Pub Date : 2023-03-14DOI: 10.48550/arXiv.2303.07858
Daniel Funke, P. Sanders
Yao graphs are geometric spanners that connect each point of a given point set to its nearest neighbor in each of $k$ cones drawn around it. Yao graphs were introduced to construct minimum spanning trees in $d$ dimensional spaces. Moreover, they are used for instance in topology control in wireless networks. An optimal Onlogn time algorithm to construct Yao graphs for given point set has been proposed in the literature but -- to the best of our knowledge -- never been implemented. Instead, algorithms with a quadratic complexity are used in popular packages to construct these graphs. In this paper we present the first implementation of the optimal Yao graph algorithm. We develop and tune the data structures required to achieve the O(n log n) bound and detail algorithmic adaptions necessary to take the original algorithm from theory to practice. We propose a priority queue data structure that separates static and dynamic events and might be of independent interest for other sweepline algorithms. Additionally, we propose a new Yao graph algorithm based on a uniform grid data structure that performs well for medium-sized inputs. We evaluate our implementations on a wide variety synthetic and real-world datasets and show that our implementation outperforms current publicly available implementations by at least an order of magnitude.
{"title":"Efficient Yao Graph Construction","authors":"Daniel Funke, P. Sanders","doi":"10.48550/arXiv.2303.07858","DOIUrl":"https://doi.org/10.48550/arXiv.2303.07858","url":null,"abstract":"Yao graphs are geometric spanners that connect each point of a given point set to its nearest neighbor in each of $k$ cones drawn around it. Yao graphs were introduced to construct minimum spanning trees in $d$ dimensional spaces. Moreover, they are used for instance in topology control in wireless networks. An optimal Onlogn time algorithm to construct Yao graphs for given point set has been proposed in the literature but -- to the best of our knowledge -- never been implemented. Instead, algorithms with a quadratic complexity are used in popular packages to construct these graphs. In this paper we present the first implementation of the optimal Yao graph algorithm. We develop and tune the data structures required to achieve the O(n log n) bound and detail algorithmic adaptions necessary to take the original algorithm from theory to practice. We propose a priority queue data structure that separates static and dynamic events and might be of independent interest for other sweepline algorithms. Additionally, we propose a new Yao graph algorithm based on a uniform grid data structure that performs well for medium-sized inputs. We evaluate our implementations on a wide variety synthetic and real-world datasets and show that our implementation outperforms current publicly available implementations by at least an order of magnitude.","PeriodicalId":9448,"journal":{"name":"Bulletin of the Society of Sea Water Science, Japan","volume":"2 1","pages":"20:1-20:20"},"PeriodicalIF":0.0,"publicationDate":"2023-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85483749","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 : 2023-02-17DOI: 10.48550/arXiv.2302.08870
Thomas Bläsius, Maximilian Katzmann, Marcus Wilhelm
We consider a variant of treewidth that we call clique-partitioned treewidth in which each bag is partitioned into cliques. This is motivated by the recent development of FPT-algorithms based on similar parameters for various problems. With this paper, we take a first step towards computing clique-partitioned tree decompositions. Our focus lies on the subproblem of computing clique partitions, i.e., for each bag of a given tree decomposition, we compute an optimal partition of the induced subgraph into cliques. The goal here is to minimize the product of the clique sizes (plus 1). We show that this problem is NP-hard. We also describe four heuristic approaches as well as an exact branch-and-bound algorithm. Our evaluation shows that the branch-and-bound solver is sufficiently efficient to serve as a good baseline. Moreover, our heuristics yield solutions close to the optimum. As a bonus, our algorithms allow us to compute first upper bounds for the clique-partitioned treewidth of real-world networks. A comparison to traditional treewidth indicates that clique-partitioned treewidth is a promising parameter for graphs with high clustering.
{"title":"Partitioning the Bags of a Tree Decomposition Into Cliques","authors":"Thomas Bläsius, Maximilian Katzmann, Marcus Wilhelm","doi":"10.48550/arXiv.2302.08870","DOIUrl":"https://doi.org/10.48550/arXiv.2302.08870","url":null,"abstract":"We consider a variant of treewidth that we call clique-partitioned treewidth in which each bag is partitioned into cliques. This is motivated by the recent development of FPT-algorithms based on similar parameters for various problems. With this paper, we take a first step towards computing clique-partitioned tree decompositions. Our focus lies on the subproblem of computing clique partitions, i.e., for each bag of a given tree decomposition, we compute an optimal partition of the induced subgraph into cliques. The goal here is to minimize the product of the clique sizes (plus 1). We show that this problem is NP-hard. We also describe four heuristic approaches as well as an exact branch-and-bound algorithm. Our evaluation shows that the branch-and-bound solver is sufficiently efficient to serve as a good baseline. Moreover, our heuristics yield solutions close to the optimum. As a bonus, our algorithms allow us to compute first upper bounds for the clique-partitioned treewidth of real-world networks. A comparison to traditional treewidth indicates that clique-partitioned treewidth is a promising parameter for graphs with high clustering.","PeriodicalId":9448,"journal":{"name":"Bulletin of the Society of Sea Water Science, Japan","volume":"7 1","pages":"3:1-3:19"},"PeriodicalIF":0.0,"publicationDate":"2023-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89840611","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 : 2023-02-14DOI: 10.48550/arXiv.2302.07168
Ernestine Großmann, J. Sauer, Christian Schulz, Patrick Steil
We present Arc-Flag TB, a journey planning algorithm for public transit networks which combines Trip-Based Public Transit Routing (TB) with the Arc-Flags speedup technique. Compared to previous attempts to apply Arc-Flags to public transit networks, which saw limited success, our approach uses stronger pruning rules to reduce the search space. Our experiments show that Arc-Flag TB achieves a speedup of up to two orders of magnitude over TB, offering query times of less than a millisecond even on large countrywide networks. Compared to the state-of-the-art speedup technique Trip-Based Public Transit Routing Using Condensed Search Trees (TB-CST), our algorithm achieves similar query times but requires significantly less additional memory. Other state-of-the-art algorithms which achieve even faster query times, e.g., Public Transit Labeling, require enormous memory usage. In contrast, Arc-Flag TB offers a tradeoff between query performance and memory usage due to the fact that the number of regions in the network partition required by our algorithm is a configurable parameter. We also identify an issue in the transfer precomputation of TB that affects both TB-CST and Arc-Flag TB, leading to incorrect answers for some queries. This has not been previously recognized by the author of TB-CST. We provide discussion on how to resolve this issue in the future. Currently, Arc-Flag TB answers 1-6% of queries incorrectly, compared to over 20% for TB-CST on some networks.
{"title":"Arc-Flags Meet Trip-Based Public Transit Routing","authors":"Ernestine Großmann, J. Sauer, Christian Schulz, Patrick Steil","doi":"10.48550/arXiv.2302.07168","DOIUrl":"https://doi.org/10.48550/arXiv.2302.07168","url":null,"abstract":"We present Arc-Flag TB, a journey planning algorithm for public transit networks which combines Trip-Based Public Transit Routing (TB) with the Arc-Flags speedup technique. Compared to previous attempts to apply Arc-Flags to public transit networks, which saw limited success, our approach uses stronger pruning rules to reduce the search space. Our experiments show that Arc-Flag TB achieves a speedup of up to two orders of magnitude over TB, offering query times of less than a millisecond even on large countrywide networks. Compared to the state-of-the-art speedup technique Trip-Based Public Transit Routing Using Condensed Search Trees (TB-CST), our algorithm achieves similar query times but requires significantly less additional memory. Other state-of-the-art algorithms which achieve even faster query times, e.g., Public Transit Labeling, require enormous memory usage. In contrast, Arc-Flag TB offers a tradeoff between query performance and memory usage due to the fact that the number of regions in the network partition required by our algorithm is a configurable parameter. We also identify an issue in the transfer precomputation of TB that affects both TB-CST and Arc-Flag TB, leading to incorrect answers for some queries. This has not been previously recognized by the author of TB-CST. We provide discussion on how to resolve this issue in the future. Currently, Arc-Flag TB answers 1-6% of queries incorrectly, compared to over 20% for TB-CST on some networks.","PeriodicalId":9448,"journal":{"name":"Bulletin of the Society of Sea Water Science, Japan","volume":"222 1","pages":"16:1-16:18"},"PeriodicalIF":0.0,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87177890","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 : 2023-02-13DOI: 10.48550/arXiv.2302.06137
Stephen Jaud, Anthony Wirth, F. Choudhury
Given a collection of $m$ sets from a universe $mathcal{U}$, the Maximum Set Coverage problem consists of finding $k$ sets whose union has largest cardinality. This problem is NP-Hard, but the solution can be approximated by a polynomial time algorithm up to a factor $1-1/e$. However, this algorithm does not scale well with the input size. In a streaming context, practical high-quality solutions are found, but with space complexity that scales linearly with respect to the size of the universe $|mathcal{U}|$. However, one randomized streaming algorithm has been shown to produce a $1-1/e-varepsilon$ approximation of the optimal solution with a space complexity that scales only poly-logarithmically with respect to $m$ and $|mathcal{U}|$. In order to achieve such a low space complexity, the authors used a technique called subsampling, based on independent-wise hash functions. This article focuses on this sublinear-space algorithm and introduces methods to reduce the time cost of subsampling. We first show how to accelerate by several orders of magnitude without altering the space complexity, number of passes and approximation quality of the original algorithm. Secondly, we derive a new lower bound for the probability of producing a $1-1/e-varepsilon$ approximation using only pairwise independence: $1-tfrac{4}{c k log m}$ compared to the original $1-tfrac{2e}{m^{ck/6}}$. Although the theoretical approximation guarantees are weaker, for large streams, our algorithm performs well in practice and present the best time-space-performance trade-off for maximum coverage in streams.
给定一个宇宙$mathcal{U}$中$m$个集合的集合，最大集合覆盖问题包括找到其并集具有最大基数的$k$个集合。这个问题是np困难的，但是解决方案可以用多项式时间算法近似到一个因子$1-1/e$。然而，该算法不能很好地随输入大小进行伸缩。在流环境中，找到了实用的高质量解决方案，但具有相对于宇宙大小线性扩展的空间复杂性$|mathcal{U}|$。然而，一种随机流算法已被证明可以产生最优解的$1-1/e-varepsilon$近似，其空间复杂度仅相对于$m$和$|mathcal{U}|$进行多对数缩放。为了实现如此低的空间复杂度，作者使用了一种基于独立哈希函数的称为子采样的技术。本文重点研究了这种次线性空间算法，并介绍了降低次采样时间成本的方法。我们首先展示了如何在不改变原始算法的空间复杂度、通过次数和近似质量的情况下加速几个数量级。其次，我们推导出仅使用成对独立产生$1-1/e-varepsilon$近似的概率的新下界:$1-tfrac{4}{c k log m}$与原始的$1-tfrac{2e}{m^{ck/6}}$相比。虽然理论上的近似保证较弱，但对于大型流，我们的算法在实践中表现良好，并且在流的最大覆盖方面提供了最佳的时间-空间性能权衡。
{"title":"Maximum Coverage in Sublinear Space, Faster","authors":"Stephen Jaud, Anthony Wirth, F. Choudhury","doi":"10.48550/arXiv.2302.06137","DOIUrl":"https://doi.org/10.48550/arXiv.2302.06137","url":null,"abstract":"Given a collection of $m$ sets from a universe $mathcal{U}$, the Maximum Set Coverage problem consists of finding $k$ sets whose union has largest cardinality. This problem is NP-Hard, but the solution can be approximated by a polynomial time algorithm up to a factor $1-1/e$. However, this algorithm does not scale well with the input size. In a streaming context, practical high-quality solutions are found, but with space complexity that scales linearly with respect to the size of the universe $|mathcal{U}|$. However, one randomized streaming algorithm has been shown to produce a $1-1/e-varepsilon$ approximation of the optimal solution with a space complexity that scales only poly-logarithmically with respect to $m$ and $|mathcal{U}|$. In order to achieve such a low space complexity, the authors used a technique called subsampling, based on independent-wise hash functions. This article focuses on this sublinear-space algorithm and introduces methods to reduce the time cost of subsampling. We first show how to accelerate by several orders of magnitude without altering the space complexity, number of passes and approximation quality of the original algorithm. Secondly, we derive a new lower bound for the probability of producing a $1-1/e-varepsilon$ approximation using only pairwise independence: $1-tfrac{4}{c k log m}$ compared to the original $1-tfrac{2e}{m^{ck/6}}$. Although the theoretical approximation guarantees are weaker, for large streams, our algorithm performs well in practice and present the best time-space-performance trade-off for maximum coverage in streams.","PeriodicalId":9448,"journal":{"name":"Bulletin of the Society of Sea Water Science, Japan","volume":"37 1","pages":"21:1-21:20"},"PeriodicalIF":0.0,"publicationDate":"2023-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74645164","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 : 2023-02-13DOI: 10.48550/arXiv.2302.06259
K. Eyubov, Marcelo Fonseca Faraj, Christian Schulz
Partitioning the vertices of a (hyper)graph into k roughly balanced blocks such that few (hyper)edges run between blocks is a key problem for large-scale distributed processing. A current trend for partitioning huge (hyper)graphs using low computational resources are streaming algorithms. In this work, we propose FREIGHT: a Fast stREamInG Hypergraph parTitioning algorithm which is an adaptation of the widely-known graph-based algorithm Fennel. By using an efficient data structure, we make the overall running of FREIGHT linearly dependent on the pin-count of the hypergraph and the memory consumption linearly dependent on the numbers of nets and blocks. The results of our extensive experimentation showcase the promising performance of FREIGHT as a highly efficient and effective solution for streaming hypergraph partitioning. Our algorithm demonstrates competitive running time with the Hashing algorithm, with a difference of a maximum factor of four observed on three fourths of the instances. Significantly, our findings highlight the superiority of FREIGHT over all existing (buffered) streaming algorithms and even the in-memory algorithm HYPE, with respect to both cut-net and connectivity measures. This indicates that our proposed algorithm is a promising hypergraph partitioning tool to tackle the challenge posed by large-scale and dynamic data processing.
{"title":"FREIGHT: Fast Streaming Hypergraph Partitioning","authors":"K. Eyubov, Marcelo Fonseca Faraj, Christian Schulz","doi":"10.48550/arXiv.2302.06259","DOIUrl":"https://doi.org/10.48550/arXiv.2302.06259","url":null,"abstract":"Partitioning the vertices of a (hyper)graph into k roughly balanced blocks such that few (hyper)edges run between blocks is a key problem for large-scale distributed processing. A current trend for partitioning huge (hyper)graphs using low computational resources are streaming algorithms. In this work, we propose FREIGHT: a Fast stREamInG Hypergraph parTitioning algorithm which is an adaptation of the widely-known graph-based algorithm Fennel. By using an efficient data structure, we make the overall running of FREIGHT linearly dependent on the pin-count of the hypergraph and the memory consumption linearly dependent on the numbers of nets and blocks. The results of our extensive experimentation showcase the promising performance of FREIGHT as a highly efficient and effective solution for streaming hypergraph partitioning. Our algorithm demonstrates competitive running time with the Hashing algorithm, with a difference of a maximum factor of four observed on three fourths of the instances. Significantly, our findings highlight the superiority of FREIGHT over all existing (buffered) streaming algorithms and even the in-memory algorithm HYPE, with respect to both cut-net and connectivity measures. This indicates that our proposed algorithm is a promising hypergraph partitioning tool to tackle the challenge posed by large-scale and dynamic data processing.","PeriodicalId":9448,"journal":{"name":"Bulletin of the Society of Sea Water Science, Japan","volume":"25 1","pages":"15:1-15:16"},"PeriodicalIF":0.0,"publicationDate":"2023-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84610096","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 : 2023-01-01DOI: 10.4230/LIPIcs.SEA.2023.17
Noah Wahl, Lars Gottesbüren
We investigate the efficacy of greedy heuristics for the judicious hypergraph partitioning problem. In contrast to balanced partitioning problems, the goal of judicious hypergraph partitioning is to minimize the maximum load over all blocks of the partition. We devise strategies for initial partitioning and FM-style post-processing. In combination with a multilevel scheme, they beat the previous state-of-the-art solver – based on greedy set covers – in both running time (two to four orders of magnitude) and solution quality (18% to 45%). A major challenge that makes local greedy approaches difficult to use for this problem is the high frequency of zero-gain moves , for which we present and evaluate counteracting mechanisms.
{"title":"Greedy Heuristics for Judicious Hypergraph Partitioning","authors":"Noah Wahl, Lars Gottesbüren","doi":"10.4230/LIPIcs.SEA.2023.17","DOIUrl":"https://doi.org/10.4230/LIPIcs.SEA.2023.17","url":null,"abstract":"We investigate the efficacy of greedy heuristics for the judicious hypergraph partitioning problem. In contrast to balanced partitioning problems, the goal of judicious hypergraph partitioning is to minimize the maximum load over all blocks of the partition. We devise strategies for initial partitioning and FM-style post-processing. In combination with a multilevel scheme, they beat the previous state-of-the-art solver – based on greedy set covers – in both running time (two to four orders of magnitude) and solution quality (18% to 45%). A major challenge that makes local greedy approaches difficult to use for this problem is the high frequency of zero-gain moves , for which we present and evaluate counteracting mechanisms.","PeriodicalId":9448,"journal":{"name":"Bulletin of the Society of Sea Water Science, Japan","volume":"134 1","pages":"17:1-17:16"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86340171","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 : 2023-01-01DOI: 10.4230/LIPIcs.SEA.2023.8
Ramtin Afshar, M. Dillencourt, M. Goodrich, Evrim Ozel
We provide and study several algorithms for sorting an array of n comparable distinct elements subject to probabilistic comparison errors. In this model, the comparison of two elements returns the wrong answer according to a fixed probability, p e < 1 / 2, and otherwise returns the correct answer. The dislocation of an element is the distance between its position in a given (current or output) array and its position in a sorted array. There are various algorithms that can be utilized for sorting or near-sorting elements subject to probabilistic comparison errors, but these algorithms are not data oblivious because they all make heavy use of noisy binary searching. In this paper, we provide new methods for sorting with comparison errors that are data oblivious while avoiding the use of noisy binary search methods. In addition, we experimentally compare our algorithms and other sorting algorithms.
我们提供并研究了几种算法来对n个可比较的不同元素进行排序，这些元素受到概率比较误差的影响。在该模型中，两个元素的比较按固定概率p e < 1 / 2返回错误答案，否则返回正确答案。元素的错位是指其在给定(当前或输出)数组中的位置与在已排序数组中的位置之间的距离。有各种各样的算法可用于排序或近似排序受概率比较错误影响的元素，但这些算法不是数据无关的，因为它们都大量使用有噪声的二进制搜索。在本文中，我们提供了一种新的排序方法，这种方法具有数据无关的比较误差，同时避免了使用有噪声的二分搜索方法。此外，我们实验比较了我们的算法和其他排序算法。
{"title":"Noisy Sorting Without Searching: Data Oblivious Sorting with Comparison Errors","authors":"Ramtin Afshar, M. Dillencourt, M. Goodrich, Evrim Ozel","doi":"10.4230/LIPIcs.SEA.2023.8","DOIUrl":"https://doi.org/10.4230/LIPIcs.SEA.2023.8","url":null,"abstract":"We provide and study several algorithms for sorting an array of n comparable distinct elements subject to probabilistic comparison errors. In this model, the comparison of two elements returns the wrong answer according to a fixed probability, p e < 1 / 2, and otherwise returns the correct answer. The dislocation of an element is the distance between its position in a given (current or output) array and its position in a sorted array. There are various algorithms that can be utilized for sorting or near-sorting elements subject to probabilistic comparison errors, but these algorithms are not data oblivious because they all make heavy use of noisy binary searching. In this paper, we provide new methods for sorting with comparison errors that are data oblivious while avoiding the use of noisy binary search methods. In addition, we experimentally compare our algorithms and other sorting algorithms.","PeriodicalId":9448,"journal":{"name":"Bulletin of the Society of Sea Water Science, Japan","volume":"161 1","pages":"8:1-8:18"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73576845","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 : 2023-01-01DOI: 10.4230/LIPIcs.SEA.2023.14
Alexander Bille, Niels Grüttemeier, Christian Komusiewicz, Nils Morawietz
We present a new simple graph-theoretic formulation of the exploratory blockmodeling problem on undirected and unweighted one-mode networks. Our formulation takes as input the network G and the maximum number t of blocks for the solution model. The task is to find a minimum-size set of edge insertions and deletions that transform the input graph G into a graph G ′ with at most t neighborhood classes. Herein, a neighborhood class is a maximal set of vertices with the same neighborhood. The neighborhood classes of G ′ directly give the blocks and block interactions of the computed blockmodel. We analyze the classic and parameterized complexity of the exploratory blockmodeling problem, provide a branch-and-bound algorithm, an ILP formulation and several heuristics. Finally, we compare our exact algorithms to previous ILP-based approaches and show that the new algorithms are faster for t ≥ 4. 2012 ACM
{"title":"A Graph-Theoretic Formulation of Exploratory Blockmodeling","authors":"Alexander Bille, Niels Grüttemeier, Christian Komusiewicz, Nils Morawietz","doi":"10.4230/LIPIcs.SEA.2023.14","DOIUrl":"https://doi.org/10.4230/LIPIcs.SEA.2023.14","url":null,"abstract":"We present a new simple graph-theoretic formulation of the exploratory blockmodeling problem on undirected and unweighted one-mode networks. Our formulation takes as input the network G and the maximum number t of blocks for the solution model. The task is to find a minimum-size set of edge insertions and deletions that transform the input graph G into a graph G ′ with at most t neighborhood classes. Herein, a neighborhood class is a maximal set of vertices with the same neighborhood. The neighborhood classes of G ′ directly give the blocks and block interactions of the computed blockmodel. We analyze the classic and parameterized complexity of the exploratory blockmodeling problem, provide a branch-and-bound algorithm, an ILP formulation and several heuristics. Finally, we compare our exact algorithms to previous ILP-based approaches and show that the new algorithms are faster for t ≥ 4. 2012 ACM","PeriodicalId":9448,"journal":{"name":"Bulletin of the Society of Sea Water Science, Japan","volume":"1 1","pages":"14:1-14:20"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90493319","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 : 2023-01-01DOI: 10.4230/LIPIcs.SEA.2023.11
Kengo Nakamura, Masaaki Nishino, Norihito Yasuda, S. Minato
The subgraph counting problem computes the number of subgraphs of a given graph that satisfy some constraints. Among various constraints imposed on a graph, those regarding the connectivity of vertices, such as “these two vertices must be connected,” have great importance since they are indispensable for determining various graph substructures, e.g., paths, Steiner trees, and rooted spanning forests. In this view, the subgraph counting problem under connectivity constraints is also important because counting such substructures often corresponds to measuring the importance of a vertex in network infrastructures. However, we must solve the subgraph counting problems multiple times to compute such an importance measure for every vertex. Conventionally, they are solved separately by constructing decision diagrams such as BDD and ZDD for each problem. However, even solving a single subgraph counting is a computationally hard task, preventing us from solving it multiple times in a reasonable time. In this paper, we propose a dynamic programming framework that simultaneously counts subgraphs for every vertex by focusing on similar connectivity constraints. Experimental results show that the proposed method solved multiple subgraph counting problems about 10–20 times faster than the existing approach for many problem settings.
{"title":"CompDP: A Framework for Simultaneous Subgraph Counting Under Connectivity Constraints","authors":"Kengo Nakamura, Masaaki Nishino, Norihito Yasuda, S. Minato","doi":"10.4230/LIPIcs.SEA.2023.11","DOIUrl":"https://doi.org/10.4230/LIPIcs.SEA.2023.11","url":null,"abstract":"The subgraph counting problem computes the number of subgraphs of a given graph that satisfy some constraints. Among various constraints imposed on a graph, those regarding the connectivity of vertices, such as “these two vertices must be connected,” have great importance since they are indispensable for determining various graph substructures, e.g., paths, Steiner trees, and rooted spanning forests. In this view, the subgraph counting problem under connectivity constraints is also important because counting such substructures often corresponds to measuring the importance of a vertex in network infrastructures. However, we must solve the subgraph counting problems multiple times to compute such an importance measure for every vertex. Conventionally, they are solved separately by constructing decision diagrams such as BDD and ZDD for each problem. However, even solving a single subgraph counting is a computationally hard task, preventing us from solving it multiple times in a reasonable time. In this paper, we propose a dynamic programming framework that simultaneously counts subgraphs for every vertex by focusing on similar connectivity constraints. Experimental results show that the proposed method solved multiple subgraph counting problems about 10–20 times faster than the existing approach for many problem settings.","PeriodicalId":9448,"journal":{"name":"Bulletin of the Society of Sea Water Science, Japan","volume":"18 1","pages":"11:1-11:20"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75895123","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 : 2023-01-01DOI: 10.4230/LIPIcs.SEA.2023.6
R. Becker, P. Crescenzi, A. Cruciani, Bojana Kodric
Identifying influential nodes in a network is arguably one of the most important tasks in graph mining and network analysis. A large variety of centrality measures, all aiming at correctly quantifying a node’s importance in the network, have been formulated in the literature. One of the most cited ones is the betweenness centrality , formally introduced by Freeman (Sociometry, 1977). On the other hand, researchers have recently been very interested in capturing the dynamic nature of real-world networks by studying temporal graphs , rather than static ones. Clearly, centrality measures, including the betweenness centrality, have also been extended to temporal graphs. Buß et al. (KDD, 2020) gave algorithms to compute various notions of temporal betweenness centrality, including the perhaps most natural one – shortest temporal betweenness . Their algorithm computes centrality values of all nodes in time O ( n 3 T 2 ), where n is the size of the network and T is the total number of time steps. For real-world networks, which easily contain tens of thousands of nodes, this complexity becomes prohibitive. Thus, it is reasonable to consider proxies for shortest temporal betweenness rankings that are more efficiently computed, and, therefore, allow for measuring the relative importance of nodes in very large temporal graphs. In this paper, we compare several such proxies on a diverse set of real-world networks. These proxies can be divided into global and local proxies. The considered global proxies include the exact algorithm for static betweenness (computed on the underlying graph), prefix foremost temporal betweenness of Buß et al., which is more efficiently computable than shortest temporal betweenness, and the recently introduced approximation approach of Santoro and Sarpe (WWW, 2022). As all of these global proxies are still expensive to compute on very large networks, we also turn to more efficiently computable local proxies. Here, we consider temporal versions of the ego-betweenness in the sense of Everett and Borgatti (Social Networks, 2005), standard degree notions, and a novel temporal degree notion termed the pass-through degree , that we introduce in this paper and which we consider to be one of our main contributions. We show that the pass-through degree, which measures the number of pairs of neighbors of a node that are temporally connected through it, can be computed in nearly linear time for all nodes in the network and we experimentally observe that it is surprisingly competitive as a proxy for shortest temporal betweenness.
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