Yi-Jun Chang, Qizheng He, Wenzheng Li, S. Pettie, Jara Uitto
The complexity of distributed edge coloring depends heavily on the palette size as a function of the maximum degree Δ. In this article, we explore the complexity of edge coloring in the LOCAL model in different palette size regimes. Our results are as follows. Lower Bounds: First, we simplify the round elimination technique of Brandt et al. [16] and prove that (2Δ −2)-edge coloring requires Ω (logΔ log n) time with high probability and Ω (logΔ n) time deterministically, even on trees. Second, we show that a natural approach to computing (Δ +1)-edge colorings (Vizing’s theorem), namely, extending an arbitrary partial coloring by iteratively recoloring subgraphs, requires Ω (Δ log n) time. Upper Bounds on General Graphs: We give a randomized edge coloring algorithm that can use palette sizes as small as Δ + Õ(√Δ), which is a natural barrier for randomized approaches. The running time of our (1+ε)Δ-edge coloring algorithm is usually dominated by O(log ε−1) calls to a distributed Lovász local lemma (LLL) algorithm. For example, using the Chung-Pettie-Su LLL algorithm, we compute a (1+ε)Δ-edge coloring in O(log n) time when ε ≥ (log3 Δ) / √ Δ , or O(logΔ n) + (log log n)3 + o(1) time when ε = Ω (1). When Δ is sublogarithmic in n the performance is improved with the Ghaffari-Harris-Kuhn LLL algorithm. Upper Bounds on Trees: We show that the Ω (logΔ log n) lower bound can be nearly matched on trees. To establish this result, we develop a new distributed Lovász local lemma algorithm for tree-structured dependency graphs, which arise naturally from O(1)-round probabilistic algorithms run on trees. Specifically, our (1+ε)Δ-edge coloring algorithm for trees takes O(log (1 / ε)) ⋅ max { log log n log log log n, loglog Δ log n} time when ε ≥ (log3 Δ) / √ Δ, or O(max { log log n log log log n, logΔ log n}) time when ε = Ω (1).
{"title":"Distributed Edge Coloring and a Special Case of the Constructive Lovász Local Lemma","authors":"Yi-Jun Chang, Qizheng He, Wenzheng Li, S. Pettie, Jara Uitto","doi":"10.1145/3365004","DOIUrl":"https://doi.org/10.1145/3365004","url":null,"abstract":"The complexity of distributed edge coloring depends heavily on the palette size as a function of the maximum degree Δ. In this article, we explore the complexity of edge coloring in the LOCAL model in different palette size regimes. Our results are as follows. Lower Bounds: First, we simplify the round elimination technique of Brandt et al. [16] and prove that (2Δ −2)-edge coloring requires Ω (logΔ log n) time with high probability and Ω (logΔ n) time deterministically, even on trees. Second, we show that a natural approach to computing (Δ +1)-edge colorings (Vizing’s theorem), namely, extending an arbitrary partial coloring by iteratively recoloring subgraphs, requires Ω (Δ log n) time. Upper Bounds on General Graphs: We give a randomized edge coloring algorithm that can use palette sizes as small as Δ + Õ(√Δ), which is a natural barrier for randomized approaches. The running time of our (1+ε)Δ-edge coloring algorithm is usually dominated by O(log ε−1) calls to a distributed Lovász local lemma (LLL) algorithm. For example, using the Chung-Pettie-Su LLL algorithm, we compute a (1+ε)Δ-edge coloring in O(log n) time when ε ≥ (log3 Δ) / √ Δ , or O(logΔ n) + (log log n)3 + o(1) time when ε = Ω (1). When Δ is sublogarithmic in n the performance is improved with the Ghaffari-Harris-Kuhn LLL algorithm. Upper Bounds on Trees: We show that the Ω (logΔ log n) lower bound can be nearly matched on trees. To establish this result, we develop a new distributed Lovász local lemma algorithm for tree-structured dependency graphs, which arise naturally from O(1)-round probabilistic algorithms run on trees. Specifically, our (1+ε)Δ-edge coloring algorithm for trees takes O(log (1 / ε)) ⋅ max { log log n log log log n, loglog Δ log n} time when ε ≥ (log3 Δ) / √ Δ, or O(max { log log n log log log n, logΔ log n}) time when ε = Ω (1).","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115953415","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}
Gopal Pandurangan, Peter Robinson, Michele Scquizzato
This article presents a randomized (Las Vegas) distributed algorithm that constructs a minimum spanning tree (MST) in weighted networks with optimal (up to polylogarithmic factors) time and message complexity. This algorithm runs in Õ(D + √ n) time and exchanges Õ(m) messages (both with high probability), where n is the number of nodes of the network, D is the hop-diameter, and m is the number of edges. This is the first distributed MST algorithm that matches simultaneously the time lower bound of Ω˜(D + √ n) [10] and the message lower bound of Ω (m) [31], which both apply to randomized Monte Carlo algorithms. The prior time and message lower bounds are derived using two completely different graph constructions; the existing lower-bound construction that shows one lower bound does not work for the other. To complement our algorithm, we present a new lower-bound graph construction for which any distributed MST algorithm requires both Ω˜(D + √ n) rounds and Ω (m) messages.
{"title":"A Time- and Message-Optimal Distributed Algorithm for Minimum Spanning Trees","authors":"Gopal Pandurangan, Peter Robinson, Michele Scquizzato","doi":"10.1145/3365005","DOIUrl":"https://doi.org/10.1145/3365005","url":null,"abstract":"This article presents a randomized (Las Vegas) distributed algorithm that constructs a minimum spanning tree (MST) in weighted networks with optimal (up to polylogarithmic factors) time and message complexity. This algorithm runs in Õ(D + √ n) time and exchanges Õ(m) messages (both with high probability), where n is the number of nodes of the network, D is the hop-diameter, and m is the number of edges. This is the first distributed MST algorithm that matches simultaneously the time lower bound of Ω˜(D + √ n) [10] and the message lower bound of Ω (m) [31], which both apply to randomized Monte Carlo algorithms. The prior time and message lower bounds are derived using two completely different graph constructions; the existing lower-bound construction that shows one lower bound does not work for the other. To complement our algorithm, we present a new lower-bound graph construction for which any distributed MST algorithm requires both Ω˜(D + √ n) rounds and Ω (m) messages.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130729848","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}
Often in a scheduling problem, there is uncertainty about the jobs to be processed. The issue of uncertainty regarding the machines has been much less studied. In this article, we study a scheduling environment in which jobs first need to be grouped into some sets before the number of machines is known, and then the sets need to be scheduled on machines without being separated. To evaluate algorithms in such an environment, we introduce the idea of an α-robust algorithm, one that is guaranteed to return a schedule on any number m of machines that is within an α factor of the optimal schedule on m machine, where the optimum is not subject to the restriction that the sets cannot be separated. Under such environment, we give a (53+ε)-robust algorithm for scheduling on parallel machines to minimize makespan and show a lower bound 43. For the special case when the jobs are infinitesimal, we give a 1.233-robust algorithm with an asymptotic lower bound of 1.207. We also study a case of fair allocation, where the objective is to minimize the difference between the maximum and minimum machine load.
{"title":"Scheduling When You Do Not Know the Number of Machines","authors":"C. Stein, Mingxian Zhong","doi":"10.1145/3340320","DOIUrl":"https://doi.org/10.1145/3340320","url":null,"abstract":"Often in a scheduling problem, there is uncertainty about the jobs to be processed. The issue of uncertainty regarding the machines has been much less studied. In this article, we study a scheduling environment in which jobs first need to be grouped into some sets before the number of machines is known, and then the sets need to be scheduled on machines without being separated. To evaluate algorithms in such an environment, we introduce the idea of an α-robust algorithm, one that is guaranteed to return a schedule on any number m of machines that is within an α factor of the optimal schedule on m machine, where the optimum is not subject to the restriction that the sets cannot be separated. Under such environment, we give a (53+ε)-robust algorithm for scheduling on parallel machines to minimize makespan and show a lower bound 43. For the special case when the jobs are infinitesimal, we give a 1.233-robust algorithm with an asymptotic lower bound of 1.207. We also study a case of fair allocation, where the objective is to minimize the difference between the maximum and minimum machine load.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125058326","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 present a factor 4/3 approximation algorithm for the problem of finding a minimum 2-edge connected spanning subgraph of a given undirected multigraph. The algorithm is based upon a reduction to a restricted class of graphs. In these graphs, the approximation algorithm constructs a 2-edge connected spanning subgraph by modifying the smallest 2-edge cover.
{"title":"A 4/3-Approximation Algorithm for the Minimum 2-Edge Connected Subgraph Problem","authors":"Christoph Hunkenschröder, S. Vempala, A. Vetta","doi":"10.1145/3341599","DOIUrl":"https://doi.org/10.1145/3341599","url":null,"abstract":"We present a factor 4/3 approximation algorithm for the problem of finding a minimum 2-edge connected spanning subgraph of a given undirected multigraph. The algorithm is based upon a reduction to a restricted class of graphs. In these graphs, the approximation algorithm constructs a 2-edge connected spanning subgraph by modifying the smallest 2-edge cover.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117008985","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 inverse geodesic length of a graph G is the sum of the inverse of the distances between all pairs of distinct vertices of G. In some domains, it is known as the Harary index or the global efficiency of the graph. We show that, if G is planar and has n vertices, then the inverse geodesic length of G can be computed in roughly O(n9/5) time. We also show that, if G has n vertices and treewidth at most k, then the inverse geodesic length of G can be computed in O(n log O(k)n) time. In both cases, we use techniques developed for computing the sum of the distances, which does not have “inverse” component, together with batched evaluations of rational functions.
{"title":"Computing the Inverse Geodesic Length in Planar Graphs and Graphs of Bounded Treewidth","authors":"Sergio Cabello","doi":"10.1145/3501303","DOIUrl":"https://doi.org/10.1145/3501303","url":null,"abstract":"The inverse geodesic length of a graph G is the sum of the inverse of the distances between all pairs of distinct vertices of G. In some domains, it is known as the Harary index or the global efficiency of the graph. We show that, if G is planar and has n vertices, then the inverse geodesic length of G can be computed in roughly O(n9/5) time. We also show that, if G has n vertices and treewidth at most k, then the inverse geodesic length of G can be computed in O(n log O(k)n) time. In both cases, we use techniques developed for computing the sum of the distances, which does not have “inverse” component, together with batched evaluations of rational functions.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"184 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122303882","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}
Massimo Cairo, P. Medvedev, Nidia Obscura Acosta, Romeo Rizzi, Alexandru I. Tomescu
In this article, we consider the following problem. Given a directed graph G, output all walks of G that are sub-walks of all closed edge-covering walks of G. This problem was first considered by Tomescu and Medvedev (RECOMB 2016), who characterized these walks through the notion of omnitig. Omnitigs were shown to be relevant for the genome assembly problem from bioinformatics, where a genome sequence must be assembled from a set of reads from a sequencing experiment. Tomescu and Medvedev (RECOMB 2016) also proposed an algorithm for listing all maximal omnitigs, by launching an exhaustive visit from every edge. In this article, we prove new insights about the structure of omnitigs and solve several open questions about them. We combine these to achieve an O(nm)-time algorithm for outputting all the maximal omnitigs of a graph (with n nodes and m edges). This is also optimal, as we show families of graphs whose total omnitig length is Ω(nm). We implement this algorithm and show that it is 9--12 times faster in practice than the one of Tomescu and Medvedev (RECOMB 2016).
{"title":"An Optimal O(nm) Algorithm for Enumerating All Walks Common to All Closed Edge-covering Walks of a Graph","authors":"Massimo Cairo, P. Medvedev, Nidia Obscura Acosta, Romeo Rizzi, Alexandru I. Tomescu","doi":"10.1145/3341731","DOIUrl":"https://doi.org/10.1145/3341731","url":null,"abstract":"In this article, we consider the following problem. Given a directed graph G, output all walks of G that are sub-walks of all closed edge-covering walks of G. This problem was first considered by Tomescu and Medvedev (RECOMB 2016), who characterized these walks through the notion of omnitig. Omnitigs were shown to be relevant for the genome assembly problem from bioinformatics, where a genome sequence must be assembled from a set of reads from a sequencing experiment. Tomescu and Medvedev (RECOMB 2016) also proposed an algorithm for listing all maximal omnitigs, by launching an exhaustive visit from every edge. In this article, we prove new insights about the structure of omnitigs and solve several open questions about them. We combine these to achieve an O(nm)-time algorithm for outputting all the maximal omnitigs of a graph (with n nodes and m edges). This is also optimal, as we show families of graphs whose total omnitig length is Ω(nm). We implement this algorithm and show that it is 9--12 times faster in practice than the one of Tomescu and Medvedev (RECOMB 2016).","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"104 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132741628","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 the problem of finding a cycle in a sparse directed graph G that is promised to be far from acyclic, meaning that the smallest feedback arc set, i.e., a subset of edges whose deletion results in an acyclic graph, in G is large. We prove an information-theoretic lower bound, showing that for N-vertex graphs with constant outdegree, any algorithm for this problem must make Ω̄(N5/9) queries to an adjacency list representation of G. In the language of property testing, our result is an Ω̄(N5/9) lower bound on the query complexity of one-sided algorithms for testing whether sparse digraphs with constant outdegree are far from acyclic. This is the first improvement on the Ω (√ N) lower bound, implicit in the work of Bender and Ron, which follows from a simple birthday paradox argument.
{"title":"A Lower Bound on Cycle-Finding in Sparse Digraphs","authors":"Xi Chen, T. Randolph, R. Servedio, Timothy Sun","doi":"10.1145/3417979","DOIUrl":"https://doi.org/10.1145/3417979","url":null,"abstract":"We consider the problem of finding a cycle in a sparse directed graph G that is promised to be far from acyclic, meaning that the smallest feedback arc set, i.e., a subset of edges whose deletion results in an acyclic graph, in G is large. We prove an information-theoretic lower bound, showing that for N-vertex graphs with constant outdegree, any algorithm for this problem must make Ω̄(N5/9) queries to an adjacency list representation of G. In the language of property testing, our result is an Ω̄(N5/9) lower bound on the query complexity of one-sided algorithms for testing whether sparse digraphs with constant outdegree are far from acyclic. This is the first improvement on the Ω (√ N) lower bound, implicit in the work of Bender and Ron, which follows from a simple birthday paradox argument.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127323100","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}
A retraction is a homomorphism from a graph G to an induced subgraph H of G that is the identity on H. In a long line of research, retractions have been studied under various algorithmic settings. Recently, the problem of approximately counting retractions was considered. We give a complete trichotomy for the complexity of approximately counting retractions to all square-free graphs (graphs that do not contain a cycle of length 4). It turns out there is a rich and interesting class of graphs for which this problem is complete in the class #BIS. As retractions generalise homomorphisms, our easiness results extend to the important problem of approximately counting homomorphisms. By giving new #BIS-easiness results, we now settle the complexity of approximately counting homomorphisms for a whole class of non-trivial graphs that were previously unresolved.
缩回是图G到G的诱导子图H的同态，该子图H是H上的恒等。在一系列的研究中，人们研究了各种算法设置下的缩回。近年来，研究了撤稿的近似计数问题。我们给出了所有无平方图(不包含长度为4的循环的图)的近似计数收缩的复杂性的完全三分法。事实证明，有一个丰富而有趣的图类，这个问题在类#BIS中是完整的。由于缩回推广了同态，我们的简单结果推广到近似计数同态的重要问题。通过给出新的# bis - easy结果，我们现在解决了一类以前未解决的非平凡图的近似计数同态的复杂性。
{"title":"The Complexity of Approximately Counting Retractions to Square-free Graphs","authors":"Jacob Focke, L. A. Goldberg, Stanislav Živný","doi":"10.1145/3458040","DOIUrl":"https://doi.org/10.1145/3458040","url":null,"abstract":"A retraction is a homomorphism from a graph G to an induced subgraph H of G that is the identity on H. In a long line of research, retractions have been studied under various algorithmic settings. Recently, the problem of approximately counting retractions was considered. We give a complete trichotomy for the complexity of approximately counting retractions to all square-free graphs (graphs that do not contain a cycle of length 4). It turns out there is a rich and interesting class of graphs for which this problem is complete in the class #BIS. As retractions generalise homomorphisms, our easiness results extend to the important problem of approximately counting homomorphisms. By giving new #BIS-easiness results, we now settle the complexity of approximately counting homomorphisms for a whole class of non-trivial graphs that were previously unresolved.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123291374","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}
An independent transversal (IT) in a graph with a given vertex partition is an independent set consisting of one vertex in each partition class. Several sufficient conditions are known for the existence of an IT in a given graph and vertex partition, which have been used over the years to solve many combinatorial problems. Some of these IT existence theorems have algorithmic proofs, but there remains a gap between the best existential bounds and the bounds obtainable by efficient algorithms. Recently, Graf and Haxell (2018) described a new (deterministic) algorithm that asymptotically closes this gap, but there are limitations on its applicability. In this article, we develop a randomized algorithm that is much more widely applicable, and demonstrate its use by giving efficient algorithms for two problems concerning the strong chromatic number of graphs.
{"title":"Algorithms for Weighted Independent Transversals and Strong Colouring","authors":"Alessandra Graf, David G. Harris, P. Haxell","doi":"10.1145/3474057","DOIUrl":"https://doi.org/10.1145/3474057","url":null,"abstract":"An independent transversal (IT) in a graph with a given vertex partition is an independent set consisting of one vertex in each partition class. Several sufficient conditions are known for the existence of an IT in a given graph and vertex partition, which have been used over the years to solve many combinatorial problems. Some of these IT existence theorems have algorithmic proofs, but there remains a gap between the best existential bounds and the bounds obtainable by efficient algorithms. Recently, Graf and Haxell (2018) described a new (deterministic) algorithm that asymptotically closes this gap, but there are limitations on its applicability. In this article, we develop a randomized algorithm that is much more widely applicable, and demonstrate its use by giving efficient algorithms for two problems concerning the strong chromatic number of graphs.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133352969","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 introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and we prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1/e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time increases, and each online vertex matches the neighbor with the highest offer at its arrival.
{"title":"Online Vertex-Weighted Bipartite Matching","authors":"Zhiyi Huang, Zhihao Gavin Tang, Xiaowei Wu, Yuhao Zhang","doi":"10.1145/3326169","DOIUrl":"https://doi.org/10.1145/3326169","url":null,"abstract":"We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and we prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1/e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time increases, and each online vertex matches the neighbor with the highest offer at its arrival.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117013428","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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