A (2 + ∊)-Approximation for Maximum Weight Matching in the Semi-Streaming Model

Paz, Ami; Schwartzman, Gregory

doi:10.1137/1.9781611974782.140

Cited by 35 publications

(58 citation statements)

References 27 publications

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“…They showed that a modification of the greedy algorithm can find a 1/6 approximation of the maximum weighted matching in the semi-streaming model. This bound later improved in a series of works [17,21,5,4,18,10] to (1/2 − ǫ) approximation. A recent work by Gamlath et al [9] shows that when edges arrive in a random order, there exists a semi-streaming algorithm that finds a 1/2 + Ω(1) approximation of the maximum weighted matching, beating the 1/2 barrier for the weighted case.…”

Section: Related Workmentioning

confidence: 96%

Approximate Maximum Matching in Random Streams

Farhadi¹,

Hajiaghayi²,

Mai³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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In this paper, we study the problem of finding a maximum matching in the semi-streaming model when edges arrive in a random order. In the semi-streaming model, an algorithm receives a stream of edges and it is allowed to have a memory ofÕ(n) 1 where n is the number of vertices in the graph. A recent inspiring work by Assadi et al. [1] shows that there exists a streaming algorithm with the approximation ratio of 2 3 that usesÕ(n 1.5 ) memory. However, the memory of their algorithm is much larger than the memory constraint of the semi-streaming algorithms. In this work, we further investigate this problem in the semi-streaming model, and we present simple algorithms for approximating maximum matching in the semi-streaming model. Our main results are as follows.• We show that there exists a single-pass deterministic semi-streaming algorithm that finds a 3 5 (= 0.6) approximation of the maximum matching in bipartite graphs usingÕ(n) memory. This result significantly outperforms the state-of-the-art result of Konrad [15] that finds a 0.539 approximation of the maximum matching usingÕ(n) memory.• By giving a black-box reduction from finding a matching in general graphs to finding a matching in bipartite graphs, we show there exists a single-pass deterministic semistreaming algorithm that finds a 6 11 (≈ 0.545) approximation of the maximum matching in general graphs, improving upon the state-of-art result 0.506 approximation by Gamlath et al. [9]. notion to hide logarithmic factors.2 With high probability

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Section: Related Workmentioning

confidence: 96%

Approximate Maximum Matching in Random Streams

Farhadi¹,

Hajiaghayi²,

Mai³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In contrast to unweighted graphs where the basic greedy algorithm gives a ( 1 /2)-approximation, it was only very recently that a ( 1 /2 − ε)-approximation streaming algorithm was given for weighted matchings [PS17]. The algorithm of Paz and Schwartzman is based on the local ratio technique, which we now describe 3 .…”

Section: Single-pass Streaming With Random Edge Arrivalsmentioning

confidence: 99%

“…The issue is that, if the edges arrive in an adversarial order, we may add all the edges to S. For dense graphs, this would lead to a memory consumption of Ω(n 2 ) instead of the wanted memory usage O(n poly(log n)) which is (roughly) linear in the output size. The main technical challenge in [PS17] is to limit the number of edges added to S; this is why that algorithm obtains a ( 1 /2 − ε)-approximation, for any ε > 0, instead of a ( 1 /2)-approximation.…”

Section: Single-pass Streaming With Random Edge Arrivalsmentioning

confidence: 99%

“…There is a large body of work devoted to (semi-)streaming algorithms for the maximum matching problem. For unweighted graphs, the basic greedy approach yields a ( 1 /2)-approximation, and for weighted graphs [PS17] recently gave a ( 1 /2 − ε)-approximation based on the local ratio technique. These are the best known algorithms that take a single pass over an adversarially ordered stream.…”

Section: Further Related Workmentioning

confidence: 99%

“…In the random-edgearrival setting -where the edges of the stream are presented in a random order -algorithms that are more advanced than the greedy algorithm overcome this barrier [KMM12]. In contrast, for weighted graphs a ( 1 /2−ε)-approximation algorithm was given only recently for adversarial streams [PS17,GW19], and here we give the first algorithm that breaks the natural "greedy" barrier of 1 /2 for random-edge-arrival streams:…”

Section: Introductionmentioning

confidence: 99%

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Weighted Matchings via Unweighted Augmentations

Gamlath

Kale

Mitrović

et al. 2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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We design a generic method for reducing the task of finding weighted matchings to that of finding short augmenting paths in unweighted graphs. This method enables us to provide efficient implementations for approximating weighted matchings in the streaming model and in the massively parallel computation (MPC) model.In the context of streaming with random edge arrivals, our techniques yield a ( 1 /2+c)-approximation algorithm thus breaking the natural barrier of 1 /2. For multi-pass streaming and the MPC model, we show that any algorithm computing a (1 − δ)-approximate unweighted matching in bipartite graphs can be translated into an algorithm that computes a (1 − ε(δ))-approximate maximum weighted matching. Furthermore, this translation incurs only a constant factor (that depends on ε > 0) overhead in the complexity. Instantiating this with the current best multi-pass streaming and MPC algorithms for unweighted matchings yields the following results for maximum weighted matchings:• A (1 − ε)-approximation streaming algorithm that uses Oε(1) passes 1 and Oε(n poly(log n)) memory. This is the first (1 − ε)-approximation streaming algorithm for weighted matchings that uses a constant number of passes (only depending on ε).• A (1 − ε)-approximation algorithm in the MPC model that uses Oε(log log n) rounds, O(m/n) machines per round, and Oε(n poly(log n)) memory per machine. This improves upon the previous best approximation guarantee of ( 1 /2 − ε) for weighted graphs. 1 We use Oε(f (n)) to denote a function that is O(f (n)) when the parameter ε is a constant.1 some of the ideas present in the streaming algorithms for unweighted matchings to achieve our result. Our techniques, perhaps surprisingly, also simplify the previous algorithms for finding unweighted matchings, and give an improved guarantee for general graphs. The idea to reduce to the problem of finding unweighted augmenting paths is rather versatile, and we use it to obtain a general reduction from weighted matchings to unweighted matchings as our second main result. We give implementations of this reduction in the models of multi-pass streaming and MPC that incur only a constant factor overhead in the complexity. In multi-pass streaming, the algorithm is (as for single-pass) restricted to use memory that is almost linear in the number of vertices and the complexity is measured in terms of the number of passes that the algorithm requires over the data stream. In MPC, parallel computation is modeled by parallel machines with sublinear memory (in the input size) and data can be transferred between machines only between two rounds of computation (see Section 2 for a precise definition). The complexity of an algorithm in the MPC model, also referred to as the round complexity, is then measured as the number of (communication) rounds used.Both the streaming model and the MPC model, which encompasses many of today's most successful parallel computing paradigms such as MapReduce and Hadoop, are motivated by the need for devising efficient algorithms for large pr...

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Semi-streaming Algorithms for Submodular Matroid Intersection

Garg

Jordan

Svensson

2021

Lecture Notes in Computer Science

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While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of 2 for the unweighted matching problem, it was only recently that Paz and Schwartzman obtained an analogous result for weighted instances. Their approach is based on the versatile local ratio technique and also applies to generalizations such as weighted hypergraph matchings. However, the framework for the analysis fails for the related problem of weighted matroid intersection and as a result the approximation guarantee for weighted instances did not match the factor 2 achieved by the greedy algorithm for unweighted instances.Our main result closes this gap by developing a semi-streaming algorithm with an approximation guarantee of 2 + ε for weighted matroid intersection, improving upon the previous best guarantee of 4 + ε. Our techniques also allow us to generalize recent results by Levin and Wajc on submodular maximization subject to matching constraints to that of matroid-intersection constraints. While our algorithm is an adaptation of the local ratio technique used in previous works, the analysis deviates significantly and relies on structural properties of matroid intersection, called kernels. Finally, we also conjecture that our algorithm gives a (k + ε) approximation for the intersection of k matroids but prove that new tools are needed in the analysis as the structural properties we use fail for k ≥ 3.

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A (2 + ∊)-Approximation for Maximum Weight Matching in the Semi-Streaming Model

Cited by 35 publications

References 27 publications

Approximate Maximum Matching in Random Streams

Approximate Maximum Matching in Random Streams

Weighted Matchings via Unweighted Augmentations

Semi-streaming Algorithms for Submodular Matroid Intersection

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