Pub Date : 2022-06-23DOI: 10.48550/arXiv.2206.11924
Krist'of B'erczi, Gergely Cs'aji, Tam'as Kir'aly
One of the most important questions in matroid optimization is to find disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures that can be formulated as special cases. B'erczi and Schwarcz showed that the problem is hard in general, therefore identifying the borderline between tractable and intractable instances is of interest. In the present paper, we study the special case when one of the matroids is a partition matroid while the other one is a graphic matroid. This setting is equivalent to the problem of packing rainbow spanning trees, an extension of the problem of packing arborescences in directed graphs which was answered by Edmonds' seminal result on disjoint arborescences. We complement his result by showing that it is NP-complete to decide whether an edge-colored graph contains two disjoint rainbow spanning trees. Our complexity result holds even for the very special case when the graph is the union of two spanning trees and each color class contains exactly two edges. As a corollary, we give a negative answer to a question on the decomposition of oriented $k$-partition-connected digraphs.
{"title":"On the complexity of packing rainbow spanning trees","authors":"Krist'of B'erczi, Gergely Cs'aji, Tam'as Kir'aly","doi":"10.48550/arXiv.2206.11924","DOIUrl":"https://doi.org/10.48550/arXiv.2206.11924","url":null,"abstract":"One of the most important questions in matroid optimization is to find disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures that can be formulated as special cases. B'erczi and Schwarcz showed that the problem is hard in general, therefore identifying the borderline between tractable and intractable instances is of interest. In the present paper, we study the special case when one of the matroids is a partition matroid while the other one is a graphic matroid. This setting is equivalent to the problem of packing rainbow spanning trees, an extension of the problem of packing arborescences in directed graphs which was answered by Edmonds' seminal result on disjoint arborescences. We complement his result by showing that it is NP-complete to decide whether an edge-colored graph contains two disjoint rainbow spanning trees. Our complexity result holds even for the very special case when the graph is the union of two spanning trees and each color class contains exactly two edges. As a corollary, we give a negative answer to a question on the decomposition of oriented $k$-partition-connected digraphs.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"1 1","pages":"113297"},"PeriodicalIF":0.0,"publicationDate":"2022-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83905263","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}
Thomassen [Problem 1 in Factorizing regular graphs, J. Combin. Theory Ser. B, 141 (2020), 343-351] asked whether every $r$-edge-connected $r$-regular graph of even order has $r-2$ pairwise disjoint perfect matchings. We show that this is not the case if $r equiv 2 text{ mod } 4$. Together with a recent result of Mattiolo and Steffen [Highly edge-connected regular graphs without large factorizable subgraphs, J. Graph Theory, 99 (2022), 107-116] this solves Thomassen's problem for all even $r$. It turns out that our methods are limited to the even case of Thomassen's problem. We then prove some equivalences of statements on pairwise disjoint perfect matchings in highly edge-connected regular graphs, where the perfect matchings contain or avoid fixed sets of edges. Based on these results we relate statements on pairwise disjoint perfect matchings of 5-edge-connected 5-regular graphs to well-known conjectures for cubic graphs, such as the Fan-Raspaud Conjecture, the Berge-Fulkerson Conjecture and the $5$-Cycle Double Cover Conjecture.
{"title":"Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs","authors":"Yulai Ma, D. Mattiolo, E. Steffen, Isaak H. Wolf","doi":"10.1137/22M1500654","DOIUrl":"https://doi.org/10.1137/22M1500654","url":null,"abstract":"Thomassen [Problem 1 in Factorizing regular graphs, J. Combin. Theory Ser. B, 141 (2020), 343-351] asked whether every $r$-edge-connected $r$-regular graph of even order has $r-2$ pairwise disjoint perfect matchings. We show that this is not the case if $r equiv 2 text{ mod } 4$. Together with a recent result of Mattiolo and Steffen [Highly edge-connected regular graphs without large factorizable subgraphs, J. Graph Theory, 99 (2022), 107-116] this solves Thomassen's problem for all even $r$. It turns out that our methods are limited to the even case of Thomassen's problem. We then prove some equivalences of statements on pairwise disjoint perfect matchings in highly edge-connected regular graphs, where the perfect matchings contain or avoid fixed sets of edges. Based on these results we relate statements on pairwise disjoint perfect matchings of 5-edge-connected 5-regular graphs to well-known conjectures for cubic graphs, such as the Fan-Raspaud Conjecture, the Berge-Fulkerson Conjecture and the $5$-Cycle Double Cover Conjecture.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"1 1","pages":"1548-1565"},"PeriodicalIF":0.0,"publicationDate":"2022-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79813609","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}
Pub Date : 2022-06-20DOI: 10.48550/arXiv.2206.09648
S. Bhore, Csaba D. Tóth
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.
{"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":"9 1","pages":"2411-2444"},"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}
Pub Date : 2022-06-20DOI: 10.48550/arXiv.2206.09662
Krist'of B'erczi, H. P. Hoang, Lilla T'othm'er'esz
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 $varepsilon>0$.
{"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":"47 1","pages":"113528"},"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}
Pub Date : 2022-06-20DOI: 10.48550/arXiv.2206.10010
B. Osting
For a regular polyhedron (or polygon) centered at the origin, the coordinates of the vertices are eigenvectors of the graph Laplacian for the skeleton of that polyhedron (or polygon) associated with the first (non-trivial) eigenvalue. In this paper, we generalize this relationship. For a given graph, we study the eigenvalue optimization problem of maximizing the first (non-trivial) eigenvalue of the graph Laplacian over non-negative edge weights. We show that the spectral realization of the graph using the eigenvectors corresponding to the solution of this problem, under certain assumptions, is a centered, unit-distance graph realization that has maximal total variance. This result gives a new method for generating unit-distance graph realizations and is based on convex duality. A drawback of this method is that the dimension of the realization is given by the multiplicity of the extremal eigenvalue, which is typically unknown prior to solving the eigenvalue optimization problem. Our results are illustrated with a number of examples.
{"title":"Extremal graph realizations and graph Laplacian eigenvalues","authors":"B. Osting","doi":"10.48550/arXiv.2206.10010","DOIUrl":"https://doi.org/10.48550/arXiv.2206.10010","url":null,"abstract":"For a regular polyhedron (or polygon) centered at the origin, the coordinates of the vertices are eigenvectors of the graph Laplacian for the skeleton of that polyhedron (or polygon) associated with the first (non-trivial) eigenvalue. In this paper, we generalize this relationship. For a given graph, we study the eigenvalue optimization problem of maximizing the first (non-trivial) eigenvalue of the graph Laplacian over non-negative edge weights. We show that the spectral realization of the graph using the eigenvectors corresponding to the solution of this problem, under certain assumptions, is a centered, unit-distance graph realization that has maximal total variance. This result gives a new method for generating unit-distance graph realizations and is based on convex duality. A drawback of this method is that the dimension of the realization is given by the multiplicity of the extremal eigenvalue, which is typically unknown prior to solving the eigenvalue optimization problem. Our results are illustrated with a number of examples.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"97 1","pages":"1630-1644"},"PeriodicalIF":0.0,"publicationDate":"2022-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87707676","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}
Pub Date : 2022-06-09DOI: 10.48550/arXiv.2206.04367
Henry Fleischmann, S. Konyagin, Steven J. Miller, E. Palsson, Ethan Pesikoff, Charles Wolf
The ErdH{o}s distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is ErdH{o}s' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear points in the plane. Recent work has introduced bounds on a wide array of variants of this problem, inspired by similar variants in the distance setting. In this short note, we improve the best known upper bound for the minimum number of distinct angles formed by $n$ points in general position from $O(n^{log_2(7)})$ to $O(n^2)$. Before this work, similar bounds relied on projections onto a generic plane from higher dimensional space. In this paper, we employ the geometric properties of a logarithmic spiral, sidestepping the need for a projection. We also apply this configuration to reduce the upper bound on the largest integer such that any set of $n$ points in general position has a subset of that size with all distinct angles. This bound is decreased from $O(n^{log_2(7)/3})$ to $O(n^{1/2})$.
{"title":"Distinct Angles in General Position","authors":"Henry Fleischmann, S. Konyagin, Steven J. Miller, E. Palsson, Ethan Pesikoff, Charles Wolf","doi":"10.48550/arXiv.2206.04367","DOIUrl":"https://doi.org/10.48550/arXiv.2206.04367","url":null,"abstract":"The ErdH{o}s distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is ErdH{o}s' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear points in the plane. Recent work has introduced bounds on a wide array of variants of this problem, inspired by similar variants in the distance setting. In this short note, we improve the best known upper bound for the minimum number of distinct angles formed by $n$ points in general position from $O(n^{log_2(7)})$ to $O(n^2)$. Before this work, similar bounds relied on projections onto a generic plane from higher dimensional space. In this paper, we employ the geometric properties of a logarithmic spiral, sidestepping the need for a projection. We also apply this configuration to reduce the upper bound on the largest integer such that any set of $n$ points in general position has a subset of that size with all distinct angles. This bound is decreased from $O(n^{log_2(7)/3})$ to $O(n^{1/2})$.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"48 1","pages":"113283"},"PeriodicalIF":0.0,"publicationDate":"2022-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89620921","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}
{"title":"Disjoint Cycles with Length Constraints in Digraphs of Large Connectivity or Large Minimum Degree","authors":"Raphael Steiner","doi":"10.1137/20m1382398","DOIUrl":"https://doi.org/10.1137/20m1382398","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"329 1","pages":"1343-1362"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76148693","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}
{"title":"Perfect Matchings in the Semirandom Graph Process","authors":"Pu Gao, Calum MacRury, P. Prałat","doi":"10.1137/21m1446939","DOIUrl":"https://doi.org/10.1137/21m1446939","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"25 1","pages":"1274-1290"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74260105","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}
Pub Date : 2022-05-23DOI: 10.4230/LIPIcs.STACS.2020.13
Dan Bergren, E. Eiben, R. Ganian, Iyad A. Kanj
We study the problem of covering a set of segments on a line with the minimum number of unit-length intervals, where an interval covers a segment if at least one of the two endpoints of the segment falls in the unit interval. We also study several variants of this problem. We show that the restrictions of the aforementioned problems to the set of instances in which all the segments have the same length are NP-hard. This result implies several NP-hardness results in the literature for variants and generalizations of the problems under consideration. We then study the parameterized complexity of the aforementioned problems. We provide tight results for most of them by showing that they are fixed-parameter tractable for the restrictions in which all the segments have the same length, and are W[1]-complete otherwise. 2012 ACM Subject Classification Theory of computation → Parameterized complexity and exact algorithms; Theory of computation → Computational geometry
{"title":"On Covering Segments with Unit Intervals","authors":"Dan Bergren, E. Eiben, R. Ganian, Iyad A. Kanj","doi":"10.4230/LIPIcs.STACS.2020.13","DOIUrl":"https://doi.org/10.4230/LIPIcs.STACS.2020.13","url":null,"abstract":"We study the problem of covering a set of segments on a line with the minimum number of unit-length intervals, where an interval covers a segment if at least one of the two endpoints of the segment falls in the unit interval. We also study several variants of this problem. We show that the restrictions of the aforementioned problems to the set of instances in which all the segments have the same length are NP-hard. This result implies several NP-hardness results in the literature for variants and generalizations of the problems under consideration. We then study the parameterized complexity of the aforementioned problems. We provide tight results for most of them by showing that they are fixed-parameter tractable for the restrictions in which all the segments have the same length, and are W[1]-complete otherwise. 2012 ACM Subject Classification Theory of computation → Parameterized complexity and exact algorithms; Theory of computation → Computational geometry","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"31 1","pages":"1200-1230"},"PeriodicalIF":0.0,"publicationDate":"2022-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73743989","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}
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