We give the first optimal bounds for returning the ℓ1-heavy hitters in a data stream of insertions, together with their approximate frequencies, closing a long line of work on this problem. For a stream of m items in { 1, 2, … , n} and parameters 0 < ε < φ ⩽ 1, let fi denote the frequency of item i, i.e., the number of times item i occurs in the stream. With arbitrarily large constant probability, our algorithm returns all items i for which fi ⩾ φ m, returns no items j for which fj ⩽ (φ −ε)m, and returns approximations f˜i with |f˜i − fi| ⩽ ε m for each item i that it returns. Our algorithm uses O(ε−1 log φ −1 + φ −1 log n + log log m) bits of space, processes each stream update in O(1) worst-case time, and can report its output in time linear in the output size. We also prove a lower bound, which implies that our algorithm is optimal up to a constant factor in its space complexity. A modification of our algorithm can be used to estimate the maximum frequency up to an additive ε m error in the above amount of space, resolving Question 3 in the IITK 2006 Workshop on Algorithms for Data Streams for the case of ℓ1-heavy hitters. We also introduce several variants of the heavy hitters and maximum frequency problems, inspired by rank aggregation and voting schemes, and show how our techniques can be applied in such settings. Unlike the traditional heavy hitters problem, some of these variants look at comparisons between items rather than numerical values to determine the frequency of an item.
{"title":"An Optimal Algorithm for ℓ1-Heavy Hitters in Insertion Streams and Related Problems","authors":"Arnab Bhattacharyya, P. Dey, David P. Woodruff","doi":"10.1145/3264427","DOIUrl":"https://doi.org/10.1145/3264427","url":null,"abstract":"We give the first optimal bounds for returning the ℓ1-heavy hitters in a data stream of insertions, together with their approximate frequencies, closing a long line of work on this problem. For a stream of m items in { 1, 2, … , n} and parameters 0 < ε < φ ⩽ 1, let fi denote the frequency of item i, i.e., the number of times item i occurs in the stream. With arbitrarily large constant probability, our algorithm returns all items i for which fi ⩾ φ m, returns no items j for which fj ⩽ (φ −ε)m, and returns approximations f˜i with |f˜i − fi| ⩽ ε m for each item i that it returns. Our algorithm uses O(ε−1 log φ −1 + φ −1 log n + log log m) bits of space, processes each stream update in O(1) worst-case time, and can report its output in time linear in the output size. We also prove a lower bound, which implies that our algorithm is optimal up to a constant factor in its space complexity. A modification of our algorithm can be used to estimate the maximum frequency up to an additive ε m error in the above amount of space, resolving Question 3 in the IITK 2006 Workshop on Algorithms for Data Streams for the case of ℓ1-heavy hitters. We also introduce several variants of the heavy hitters and maximum frequency problems, inspired by rank aggregation and voting schemes, and show how our techniques can be applied in such settings. Unlike the traditional heavy hitters problem, some of these variants look at comparisons between items rather than numerical values to determine the frequency of an item.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115294819","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}
Marek Cygan, Pawel Komosa, D. Lokshtanov, Michal Pilipczuk, Marcin Pilipczuk, Saket Saurabh
We show an algorithm that, given an n-vertex graph G and a parameter k, in time 2O(k log k) n O(1) finds a tree decomposition of G with the following properties: — every adhesion of the tree decomposition is of size at most k, and — every bag of the tree decomposition is (i,i)-unbreakable in G for every 1 ⩽ i ⩽ k. Here, a set X ⊆ V(G) is (a,b)-unbreakable in G if for every separation (A,B) of order at most b in G, we have |A cap X| ⩽ a or |B ∩ X| ⩽ a. The resulting tree decomposition has arguably best possible adhesion size bounds and unbreakability guarantees. Furthermore, the parametric factor in the running time bound is significantly smaller than in previous similar constructions. These improvements allow us to present parameterized algorithms for MINIMUM BISECTION, STEINER CUT, and STEINER MULTICUT with improved parameteric factor in the running time bound. The main technical insight is to adapt the notion of lean decompositions of Thomas and the subsequent construction algorithm of Bellenbaum and Diestel to the parameterized setting.
我们给出了一个算法，给定一个n顶点图G和一个参数k，在时间2O(k log k) n O(1)中找到G的树分解，它具有以下性质:——每个树分解的附着力大小最多k,每袋,树的分解是(我)牢不可破的G每1⩽我⩽k。这里,一组X⊆V (G) (a, b)牢不可破的G如果每个订单最多的分离(a, b) b G,我们有X | | 帽⩽a或b∩X | |⩽。结果树分解无疑最好的附着力大小范围和unbreakability担保。此外，在运行时间范围内的参数因子明显小于以前的类似结构。这些改进使我们能够在运行时间范围内提出最小平分、STEINER CUT和STEINER multiccut的参数化算法，并改进了参数化因子。主要的技术见解是将Thomas的精益分解概念以及随后的Bellenbaum和Diestel的构造算法适应于参数化设置。
{"title":"Randomized Contractions Meet Lean Decompositions","authors":"Marek Cygan, Pawel Komosa, D. Lokshtanov, Michal Pilipczuk, Marcin Pilipczuk, Saket Saurabh","doi":"10.1145/3426738","DOIUrl":"https://doi.org/10.1145/3426738","url":null,"abstract":"We show an algorithm that, given an n-vertex graph G and a parameter k, in time 2O(k log k) n O(1) finds a tree decomposition of G with the following properties: — every adhesion of the tree decomposition is of size at most k, and — every bag of the tree decomposition is (i,i)-unbreakable in G for every 1 ⩽ i ⩽ k. Here, a set X ⊆ V(G) is (a,b)-unbreakable in G if for every separation (A,B) of order at most b in G, we have |A cap X| ⩽ a or |B ∩ X| ⩽ a. The resulting tree decomposition has arguably best possible adhesion size bounds and unbreakability guarantees. Furthermore, the parametric factor in the running time bound is significantly smaller than in previous similar constructions. These improvements allow us to present parameterized algorithms for MINIMUM BISECTION, STEINER CUT, and STEINER MULTICUT with improved parameteric factor in the running time bound. The main technical insight is to adapt the notion of lean decompositions of Thomas and the subsequent construction algorithm of Bellenbaum and Diestel to the parameterized setting.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122017373","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}
We continue developing the information theory of structured data. In this article, we study models generating d-ary trees (d ≥ 2) and trees with unrestricted degree. We first compute the entropy which gives us the fundamental lower bound on compression of such trees. Then we present efficient compression algorithms based on arithmetic encoding that achieve the entropy within a constant number of bits. A naïve implementation of these algorithms has a prohibitive time complexity of O(nd) elementary arithmetic operations (each corresponding to a number f(n, d) of bit operations), but our efficient algorithms run in O(n2) of these operations, where n is the number of nodes. It turns out that extending source coding (i.e., compression) from sequences to advanced data structures such as degree-unconstrained trees is mathematically quite challenging and leads to recurrences that find ample applications in the information theory of general structures (e.g., to analyze the information content of degree-unconstrained non-plane trees).
{"title":"Entropy and Optimal Compression of Some General Plane Trees","authors":"Z. Golebiewski, A. Magner, W. Szpankowski","doi":"10.1145/3275444","DOIUrl":"https://doi.org/10.1145/3275444","url":null,"abstract":"We continue developing the information theory of structured data. In this article, we study models generating d-ary trees (d ≥ 2) and trees with unrestricted degree. We first compute the entropy which gives us the fundamental lower bound on compression of such trees. Then we present efficient compression algorithms based on arithmetic encoding that achieve the entropy within a constant number of bits. A naïve implementation of these algorithms has a prohibitive time complexity of O(nd) elementary arithmetic operations (each corresponding to a number f(n, d) of bit operations), but our efficient algorithms run in O(n2) of these operations, where n is the number of nodes. It turns out that extending source coding (i.e., compression) from sequences to advanced data structures such as degree-unconstrained trees is mathematically quite challenging and leads to recurrences that find ample applications in the information theory of general structures (e.g., to analyze the information content of degree-unconstrained non-plane trees).","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125913737","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}
Marthe Bonamy, Oscar Defrain, Marc Heinrich, Michal Pilipczuk, Jean-Florent Raymond
It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this article we investigate this problem in graph classes defined by forbidding an induced subgraph. In particular, we provide output-polynomial time algorithms for Kt-free graphs and for several related graph classes. This answers a question of Kanté et al. about enumeration in bipartite graphs.
{"title":"Enumerating Minimal Dominating Sets in Kt-free Graphs and Variants","authors":"Marthe Bonamy, Oscar Defrain, Marc Heinrich, Michal Pilipczuk, Jean-Florent Raymond","doi":"10.1145/3386686","DOIUrl":"https://doi.org/10.1145/3386686","url":null,"abstract":"It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this article we investigate this problem in graph classes defined by forbidding an induced subgraph. In particular, we provide output-polynomial time algorithms for Kt-free graphs and for several related graph classes. This answers a question of Kanté et al. about enumeration in bipartite graphs.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117251266","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}
Unaggregated data, in a streamed or distributed form, are prevalent and come from diverse sources such as interactions of users with web services and IP traffic. Data elements have keys (cookies, users, queries), and elements with different keys interleave. Analytics on such data typically utilizes statistics expressed as a sum over keys in a specified segment of a function f applied to the frequency (the total number of occurrences) of the key. In particular, Distinct is the number of active keys in the segment, Sum is the sum of their frequencies, and both are special cases of frequency cap statistics, which cap the frequency by a parameter T. Random samples can be very effective for quick and efficient estimation of statistics at query time. Ideally, to estimate statistics for a given function f, our sample would include a key with frequency w with probability roughly proportional to f(w). The challenge is that while such “gold-standard” samples can be easily computed after aggregating the data (computing the set of key-frequency pairs), this aggregation is costly: It requires structure of size that is proportional to the number of active keys, which can be very large. We present a sampling framework for unaggregated data that uses a single pass (for streams) or two passes (for distributed data) and structure size proportional to the desired sample size. Our design unifies classic solutions for Distinct and Sum. Specifically, our ℓ-capped samples provide nonnegative unbiased estimates of any monotone non-decreasing frequency statistics and statistical guarantees on quality that are close to gold standard for cap statistics with T=Θ (ℓ). Furthermore, our multi-objective samples provide these statistical guarantees on quality for all concave sub-linear statistics (the nonnegative span of cap functions) while incurring only a logarithmic overhead on sample size.
{"title":"Stream Sampling Framework and Application for Frequency Cap Statistics","authors":"E. Cohen","doi":"10.1145/3234338","DOIUrl":"https://doi.org/10.1145/3234338","url":null,"abstract":"Unaggregated data, in a streamed or distributed form, are prevalent and come from diverse sources such as interactions of users with web services and IP traffic. Data elements have keys (cookies, users, queries), and elements with different keys interleave. Analytics on such data typically utilizes statistics expressed as a sum over keys in a specified segment of a function f applied to the frequency (the total number of occurrences) of the key. In particular, Distinct is the number of active keys in the segment, Sum is the sum of their frequencies, and both are special cases of frequency cap statistics, which cap the frequency by a parameter T. Random samples can be very effective for quick and efficient estimation of statistics at query time. Ideally, to estimate statistics for a given function f, our sample would include a key with frequency w with probability roughly proportional to f(w). The challenge is that while such “gold-standard” samples can be easily computed after aggregating the data (computing the set of key-frequency pairs), this aggregation is costly: It requires structure of size that is proportional to the number of active keys, which can be very large. We present a sampling framework for unaggregated data that uses a single pass (for streams) or two passes (for distributed data) and structure size proportional to the desired sample size. Our design unifies classic solutions for Distinct and Sum. Specifically, our ℓ-capped samples provide nonnegative unbiased estimates of any monotone non-decreasing frequency statistics and statistical guarantees on quality that are close to gold standard for cap statistics with T=Θ (ℓ). Furthermore, our multi-objective samples provide these statistical guarantees on quality for all concave sub-linear statistics (the nonnegative span of cap functions) while incurring only a logarithmic overhead on sample size.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123116316","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}
T-H. Hubert Chan, Zhiyi Huang, S. Jiang, N. Kang, Zhihao Gavin Tang
We study the online submodular maximization problem with free disposal under a matroid constraint. Elements from some ground set arrive one by one in rounds, and the algorithm maintains a feasible set that is independent in the underlying matroid. In each round when a new element arrives, the algorithm may accept the new element into its feasible set and possibly remove elements from it, provided that the resulting set is still independent. The goal is to maximize the value of the final feasible set under some monotone submodular function, to which the algorithm has oracle access. For k-uniform matroids, we give a deterministic algorithm with competitive ratio at least 0.2959, and the ratio approaches 1/α∞≈ 0.3178 as k approaches infinity, improving the previous best ratio of 0.25 by Chakrabarti and Kale (IPCO 2014), Buchbinder et al. (SODA 2015), and Chekuri et al. (ICALP 2015). We also show that our algorithm is optimal among a class of deterministic monotone algorithms that accept a new arriving element only if the objective is strictly increased. Further, we prove that no deterministic monotone algorithm can be strictly better than 0.25-competitive even for partition matroids, the most modest generalization of k-uniform matroids, matching the competitive ratio by Chakrabarti and Kale (IPCO 2014) and Chekuri et al. (ICALP 2015). Interestingly, we show that randomized algorithms are strictly more powerful by giving a (non-monotone) randomized algorithm for partition matroids with ratio 1/α∞≈ 0.3178.
{"title":"Online Submodular Maximization with Free Disposal","authors":"T-H. Hubert Chan, Zhiyi Huang, S. Jiang, N. Kang, Zhihao Gavin Tang","doi":"10.1145/3242770","DOIUrl":"https://doi.org/10.1145/3242770","url":null,"abstract":"We study the online submodular maximization problem with free disposal under a matroid constraint. Elements from some ground set arrive one by one in rounds, and the algorithm maintains a feasible set that is independent in the underlying matroid. In each round when a new element arrives, the algorithm may accept the new element into its feasible set and possibly remove elements from it, provided that the resulting set is still independent. The goal is to maximize the value of the final feasible set under some monotone submodular function, to which the algorithm has oracle access. For k-uniform matroids, we give a deterministic algorithm with competitive ratio at least 0.2959, and the ratio approaches 1/α∞≈ 0.3178 as k approaches infinity, improving the previous best ratio of 0.25 by Chakrabarti and Kale (IPCO 2014), Buchbinder et al. (SODA 2015), and Chekuri et al. (ICALP 2015). We also show that our algorithm is optimal among a class of deterministic monotone algorithms that accept a new arriving element only if the objective is strictly increased. Further, we prove that no deterministic monotone algorithm can be strictly better than 0.25-competitive even for partition matroids, the most modest generalization of k-uniform matroids, matching the competitive ratio by Chakrabarti and Kale (IPCO 2014) and Chekuri et al. (ICALP 2015). Interestingly, we show that randomized algorithms are strictly more powerful by giving a (non-monotone) randomized algorithm for partition matroids with ratio 1/α∞≈ 0.3178.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124143030","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}
Krzysztof Onak, B. Schieber, Shay Solomon, Nicole Wein
We consider the problem of maintaining a maximal independent set in a dynamic graph subject to edge insertions and deletions. Recently, Assadi et al. (at STOC’18) showed that a maximal independent set can be maintained in sublinear (in the dynamically changing number of edges) amortized update time. In this article, we significantly improve the update time for uniformly sparse graphs. Specifically, for graphs with arboricity α, the amortized update time of our algorithm is O(α2 ⋅ log2 n), where n is the number of vertices. For low arboricity graphs, which include, for example, minor-free graphs and some classes of “real-world” graphs, our update time is polylogarithmic. Our update time improves the result of Assadi et al. for all graphs with arboricity bounded by m3/8−ε, for any constant ε > 0. This covers much of the range of possible values for arboricity, as the arboricity of a general graph cannot exceed m1/2.
{"title":"Fully Dynamic MIS in Uniformly Sparse Graphs","authors":"Krzysztof Onak, B. Schieber, Shay Solomon, Nicole Wein","doi":"10.1145/3378025","DOIUrl":"https://doi.org/10.1145/3378025","url":null,"abstract":"We consider the problem of maintaining a maximal independent set in a dynamic graph subject to edge insertions and deletions. Recently, Assadi et al. (at STOC’18) showed that a maximal independent set can be maintained in sublinear (in the dynamically changing number of edges) amortized update time. In this article, we significantly improve the update time for uniformly sparse graphs. Specifically, for graphs with arboricity α, the amortized update time of our algorithm is O(α2 ⋅ log2 n), where n is the number of vertices. For low arboricity graphs, which include, for example, minor-free graphs and some classes of “real-world” graphs, our update time is polylogarithmic. Our update time improves the result of Assadi et al. for all graphs with arboricity bounded by m3/8−ε, for any constant ε > 0. This covers much of the range of possible values for arboricity, as the arboricity of a general graph cannot exceed m1/2.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114378552","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}
We consider a natural generalization of the classical multiple knapsack problem in which instead of packing single items we are packing groups of items. In this problem, we have multiple knapsacks and a set of items partitioned into groups. Each item has an individual weight, while the profit is associated with groups rather than items. The profit of a group can be attained if and only if every item of this group is packed. Such a general model finds applications in various practical problems, e.g., delivering bundles of goods. The tractability of this problem relies heavily on how large a group could be. Deciding if a group of items of total weight 2 could be packed into two knapsacks of unit capacity is already NP-hard and it thus rules out a constant-approximation algorithm for this problem in general. We then focus on the parameterized version where the total weight of items in each group is bounded by a factor δ of the total capacity of all knapsacks. Both approximation and inapproximability results with respect to δ are derived. We also show that, depending on whether the number of knapsacks is a constant or part of the input, the approximation ratio for the problem, as a function on δ, changes substantially, which has a clear difference from the classical multiple knapsack problem.
{"title":"Packing Groups of Items into Multiple Knapsacks","authors":"Lin Chen, Guochuan Zhang","doi":"10.1145/3233524","DOIUrl":"https://doi.org/10.1145/3233524","url":null,"abstract":"We consider a natural generalization of the classical multiple knapsack problem in which instead of packing single items we are packing groups of items. In this problem, we have multiple knapsacks and a set of items partitioned into groups. Each item has an individual weight, while the profit is associated with groups rather than items. The profit of a group can be attained if and only if every item of this group is packed. Such a general model finds applications in various practical problems, e.g., delivering bundles of goods. The tractability of this problem relies heavily on how large a group could be. Deciding if a group of items of total weight 2 could be packed into two knapsacks of unit capacity is already NP-hard and it thus rules out a constant-approximation algorithm for this problem in general. We then focus on the parameterized version where the total weight of items in each group is bounded by a factor δ of the total capacity of all knapsacks. Both approximation and inapproximability results with respect to δ are derived. We also show that, depending on whether the number of knapsacks is a constant or part of the input, the approximation ratio for the problem, as a function on δ, changes substantially, which has a clear difference from the classical multiple knapsack problem.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124195396","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}
Given an n-vertex undirected graph G = (V,E) and positive edge weights {we}e∈E, a linear arrangement is a permutation π : V → {1, 2, …, n}. The value of the arrangement is val(G, π) := 1/n∑ e ={u, v} ∈ E we|π(u) − π (v)|. In the minimum linear arrangement problem, the goal is to find a linear arrangement π * that achieves val(G, π*) = MLA(G) := min π val(G, π). In this article, we show that for any ε > 0 and positive integer r, there is an nO(r/ϵ)-time randomized algorithm that, given a graph G, returns a linear arrangement π, such that val(G, π) ≤ (1 + 2/(1 − ε)λr(L)) MLA(G) + O(√log n/n ∑ e ∈ E we) with high probability, where L is the normalized Laplacian of G and λr(L) is the rth smallest eigenvalue of L. Our algorithm gives a constant factor approximation for regular graphs that are weak expanders.
{"title":"Approximation Guarantees for the Minimum Linear Arrangement Problem by Higher Eigenvalues","authors":"Suguru Tamaki, Yuichi Yoshida","doi":"10.1145/3228342","DOIUrl":"https://doi.org/10.1145/3228342","url":null,"abstract":"Given an n-vertex undirected graph G = (V,E) and positive edge weights {we}e∈E, a linear arrangement is a permutation π : V → {1, 2, …, n}. The value of the arrangement is val(G, π) := 1/n∑ e ={u, v} ∈ E we|π(u) − π (v)|. In the minimum linear arrangement problem, the goal is to find a linear arrangement π * that achieves val(G, π*) = MLA(G) := min π val(G, π). In this article, we show that for any ε > 0 and positive integer r, there is an nO(r/ϵ)-time randomized algorithm that, given a graph G, returns a linear arrangement π, such that val(G, π) ≤ (1 + 2/(1 − ε)λr(L)) MLA(G) + O(√log n/n ∑ e ∈ E we) with high probability, where L is the normalized Laplacian of G and λr(L) is the rth smallest eigenvalue of L. Our algorithm gives a constant factor approximation for regular graphs that are weak expanders.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"544 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116509902","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}
How efficiently can we find an unknown graph using distance or shortest path queries between its vertices? We assume that the unknown graph G is connected, unweighted, and has bounded degree. In the reconstruction problem, the goal is to find the graph G. In the verification problem, we are given a hypothetical graph Ĝ and want to check whether G is equal to Ĝ. We provide a randomized algorithm for reconstruction using Õ(n3/2) distance queries, based on Voronoi cell decomposition. Next, we analyze natural greedy algorithms for reconstruction using a shortest path oracle and also for verification using either oracle, and show that their query complexity is n1+o(1). We further improve the query complexity when the graph is chordal or outerplanar. Finally, we show some lower bounds, and consider an approximate version of the reconstruction problem.
{"title":"Graph Reconstruction and Verification","authors":"Sampath Kannan, Claire Mathieu, Hang Zhou","doi":"10.1145/3199606","DOIUrl":"https://doi.org/10.1145/3199606","url":null,"abstract":"How efficiently can we find an unknown graph using distance or shortest path queries between its vertices? We assume that the unknown graph G is connected, unweighted, and has bounded degree. In the reconstruction problem, the goal is to find the graph G. In the verification problem, we are given a hypothetical graph Ĝ and want to check whether G is equal to Ĝ. We provide a randomized algorithm for reconstruction using Õ(n3/2) distance queries, based on Voronoi cell decomposition. Next, we analyze natural greedy algorithms for reconstruction using a shortest path oracle and also for verification using either oracle, and show that their query complexity is n1+o(1). We further improve the query complexity when the graph is chordal or outerplanar. Finally, we show some lower bounds, and consider an approximate version of the reconstruction problem.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126293528","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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