The Steiner tree problem is one of the most prominent problems in network design. Given an edge-weighted undirected graph and a subset of the vertices, called terminals, the task is to compute a minimum-weight tree containing all terminals (and possibly further vertices). The best-known approximation algorithms for Steiner tree involve enumeration of a (polynomial but) very large number of candidate components and are therefore slow in practice. A promising ingredient for the design of fast and accurate approximation algorithms for Steiner tree is the bidirected cut relaxation (BCR): bidirect all edges, choose an arbitrary terminal as a root, and enforce that each cut containing some terminal but not the root has one unit of fractional edges leaving it. BCR is known to be integral in the spanning tree case [Edmonds'67], i.e., when all the vertices are terminals. For general instances, however, it was not even known whether the integrality gap of BCR is better than the integrality gap of the natural undirected relaxation, which is exactly 2. We resolve this question by proving an upper bound of 1.9988 on the integrality gap of BCR.
{"title":"The Bidirected Cut Relaxation for Steiner Tree has Integrality Gap Smaller than 2","authors":"Jarosław Byrka, Fabrizio Grandoni, Vera Traub","doi":"arxiv-2407.19905","DOIUrl":"https://doi.org/arxiv-2407.19905","url":null,"abstract":"The Steiner tree problem is one of the most prominent problems in network\u0000design. Given an edge-weighted undirected graph and a subset of the vertices,\u0000called terminals, the task is to compute a minimum-weight tree containing all\u0000terminals (and possibly further vertices). The best-known approximation\u0000algorithms for Steiner tree involve enumeration of a (polynomial but) very\u0000large number of candidate components and are therefore slow in practice. A promising ingredient for the design of fast and accurate approximation\u0000algorithms for Steiner tree is the bidirected cut relaxation (BCR): bidirect\u0000all edges, choose an arbitrary terminal as a root, and enforce that each cut\u0000containing some terminal but not the root has one unit of fractional edges\u0000leaving it. BCR is known to be integral in the spanning tree case [Edmonds'67],\u0000i.e., when all the vertices are terminals. For general instances, however, it\u0000was not even known whether the integrality gap of BCR is better than the\u0000integrality gap of the natural undirected relaxation, which is exactly 2. We\u0000resolve this question by proving an upper bound of 1.9988 on the integrality\u0000gap of BCR.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"69 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141871555","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}
Nicolás Álvarez, Verónica Becher, Martín Mereb, Ivo Pajor, Carlos Miguel Soto
The discrepancy of a binary string is the maximum (absolute) difference between the number of ones and the number of zeroes over all possible substrings of the given binary string. In this note we determine the minimal discrepancy that a binary de Bruijn sequence of order $n$ can achieve, which is $n$. This was an open problem until now. We give an algorithm that constructs a binary de Bruijn sequence with minimal discrepancy. A slight modification of this algorithm deals with arbitrary alphabets and yields de Bruijn sequences of order $n$ with discrepancy at most $1$ above the trivial lower bound $n$.
{"title":"De Bruijn Sequences with Minimum Discrepancy","authors":"Nicolás Álvarez, Verónica Becher, Martín Mereb, Ivo Pajor, Carlos Miguel Soto","doi":"arxiv-2407.17367","DOIUrl":"https://doi.org/arxiv-2407.17367","url":null,"abstract":"The discrepancy of a binary string is the maximum (absolute) difference\u0000between the number of ones and the number of zeroes over all possible\u0000substrings of the given binary string. In this note we determine the minimal\u0000discrepancy that a binary de Bruijn sequence of order $n$ can achieve, which is\u0000$n$. This was an open problem until now. We give an algorithm that constructs a\u0000binary de Bruijn sequence with minimal discrepancy. A slight modification of\u0000this algorithm deals with arbitrary alphabets and yields de Bruijn sequences of\u0000order $n$ with discrepancy at most $1$ above the trivial lower bound $n$.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771775","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}
In the minimum spanning tree (MST) interdiction problem, we are given a graph $G=(V,E)$ with edge weights, and want to find some $Xsubseteq E$ satisfying a knapsack constraint such that the MST weight in $(V,Esetminus X)$ is maximized. Since MSTs of $G$ are the minimum weight bases in the graphic matroid of $G$, this problem is a special case of matroid interdiction on a matroid $M=(E,mathcal{I})$, in which the objective is instead to maximize the minimum weight of a basis of $M$ which is disjoint from $X$. By reduction from 0-1 knapsack, matroid interdiction is NP-complete, even for uniform matroids. We develop a new exact algorithm to solve the matroid interdiction problem. One of the key components of our algorithm is a dynamic programming upper bound which only requires that a simpler discrete derivative problem can be calculated/approximated for the given matroid. Our exact algorithm then uses this bound within a custom branch-and-bound algorithm. For different matroids, we show how this discrete derivative can be calculated/approximated. In particular, for partition matroids, this yields a pseudopolynomial time algorithm. For graphic matroids, an approximation can be obtained by solving a sequence of minimum cut problems, which we apply to the MST interdiction problem. The running time of our algorithm is asymptotically faster than the best known MST interdiction algorithm, up to polylog factors. Furthermore, our algorithm achieves state-of-the-art computational performance: we solved all available instances from the literature, and in many cases reduced the best running time from hours to seconds.
{"title":"Interdiction of minimum spanning trees and other matroid bases","authors":"Noah Weninger, Ricardo Fukasawa","doi":"arxiv-2407.14906","DOIUrl":"https://doi.org/arxiv-2407.14906","url":null,"abstract":"In the minimum spanning tree (MST) interdiction problem, we are given a graph\u0000$G=(V,E)$ with edge weights, and want to find some $Xsubseteq E$ satisfying a\u0000knapsack constraint such that the MST weight in $(V,Esetminus X)$ is\u0000maximized. Since MSTs of $G$ are the minimum weight bases in the graphic\u0000matroid of $G$, this problem is a special case of matroid interdiction on a\u0000matroid $M=(E,mathcal{I})$, in which the objective is instead to maximize the\u0000minimum weight of a basis of $M$ which is disjoint from $X$. By reduction from\u00000-1 knapsack, matroid interdiction is NP-complete, even for uniform matroids. We develop a new exact algorithm to solve the matroid interdiction problem.\u0000One of the key components of our algorithm is a dynamic programming upper bound\u0000which only requires that a simpler discrete derivative problem can be\u0000calculated/approximated for the given matroid. Our exact algorithm then uses\u0000this bound within a custom branch-and-bound algorithm. For different matroids,\u0000we show how this discrete derivative can be calculated/approximated. In\u0000particular, for partition matroids, this yields a pseudopolynomial time\u0000algorithm. For graphic matroids, an approximation can be obtained by solving a\u0000sequence of minimum cut problems, which we apply to the MST interdiction\u0000problem. The running time of our algorithm is asymptotically faster than the\u0000best known MST interdiction algorithm, up to polylog factors. Furthermore, our\u0000algorithm achieves state-of-the-art computational performance: we solved all\u0000available instances from the literature, and in many cases reduced the best\u0000running time from hours to seconds.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771810","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}
Valérie Gillot, Philippe Langevin, Alexandr Polujan
In the BFA 2023 conference paper, A. Polujan, L. Mariot and S. Picek exhibited the first example of a non-normal but weakly normal bent function in dimension 8. In this note, we present numerical approaches based on the classification of Boolean spaces to explore in detail the normality of bent functions of 8 variables and we complete S. Dubuc s results for dimensions less or equal to 7. Based on our investigations, we show that all bent functions in 8 variables are normal or weakly normal. Finally, we conjecture that more generally all Boolean functions of degree at most 4 in 8 variables are normal or weakly normal.
{"title":"On the normality of Boolean quadrics","authors":"Valérie Gillot, Philippe Langevin, Alexandr Polujan","doi":"arxiv-2407.14038","DOIUrl":"https://doi.org/arxiv-2407.14038","url":null,"abstract":"In the BFA 2023 conference paper, A. Polujan, L. Mariot and S. Picek\u0000exhibited the first example of a non-normal but weakly normal bent function in\u0000dimension 8. In this note, we present numerical approaches based on the\u0000classification of Boolean spaces to explore in detail the normality of bent\u0000functions of 8 variables and we complete S. Dubuc s results for dimensions less\u0000or equal to 7. Based on our investigations, we show that all bent functions in\u00008 variables are normal or weakly normal. Finally, we conjecture that more\u0000generally all Boolean functions of degree at most 4 in 8 variables are normal\u0000or weakly normal.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741050","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}
The Storage Location Assignment Problem (SLAP) and the Picker Routing Problem (PRP) have received significant attention in the literature due to their pivotal role in the performance of the Order Picking (OP) activity, the most resource-intensive process of warehousing logistics. The two problems are traditionally considered at different decision-making levels: tactical for the SLAP, and operational for the PRP. However, this paradigm has been challenged by the emergence of modern practices in e-commerce warehouses, where storage decisions are more dynamic and are made at an operational level, making the integration of the SLAP and PRP pertinent to consider. Despite its practical significance, the joint optimization of both operations, called the Storage Location Assignment and Picker Routing Problem (SLAPRP), has received limited attention. Scholars have investigated several variants of the SLAPRP, including different warehouse layouts and routing policies. Nevertheless, the available computational results suggest that each variant requires an ad hoc formulation. Moreover, achieving a complete integration of the two problems, where the routing is solved optimally, remains out of reach for commercial solvers. In this paper, we propose an exact solution framework that addresses a broad class of variants of the SLAPRP, including all the previously existing ones. This paper proposes a Branch-Cut-and-Price framework based on a novel formulation with an exponential number of variables, which is strengthened with a novel family of non-robust valid inequalities. We have developed an ad-hoc branching scheme to break symmetries and maintain the size of the enumeration tree manageable. Computational experiments show that our framework can effectively solve medium-sized instances of several SLAPRP variants and outperforms the state-of-the-art methods from the literature.
{"title":"The Storage Location Assignment and Picker Routing Problem: A Generic Branch-Cut-and-Price Algorithm","authors":"Thibault Prunet, Nabil Absi, Diego Cattaruzza","doi":"arxiv-2407.13570","DOIUrl":"https://doi.org/arxiv-2407.13570","url":null,"abstract":"The Storage Location Assignment Problem (SLAP) and the Picker Routing Problem\u0000(PRP) have received significant attention in the literature due to their\u0000pivotal role in the performance of the Order Picking (OP) activity, the most\u0000resource-intensive process of warehousing logistics. The two problems are\u0000traditionally considered at different decision-making levels: tactical for the\u0000SLAP, and operational for the PRP. However, this paradigm has been challenged\u0000by the emergence of modern practices in e-commerce warehouses, where storage\u0000decisions are more dynamic and are made at an operational level, making the\u0000integration of the SLAP and PRP pertinent to consider. Despite its practical\u0000significance, the joint optimization of both operations, called the Storage\u0000Location Assignment and Picker Routing Problem (SLAPRP), has received limited\u0000attention. Scholars have investigated several variants of the SLAPRP, including\u0000different warehouse layouts and routing policies. Nevertheless, the available\u0000computational results suggest that each variant requires an ad hoc formulation.\u0000Moreover, achieving a complete integration of the two problems, where the\u0000routing is solved optimally, remains out of reach for commercial solvers. In this paper, we propose an exact solution framework that addresses a broad\u0000class of variants of the SLAPRP, including all the previously existing ones.\u0000This paper proposes a Branch-Cut-and-Price framework based on a novel\u0000formulation with an exponential number of variables, which is strengthened with\u0000a novel family of non-robust valid inequalities. We have developed an ad-hoc\u0000branching scheme to break symmetries and maintain the size of the enumeration\u0000tree manageable. Computational experiments show that our framework can\u0000effectively solve medium-sized instances of several SLAPRP variants and\u0000outperforms the state-of-the-art methods from the literature.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741051","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}
Arnold Filtser, Tobias Friedrich, Davis Issac, Nikhil Kumar, Hung Le, Nadym Mallek, Ziena Zeif
A $(beta,delta,Delta)$-padded decomposition of an edge-weighted graph $G = (V,E,w)$ is a stochastic decomposition into clusters of diameter at most $Delta$ such that for every vertex $vin V$, the probability that $rm{ball}_G(v,gammaDelta)$ is entirely contained in the cluster containing $v$ is at least $e^{-betagamma}$ for every $gamma in [0,delta]$. Padded decompositions have been studied for decades and have found numerous applications, including metric embedding, multicommodity flow-cut gap, muticut, and zero extension problems, to name a few. In these applications, parameter $beta$, called the padding parameter, is the most important parameter since it decides either the distortion or the approximation ratios. For general graphs with $n$ vertices, $beta = Theta(log n)$. Klein, Plotkin, and Rao showed that $K_r$-minor-free graphs have padding parameter $beta = O(r^3)$, which is a significant improvement over general graphs when $r$ is a constant. A long-standing conjecture is to construct a padded decomposition for $K_r$-minor-free graphs with padding parameter $beta = O(log r)$. Despite decades of research, the best-known result is $beta = O(r)$, even for graphs with treewidth at most $r$. In this work, we make significant progress toward the aforementioned conjecture by showing that graphs with treewidth $rm{tw}$ admit a padded decomposition with padding parameter $O(log rm{tw})$, which is tight. As corollaries, we obtain an exponential improvement in dependency on treewidth in a host of algorithmic applications: $O(sqrt{ log n cdot log(rm{tw})})$ flow-cut gap, max flow-min multicut ratio of $O(log(rm{tw}))$, an $O(log(rm{tw}))$ approximation for the 0-extension problem, an $ell^{O(log n)}_infty$ embedding with distortion $O(log rm{tw})$, and an $O(log rm{tw})$ bound for integrality gap for the uniform sparsest cut.
{"title":"Optimal Padded Decomposition For Bounded Treewidth Graphs","authors":"Arnold Filtser, Tobias Friedrich, Davis Issac, Nikhil Kumar, Hung Le, Nadym Mallek, Ziena Zeif","doi":"arxiv-2407.12230","DOIUrl":"https://doi.org/arxiv-2407.12230","url":null,"abstract":"A $(beta,delta,Delta)$-padded decomposition of an edge-weighted graph $G =\u0000(V,E,w)$ is a stochastic decomposition into clusters of diameter at most\u0000$Delta$ such that for every vertex $vin V$, the probability that\u0000$rm{ball}_G(v,gammaDelta)$ is entirely contained in the cluster containing\u0000$v$ is at least $e^{-betagamma}$ for every $gamma in [0,delta]$. Padded\u0000decompositions have been studied for decades and have found numerous\u0000applications, including metric embedding, multicommodity flow-cut gap, muticut,\u0000and zero extension problems, to name a few. In these applications, parameter\u0000$beta$, called the padding parameter, is the most important parameter since it\u0000decides either the distortion or the approximation ratios. For general graphs\u0000with $n$ vertices, $beta = Theta(log n)$. Klein, Plotkin, and Rao showed\u0000that $K_r$-minor-free graphs have padding parameter $beta = O(r^3)$, which is\u0000a significant improvement over general graphs when $r$ is a constant. A\u0000long-standing conjecture is to construct a padded decomposition for\u0000$K_r$-minor-free graphs with padding parameter $beta = O(log r)$. Despite\u0000decades of research, the best-known result is $beta = O(r)$, even for graphs\u0000with treewidth at most $r$. In this work, we make significant progress toward\u0000the aforementioned conjecture by showing that graphs with treewidth $rm{tw}$\u0000admit a padded decomposition with padding parameter $O(log rm{tw})$, which is\u0000tight. As corollaries, we obtain an exponential improvement in dependency on\u0000treewidth in a host of algorithmic applications: $O(sqrt{ log n cdot\u0000log(rm{tw})})$ flow-cut gap, max flow-min multicut ratio of\u0000$O(log(rm{tw}))$, an $O(log(rm{tw}))$ approximation for the 0-extension\u0000problem, an $ell^{O(log n)}_infty$ embedding with distortion $O(log\u0000rm{tw})$, and an $O(log rm{tw})$ bound for integrality gap for the uniform\u0000sparsest cut.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741052","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}
To solve many problems on graphs, graph traversals are used, the usual variants of which are the depth-first search and the breadth-first search. Implementing a graph traversal we consequently reach all vertices of the graph that belong to a connected component. The breadth-first search is the usual choice when constructing efficient algorithms for finding connected components of a graph. Methods of simple iteration for solving systems of linear equations with modified graph adjacency matrices and with the properly specified right-hand side can be considered as graph traversal algorithms. These traversal algorithms, generally speaking, turn out to be non-equivalent neither to the depth-first search nor the breadth-first search. The example of such a traversal algorithm is the one associated with the Gauss-Seidel method. For an arbitrary connected graph, to visit all its vertices, the algorithm requires not more iterations than that is required for BFS. For a large number of instances of the problem, fewer iterations will be required.
{"title":"Finding connected components of a graph using traversals associated with iterative methods for solving systems of linear equations","authors":"A. V. Prolubnikov","doi":"arxiv-2407.10790","DOIUrl":"https://doi.org/arxiv-2407.10790","url":null,"abstract":"To solve many problems on graphs, graph traversals are used, the usual\u0000variants of which are the depth-first search and the breadth-first search.\u0000Implementing a graph traversal we consequently reach all vertices of the graph\u0000that belong to a connected component. The breadth-first search is the usual\u0000choice when constructing efficient algorithms for finding connected components\u0000of a graph. Methods of simple iteration for solving systems of linear equations\u0000with modified graph adjacency matrices and with the properly specified\u0000right-hand side can be considered as graph traversal algorithms. These\u0000traversal algorithms, generally speaking, turn out to be non-equivalent neither\u0000to the depth-first search nor the breadth-first search. The example of such a\u0000traversal algorithm is the one associated with the Gauss-Seidel method. For an\u0000arbitrary connected graph, to visit all its vertices, the algorithm requires\u0000not more iterations than that is required for BFS. For a large number of\u0000instances of the problem, fewer iterations will be required.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"62 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719605","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}
In 2022, Chen et al. proposed an algorithm in cite{main} that solves the min cost flow problem in $m^{1 + o(1)} log U log C$ time, where $m$ is the number of edges in the graph, $U$ is an upper bound on capacities and $C$ is an upper bound on costs. However, as far as the authors of cite{main} know, no one has implemented their algorithm to date. In this paper, we discuss implementations of several key portions of the algorithm given in cite{main}, including the justifications for specific implementation choices. For the portions of the algorithm that we do not implement, we provide stubs. We then go through the entire algorithm and calculate the $m^{o(1)}$ term more precisely. Finally, we conclude with potential directions for future work in this area.
{"title":"Partial Implementation of Max Flow and Min Cost Flow in Almost-Linear Time","authors":"Nithin Kavi","doi":"arxiv-2407.10034","DOIUrl":"https://doi.org/arxiv-2407.10034","url":null,"abstract":"In 2022, Chen et al. proposed an algorithm in cite{main} that solves the min\u0000cost flow problem in $m^{1 + o(1)} log U log C$ time, where $m$ is the number\u0000of edges in the graph, $U$ is an upper bound on capacities and $C$ is an upper\u0000bound on costs. However, as far as the authors of cite{main} know, no one has\u0000implemented their algorithm to date. In this paper, we discuss implementations\u0000of several key portions of the algorithm given in cite{main}, including the\u0000justifications for specific implementation choices. For the portions of the\u0000algorithm that we do not implement, we provide stubs. We then go through the\u0000entire algorithm and calculate the $m^{o(1)}$ term more precisely. Finally, we\u0000conclude with potential directions for future work in this area.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"62 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719604","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}
This work deals with a problem of assigning periodic tasks to employees in such a way that each employee performs each task with the same frequency in the long term. The motivation comes from a collaboration with the SNCF, the main French railway company. An almost complete solution is provided under the form of a necessary and sufficient condition that can be checked in polynomial time. A complementary discussion about possible extensions is also proposed.
{"title":"Balanced assignments of periodic tasks","authors":"Héloïse Gachet, Frédéric Meunier","doi":"arxiv-2407.05485","DOIUrl":"https://doi.org/arxiv-2407.05485","url":null,"abstract":"This work deals with a problem of assigning periodic tasks to employees in\u0000such a way that each employee performs each task with the same frequency in the\u0000long term. The motivation comes from a collaboration with the SNCF, the main\u0000French railway company. An almost complete solution is provided under the form\u0000of a necessary and sufficient condition that can be checked in polynomial time.\u0000A complementary discussion about possible extensions is also proposed.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573797","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}
Markus Chimani, Torben Donzelmann, Nick Kloster, Melissa Koch, Jan-Jakob Völlering, Mirko H. Wagner
Beyond planarity concepts (prominent examples include k-planarity or fan-planarity) apply certain restrictions on the allowed patterns of crossings in drawings. It is natural to ask, how much the number of crossings may increase over the traditional (unrestricted) crossing number. Previous approaches to bound such ratios, e.g. [arXiv:1908.03153, arXiv:2105.12452], require very specialized constructions and arguments for each considered beyond planarity concept, and mostly only yield asymptotically non-tight bounds. We propose a very general proof framework that allows us to obtain asymptotically tight bounds, and where the concept-specific parts of the proof typically boil down to a couple of lines. We show the strength of our approach by giving improved or first bounds for several beyond planarity concepts.
除了平面性概念（著名的例子包括 k 平面性或扇形平面性）之外，还对图纸中允许的交叉模式施加了某些限制。我们自然会问，与传统的（无限制的）交叉数量相比，交叉数量可能会增加多少。以前约束这种比率的方法，例如[arXiv:1908.03153, arXiv:2105.12452]，需要为每个考虑的超越平面概念进行非常专门的构造和论证，而且大多只能得到渐近的非严密约束。我们提出了一个非常通用的证明框架，它允许我们获得渐近严密的边界，而且证明中与概念相关的部分通常只需几行。我们通过给出几个超越平面性概念的改进或首次边界，展示了我们方法的优势。
{"title":"Crossing Numbers of Beyond Planar Graphs Re-revisited: A Framework Approach","authors":"Markus Chimani, Torben Donzelmann, Nick Kloster, Melissa Koch, Jan-Jakob Völlering, Mirko H. Wagner","doi":"arxiv-2407.05057","DOIUrl":"https://doi.org/arxiv-2407.05057","url":null,"abstract":"Beyond planarity concepts (prominent examples include k-planarity or\u0000fan-planarity) apply certain restrictions on the allowed patterns of crossings\u0000in drawings. It is natural to ask, how much the number of crossings may\u0000increase over the traditional (unrestricted) crossing number. Previous\u0000approaches to bound such ratios, e.g. [arXiv:1908.03153, arXiv:2105.12452],\u0000require very specialized constructions and arguments for each considered beyond\u0000planarity concept, and mostly only yield asymptotically non-tight bounds. We\u0000propose a very general proof framework that allows us to obtain asymptotically\u0000tight bounds, and where the concept-specific parts of the proof typically boil\u0000down to a couple of lines. We show the strength of our approach by giving\u0000improved or first bounds for several beyond planarity concepts.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573799","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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