Time-Optimal Sublinear Algorithms for Matching and Vertex Cover

Behnezhad, Soheil

doi:10.1109/focs52979.2021.00089

Cited by 16 publications

(59 citation statements)

References 30 publications

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“…The maximum path cover in a graph is a collection of vertex disjoint paths with the maximum number of edges in it. The (almost) 1/2-approximate maximum matching size estimator of Behnezhad [2] immediately implies an (almost) 1/4-approximation for the maximum path cover problem in O(n) time. 2 This can be improved to an (almost) (3/8 = .375)approximation using the matching-pair idea of Chen, Kannan, and Khanna [8] in O(n √ n)-time.…”

Section: Maximum Path Covermentioning

confidence: 95%

“…The (almost) 1/2-approximate maximum matching size estimator of Behnezhad [2] immediately implies an (almost) 1/4-approximation for the maximum path cover problem in O(n) time. 2 This can be improved to an (almost) (3/8 = .375)approximation using the matching-pair idea of Chen, Kannan, and Khanna [8] in O(n √ n)-time. 3 Our first main contribution is an improvement over both of these results:…”

Section: Maximum Path Covermentioning

confidence: 95%

“…It is also worth noting that in bounding the running time of our algorithm in Result 1, we use connections to parallel algorithms. Such a connection was previously only used for matchings [2].…”

Section: Maximum Path Covermentioning

confidence: 99%

“…Open Problem 71 on sublinear.info [14]. 2 The application of sublinear time maximum matching algorithms for approximating maximum path cover was first proposed by Gupta and Onak. See [14].…”

Section: Maximum Path Covermentioning

confidence: 99%

See 3 more Smart Citations

Sublinear Algorithms for TSP via Path Covers

Behnezhad¹,

Roghani²,

Rubinstein³

et al. 2023

Preprint

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We study sublinear time algorithms for the traveling salesman problem (TSP). First, we focus on the closely related maximum path cover problem, which asks for a collection of vertex disjoint paths that include the maximum number of edges. We show that for any fixed ε > 0, there is an algorithm that (1/2 − ε)-approximates the maximum path cover size of an n-vertextime algorithm of Chen, Kannan, and Khanna [ICALP'20]. Equipped with our path cover algorithm, we give O(n) time algorithms that estimate the cost of graphic TSP and (1, 2)-TSP up to factors of 1.83 and (1.5 + ε), respectively. Our algorithm for graphic TSP improves over a 1.92-approximate O(n) time algorithm due to [CHK ICALP'20, Behnezhad FOCS'21]. Our algorithm for (1, 2)-TSP improves over a folklore (1.75 + ε)-approximate O(n)-time algorithm, as well as a (1.625 + ε)-approximate O(n √ n)-time algorithm of [CHK ICALP'20].Our analysis of the running time uses connections to parallel algorithms and is informationtheoretically optimal up to poly log n factors. Additionally, we show that our approximation guarantees for path cover and (1, 2)-TSP hit a natural barrier: We show better approximations require better sublinear time algorithms for the well-studied maximum matching problem.

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Section: Maximum Path Covermentioning

confidence: 95%

Section: Maximum Path Covermentioning

confidence: 95%

Section: Maximum Path Covermentioning

confidence: 99%

Section: Maximum Path Covermentioning

confidence: 99%

See 2 more Smart Citations

Sublinear Algorithms for TSP via Path Covers

Behnezhad¹,

Roghani²,

Rubinstein³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…In the literature, the first two types of queries form the adjacency list query model, while all three types of queries form the adjacency matrix query model. Under these models, a variety of graph estimation problems have been well studied, including edge counting and sampling [ER18, GR08, Ses, TT22], subgraph counting [ABG + 18, BER21, ERS20], vertex cover [Beh22,ORRR12], and beyond [Ron19].…”

Section: Introductionmentioning

confidence: 99%

Non-Adaptive Edge Counting and Sampling via Bipartite Independent Set Queries

Addanki¹,

McGregor²,

Musco³

2022

Preprint

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We study the problem of estimating the number of edges in an n-vertex graph, accessed via the Bipartite Independent Set query model introduced by Beame et al. (ITCS '18). In this model, each query returns a Boolean, indicating the existence of at least one edge between two specified sets of nodes. We present a non-adaptive algorithm that returns a (1 ± ǫ) relative error approximation to the number of edges, with query complexity O(ǫ −5 log 5 n), where O(•) hides poly(log log n) dependencies. This is the first non-adaptive algorithm in this setting achieving poly(1/ǫ, log n) query complexity. Prior work requires Ω(log 2 n) rounds of adaptivity. We avoid this by taking a fundamentally different approach, inspired by work on single-pass streaming algorithms. Moreover, for constant ǫ, our query complexity significantly improves on the best known adaptive algorithm due to Bhattacharya et al. (STACS '22), which requires O(ǫ −2 log 11 n) queries. Building on our edge estimation result, we give the first non-adaptive algorithm for outputting a nearly uniformly sampled edge with query complexity O(ǫ −6 log 6 n), improving on the works of Dell et al. (SODA '20) and Bhattacharya et al. (STACS '22), which require Ω(log 3 n) rounds of adaptivity. Finally, as a consequence of our edge sampling algorithm, we obtain a O(n log 8 n) query algorithm for connectivity, using two rounds of adaptivity. This improves on a three-round algorithm of Assadi et al. (ESA '21) and is tight; there is no non-adaptive algorithm for connectivity making o(n 2 ) queries.

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Accelerating approximate matrix multiplication for near-sparse matrices on GPUs

Liu

Yang

et al. 2022

J Supercomput

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We study connections between the problem of fully dynamic (1 − ǫ)-approximate maximum bipartite matching, and the dual (1 + ǫ)-approximate vertex cover problem, with the online matrix-vector (OMv) conjecture which has recently been used in several fine-grained hardness reductions. We prove that there is an online algorithm that maintains a (1 + ǫ)-approximate vertex cover in amortized n 1−c ǫ −C time for constants c, C > 0 for fully dynamic updates if and only if the OMv conjecture is false. Similarly, we prove that there is an online algorithm that maintains a (1 − ǫ)-approximate maximum matching in amortized n 1−c ǫ −C time if and only if there is a nontrivial algorithm for another dynamic problem, which we call dynamic approximate OMv, that has seemingly no matching structure. This provides some evidence against achieving amortized sublinear update times for approximate fully dynamic matching and vertex cover.Leveraging these connections, we obtain faster algorithms for approximate fully dynamic matching in both the online and offline settings.• We give a randomized offline algorithm for (1 − ǫ)-approximate maximum matching with amortized runtime O(n .58 ǫ −O(1) ) by using fast matrix multiplication, significantly improving over the runtimes achieved via online algorithms mentioned above. This mirrors the situation with OMv, where an offline algorithm exactly corresponds to fast matrix multiplication. We also give an offline algorithm that maintains a (1 + ǫ)-approximate vertex cover in amortized O(n .723 ǫ −O(1) ) time.

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Time-Optimal Sublinear Algorithms for Matching and Vertex Cover

Cited by 16 publications

References 30 publications

Sublinear Algorithms for TSP via Path Covers

Sublinear Algorithms for TSP via Path Covers

Non-Adaptive Edge Counting and Sampling via Bipartite Independent Set Queries

Accelerating approximate matrix multiplication for near-sparse matrices on GPUs

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