SIAM J. Discret. Math.最新文献_第7页

On the complexity of packing rainbow spanning trees 关于填充彩虹生成树的复杂性

SIAM J. Discret. Math. Pub Date : 2022-06-23 DOI: 10.48550/arXiv.2206.11924

Krist'of B'erczi, Gergely Cs'aji, Tam'as Kir'aly

引用次数: 2

Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs r边连通r正则图的两两不相交完美匹配

SIAM J. Discret. Math. Pub Date : 2022-06-22 DOI: 10.1137/22M1500654

Yulai Ma, D. Mattiolo, E. Steffen, Isaak H. Wolf

引用次数: 2

Euclidean Steiner Spanners: Light and Sparse 欧几里得斯坦纳扳手:轻而稀疏

SIAM J. Discret. Math. Pub Date : 2022-06-20 DOI: 10.48550/arXiv.2206.09648

S. Bhore, Csaba D. Tóth

{"title":"Euclidean Steiner Spanners: Light and Sparse","authors":"S. Bhore, Csaba D. Tóth","doi":"10.48550/arXiv.2206.09648","DOIUrl":"https://doi.org/10.48550/arXiv.2206.09648","url":null,"abstract":"Lightness and sparsity are two natural parameters for Euclidean $(1+varepsilon)$-spanners. Classical results show that, when the dimension $din mathbb{N}$ and $varepsilon>0$ are constant, every set $S$ of $n$ points in $d$-space admits an $(1+varepsilon)$-spanners with $O(n)$ edges and weight proportional to that of the Euclidean MST of $S$. In a recent breakthrough, Le and Solomon (2019) established the precise dependencies on $varepsilon>0$, for constant $din mathbb{N}$, of the minimum lightness and sparsity of $(1+varepsilon)$-spanners, and observed that Steiner points can substantially improve the lightness and sparsity of a $(1+varepsilon)$-spanner. They gave upper bounds of $tilde{O}(varepsilon^{-(d+1)/2})$ for the minimum lightness in dimensions $dgeq 3$, and $tilde{O}(varepsilon^{-(d-1)/2})$ for the minimum sparsity in $d$-space for all $dgeq 1$. In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner $(1+varepsilon)$-spanners. We establish lower bounds of $Omega(varepsilon^{-d/2})$ for the lightness and $Omega(varepsilon^{-(d-1)/2})$ for the sparsity of such spanners in Euclidean $d$-space for all constant $dgeq 2$. Our lower bound constructions generalize previous constructions by Le and Solomon, but the analysis substantially simplifies previous work, using new geometric insight, focusing on the directions of edges. Next, we show that for every finite set of points in the plane and every $varepsilonin (0,1]$, there exists a Euclidean Steiner $(1+varepsilon)$-spanner of lightness $O(varepsilon^{-1})$; this matches the lower bound for $d=2$. We generalize the notion of shallow light trees, which may be of independent interest, and use directional spanners and a modified window partitioning scheme to achieve a tight weight analysis.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76171253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 3

On approximating the rank of graph divisors 关于图因子秩的逼近

SIAM J. Discret. Math. Pub Date : 2022-06-20 DOI: 10.48550/arXiv.2206.09662

Krist'of B'erczi, H. P. Hoang, Lilla T'othm'er'esz

{"title":"On approximating the rank of graph divisors","authors":"Krist'of B'erczi, H. P. Hoang, Lilla T'othm'er'esz","doi":"10.48550/arXiv.2206.09662","DOIUrl":"https://doi.org/10.48550/arXiv.2206.09662","url":null,"abstract":"Baker and Norine initiated the study of graph divisors as a graph-theoretic analogue of the Riemann-Roch theory for Riemann surfaces. One of the key concepts of graph divisor theory is the {it rank} of a divisor on a graph. The importance of the rank is well illustrated by Baker's {it Specialization lemma}, stating that the dimension of a linear system can only go up under specialization from curves to graphs, leading to a fruitful interaction between divisors on graphs and curves. Due to its decisive role, determining the rank is a central problem in graph divisor theory. Kiss and T'othm'eresz reformulated the problem using chip-firing games, and showed that computing the rank of a divisor on a graph is NP-hard via reduction from the Minimum Feedback Arc Set problem. In this paper, we strengthen their result by establishing a connection between chip-firing games and the Minimum Target Set Selection problem. As a corollary, we show that the rank is difficult to approximate to within a factor of $O(2^{log^{1-varepsilon}n})$ for any $varepsilon>0$ unless $P=NP$. Furthermore, assuming the Planted Dense Subgraph Conjecture, the rank is difficult to approximate to within a factor of $O(n^{1/4-varepsilon})$ for any $varepsilon>0$.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82086684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Extremal graph realizations and graph Laplacian eigenvalues 极值图实现和图拉普拉斯特征值

SIAM J. Discret. Math. Pub Date : 2022-06-20 DOI: 10.48550/arXiv.2206.10010

B. Osting

引用次数: 0

Distinct Angles in General Position 一般位置上的不同角度

SIAM J. Discret. Math. Pub Date : 2022-06-09 DOI: 10.48550/arXiv.2206.04367

Henry Fleischmann, S. Konyagin, Steven J. Miller, E. Palsson, Ethan Pesikoff, Charles Wolf

引用次数: 4

Disjoint Cycles with Length Constraints in Digraphs of Large Connectivity or Large Minimum Degree 大连通性或大最小度有向图中具有长度约束的不相交环

SIAM J. Discret. Math. Pub Date : 2022-06-01 DOI: 10.1137/20m1382398

Raphael Steiner