Parameterized Streaming: Maximal Matching and Vertex Cover

Chitnis, Rajesh; Cormode, Graham; Hajiaghayi, Mohammad Taghi; Monemizadeh, Morteza

doi:10.1137/1.9781611973730.82

Cited by 43 publications

(56 citation statements)

References 25 publications

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“…Most relevant to our work are the linear sketches of [20] for computing an exact minimum vertex cover or maximum matching in O(opt 2 ) space (opt is the size of the solution), and linear sketches of [10,20] for α-approximating maximum matching in O(n 2 /α 3 ) space. These results are proven to be tight by [21], and [10], respectively. Finally, [10] also studied the simultaneous communication complexity of bipartite matching in the vertex-partition model and proved that obtaining better than an O( √ k)-approximation in this model requires strictly more than O(n) communication from each player (see [10] for more details on this model).…”

Section: Further Related Workmentioning

confidence: 74%

Randomized Composable Coresets for Matching and Vertex Cover

Assadi

Khanna

2017

Proceedings of the 29th ACM Symposium on Parallelism in Algorithms and Architectures

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A common approach for designing scalable algorithms for massive data sets is to distribute the computation across, say k, machines and process the data using limited communication between them. A particularly appealing framework here is the simultaneous communication model whereby each machine constructs a small representative summary of its own data and one obtains an approximate/exact solution from the union of the representative summaries. If the representative summaries needed for a problem are small, then this results in a communication-efficient and round-optimal (requiring essentially no interaction between the machines) protocol. Some well-known examples of techniques for creating summaries include sampling, linear sketching, and composable coresets. These techniques have been successfully used to design communication efficient solutions for many fundamental graph problems. However, two prominent problems are notably absent from the list of successes, namely, the maximum matching problem and the minimum vertex cover problem. Indeed, it was shown recently that for both these problems, even achieving a modest approximation factor of polylog(n) requires using representative summaries of size Ω(n 2 ) i.e. essentially no better summary exists than each machine simply sending its entire input graph.The main insight of our work is that the intractability of matching and vertex cover in the simultaneous communication model is inherently connected to an adversarial partitioning of the underlying graph across machines. We show that when the underlying graph is randomly partitioned across machines, both these problems admit randomized composable coresets of size O(n) that yield an O(1)-approximate solution 1 . In other words, a small subgraph of the input graph at each machine can be identified as its representative summary and the final answer then is obtained by simply running any maximum matching or minimum vertex cover algorithm on these combined subgraphs. This results in an O(1)-approximation simultaneous protocol for these problems with O(nk) total communication when the input is randomly partitioned across k machines. We also prove our results are optimal in a very strong sense: we not only rule out existence of smaller randomized composable coresets for these problems but in fact show that our O(nk) bound for total communication is optimal for any simultaneous communication protocol (i.e. not only for randomized coresets) for these two problems. Finally, by a standard application of composable coresets, our results also imply MapReduce algorithms with the same approximation guarantee in one or two rounds of communication, improving the previous best known round complexity for these problems.

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Section: Further Related Workmentioning

confidence: 74%

Randomized Composable Coresets for Matching and Vertex Cover

Assadi

Khanna

2017

Proceedings of the 29th ACM Symposium on Parallelism in Algorithms and Architectures

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“…We show that in order to solve the LDS problem, one needs to essentially store the entire graph. We prove the following theorem by using similar approach used in [4].…”

Section: Streaming Lower Bound For Ldsmentioning

confidence: 99%

Algorithm and Hardness Results on Liar’s Dominating Set and $$\varvec{k}$$-tuple Dominating Set

Banerjee

Bhore²

2019

Lecture Notes in Computer Science

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Given a graph G = (V, E), the dominating set problem asks for a minimum subset of vertices D ⊆ V such that every vertex u ∈ V \D is adjacent to at least one vertex v ∈ D. That is, the set D satisfies the condition that |NIn this paper, we study two variants of the classical dominating set problem: k-tuple dominating set (k-DS) problem and Liar's dominating set (LDS) problem, and obtain several algorithmic and hardness results.On the algorithmic side, we present a constant factor ( 11 2 )-approximation algorithm for the Liar's dominating set problem on unit disk graphs. Then, we obtain a PTAS for the k-tuple dominating set problem on unit disk graphs. On the hardness side, we show a Ω(n 2 ) bits lower bound for the space complexity of any (randomized) streaming algorithm for Liar's dominating set problem as well as for the k-tuple dominating set problem. Furthermore, we prove that the Liar's dominating set problem on bipartite graphs is W[2]-hard.

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“…The only time a randomized data structure is made use of is in the algorithm to find a matching, where we probe the k-sample algorithms. This greatly simplifies the analysis as compared to that in [2] …”

Section: From Lemma 4 We Can Conclude the Following Theoremmentioning

confidence: 99%

“…An ongoing line of research asks what can be computed using resources much less than simply storing the data in full. In our recent work [2], we studied graph streaming problems in the parameterized setting: we seek solutions provided that * R.C. was supported by a postdoctoral fellowship from I-CORE ALGO.…”

Section: Introductionmentioning

confidence: 99%