Maximum Matching in Two, Three, and a Few More Passes Over Graph Streams

Kale, Sagar; Tirodkar, Sumedh

doi:10.4230/lipics.approx-random.2017.15

Cited by 3 publications

(6 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…deterministic Greedy, folklore 2 2 − √ 2 ≈ 0.5857 randomized Konrad [21] [22,13,18] 3 0.6067 randomized Konrad [21] [ 13,18] 2 3ǫ 2 3 − ǫ deterministic Kale and Tirodkar [18] [14] O( 1 ǫ 2 log log ǫ) 1 − ǫ deterministic Ahn and Guha [2] [12] General Graphs 1…”

Section: # Passes Approximation Det/randmentioning

confidence: 99%

“…1. A 3-pass 0.6-approximation streaming algorithm for bipartite graphs that is implicit in [14], explicitly mentioned in [22], and analyzed in [18]. Given a bipartite graph G = (A, B, E), the algorithm first computes a maximal matching in G, i.e., M ← Greedy(G).…”

Section: # Passes Approximation Det/randmentioning

confidence: 99%

“…Given a bipartite graph G = (A, B, E), the algorithm first computes a maximal matching in G, i.e., M ← Greedy(G). Then, the algorithm attempts to find length-3 augmenting paths by invoking Greedy twice more: showed that M ∪ M L ∪ M R contains a 0.6-approximate matching [18]. We will denote this algorithm by 3RoundMatch.…”

Section: # Passes Approximation Det/randmentioning

confidence: 99%

“…The Greedy matching algorithm is the key building block of most published streaming algorithms for approximate Maximum Matching [14,23,12,22,2,13,18,21]. Given a graph G = (V, E), Greedy scans the set of edges E in arbitrary order and inserts the current edge e ∈ E into an initially empty matching M if possible, i.e., if both endpoints of e are not yet matched by an edge in M .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Constructing Large Matchings via Query Access to a Maximal Matching Oracle

Khalil¹,

Konrad²

2020

Preprint

View full text Add to dashboard Cite

Multi-pass streaming algorithm for Maximum Matching have been studied since more than 15 years and various algorithmic results are known today, including 2-pass streaming algorithms that break the 1/2-approximation barrier, and (1 − ǫ)-approximation streaming algorithms that run in O(poly 1 ǫ ) passes in bipartite graphs and in O(passes in general graphs, where n is the number of vertices of the input graph. However, proving impossibility results for such algorithms has so far been elusive, and, for example, even the existence of 2-pass small space streaming algorithms with approximation factor 0.999 has not yet been ruled out.The key building block of all multi-pass streaming algorithms for Maximum Matching is the Greedy matching algorithm. Our aim is to understand the limitations of this approach: How many passes are required if the algorithm solely relies on the invocation of the Greedy algorithm? In this paper, we initiate the study of lower bounds for restricted families of multi-pass streaming algorithms for Maximum Matching. We focus on the simple yet powerful class of algorithms that in each pass run Greedy on a vertex-induced subgraph of the input graph. In bipartite graphs, we show that 3 passes are necessary and sufficient to improve on the trivial approximation factor of 1/2: We give a lower bound of 0.6 on the approximation ratio of such algorithms, which is optimal. We further show that Ω( 1 ǫ ) passes are required for computing a (1 − ǫ)-approximation, even in bipartite graphs. Last, the considered class of algorithms is not well-suited to general graphs: We show that Ω(n) passes are required in order to improve on the trivial approximation factor of 1/2.

show abstract

Section: # Passes Approximation Det/randmentioning

confidence: 99%

Section: # Passes Approximation Det/randmentioning

confidence: 99%

Section: # Passes Approximation Det/randmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Constructing Large Matchings via Query Access to a Maximal Matching Oracle

Khalil¹,

Konrad²

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…The insertion-only model has received significant attention, and many problems, such as matchings (e.g. [26,13,21,27,18,24,31,12]), independent sets (e.g. [15,14,9,10]), and subgraph counting (e.g.…”

Section: Introductionmentioning

confidence: 99%

Optimal Lower Bounds for Matching and Vertex Cover in Dynamic Graph Streams

Dark,

Konrad

2020

Preprint

View full text Add to dashboard Cite

In this paper, we give simple optimal lower bounds on the one-way two-party communication complexity of approximate Maximum Matching and Minimum Vertex Cover with deletions. In our model, Alice holds a set of edges and sends a single message to Bob. Bob holds a set of edge deletions, which form a subset of Alice's edges, and needs to report a large matching or a small vertex cover in the graph spanned by the edges that are not deleted. Our results imply optimal space lower bounds for insertion-deletion streaming algorithms for Maximum Matching and Minimum Vertex Cover. Previously, Assadi et al. [SODA 2016] gave an optimal space lower bound for insertion-deletion streaming algorithms for Maximum Matching via the simultaneous model of communication. Our lower bound is simpler and stronger in several aspects: The lower bound of Assadi et al. only holds for algorithms that (1) are able to process streams that contain a triple exponential number of deletions in n, the number of vertices of the input graph; (2) are able to process multi-graphs; and(3) never output edges that do not exist in the input graph when the randomized algorithm errs. In contrast, our lower bound even holds for algorithms that (1) rely on short (O(n 2 )-length) input streams; (2) are only able to process simple graphs; and (3) may output non-existing edges when the algorithm errs.

show abstract

Maximum Matching Sans Maximal Matching: A New Approach for Finding Maximum Matchings in the Data Stream Model

Feldman,

Szarf

2023

Algorithmica

View full text Add to dashboard Cite

The problem of finding a maximum size matching in a graph (known as the maximum matching problem) is one of the most classical problems in computer science. Despite a significant body of work dedicated to the study of this problem in the data stream model, the state-of-the-art single-pass semi-streaming algorithm for it is still a simple greedy algorithm that computes a maximal matching, and this way obtains 1 /2-approximation. Some previous works described two/three-pass algorithms that improve over this approximation ratio by using their second and third passes to improve the above mentioned maximal matching. One contribution of this paper continues this line of work by presenting new three-pass semi-streaming algorithms that work along these lines and obtain improved approximation ratios of 0.6111 and 0.5694 for triangle-free and general graphs, respectively.Unfortunately, a recent work [30] shows that the strategy of constructing a maximal matching in the first pass and then improving it in further passes has limitations. Additionally, this technique is unlikely to get us closer to single-pass semi-streaming algorithms obtaining a better than 1 /2approximation. Therefore, it is interesting to come up with algorithms that do something else with their first pass (we term such algorithms non-maximal-matching-first algorithms). No such algorithms are currently known (to the best of our knowledge), and the main contribution of this paper is describing such algorithms that obtain approximation ratios of 0.5384 and 0.5555 in two and three passes, respectively, for general graphs (the result for three passes improves over the previous state-of-the-art, but is worse than the result of this paper mentioned in the previous paragraph for general graphs). The improvements obtained by these results are, unfortunately, numerically not very impressive, but the main importance (in our opinion) of these results is in demonstrating the potential of non-maximal-matching-first algorithms.

show abstract

Maximum Matching in Two, Three, and a Few More Passes Over Graph Streams

Cited by 3 publications

References 0 publications

Constructing Large Matchings via Query Access to a Maximal Matching Oracle

Constructing Large Matchings via Query Access to a Maximal Matching Oracle

Optimal Lower Bounds for Matching and Vertex Cover in Dynamic Graph Streams

Maximum Matching Sans Maximal Matching: A New Approach for Finding Maximum Matchings in the Data Stream Model

Contact Info

Product

Resources

About