\\mathbb {R} \\\\) , \\\\(W \\\\in \\\\mathbb {R} \\\\) and a parameter \\\\(k \\\\in \\\\mathbb {N} \\\\) , the objective is to decide if there is a subfamily \\\\({\\\\mathcal {S}}^{\\\\prime } \\\\subseteq {\\\\mathcal {S}} \\\\) of k disjoint sets and total weight at least W. We present a deterministic parameterized algorithm for this problem that runs in time O*(8.097k), where O* hides factors polynomial in the input size. This substantially improves upon the previously best deterministic algorithm for Weighted 3-Set k-Packing, which runs in time O*(12.155k) [SIDMA 2015], and was also the best deterministic algorithm for the unweighted version of this problem. Our algorithm is based on a novel application of the method of representative sets that might be of independent interest.\",\"PeriodicalId\":44045,\"journal\":{\"name\":\"ACM Transactions on Computation Theory\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2023-08-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACM Transactions on Computation Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/3599722\",\"RegionNum\":0,\"RegionCategory\":null,\"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":"ACM Transactions on Computation Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3599722","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
Forgetfulness Can Make You Faster: An O*(8.097k)-Time Algorithm for Weighted 3-Set k-Packing
In this paper, we study the classic Weighted 3-Set k-Packing problem: given a universe U, a family \(\mathcal {S} \) of subsets of size 3 of U, a weight function \(w : {\mathcal {S}} \rightarrow \mathbb {R} \) , \(W \in \mathbb {R} \) and a parameter \(k \in \mathbb {N} \) , the objective is to decide if there is a subfamily \({\mathcal {S}}^{\prime } \subseteq {\mathcal {S}} \) of k disjoint sets and total weight at least W. We present a deterministic parameterized algorithm for this problem that runs in time O*(8.097k), where O* hides factors polynomial in the input size. This substantially improves upon the previously best deterministic algorithm for Weighted 3-Set k-Packing, which runs in time O*(12.155k) [SIDMA 2015], and was also the best deterministic algorithm for the unweighted version of this problem. Our algorithm is based on a novel application of the method of representative sets that might be of independent interest.