# Tight Bounds for Linear Sketches of Approximate Matchings

@article{Assadi2015TightBF, title={Tight Bounds for Linear Sketches of Approximate Matchings}, author={Sepehr Assadi and Sanjeev Khanna and Y. Li and Grigory Yaroslavtsev}, journal={ArXiv}, year={2015}, volume={abs/1505.01467} }

We resolve the space complexity of linear sketches for approximating the maximum matching problem in dynamic graph streams where the stream may include both edge insertion and deletion. Specifically, we show that for any $\epsilon > 0$, there exists a one-pass streaming algorithm, which only maintains a linear sketch of size $\tilde{O}(n^{2-3\epsilon})$ bits and recovers an $n^\epsilon$-approximate maximum matching in dynamic graph streams, where $n$ is the number of vertices in the graph. In…

## 19 Citations

### Maximum Matching in Turnstile Streams

- Computer ScienceESA
- 2015

In the one-pass turnstile streaming model, in order to compute a O(n e )-approximation, space Ω(n3/2 − 4e) is required for constant error randomized algorithms, and, up to logarithmic factors, space \(\tilde{\mathrm{O}}( n^{2-2\epsilon} )\) is sufficient.

### Kernelization via Sampling with Applications to Finding Matchings and Related Problems in Dynamic Graph Streams

- Computer Science, MathematicsSODA
- 2016

This paper presents a simple but powerful subgraph sampling primitive that is applicable in a variety of computational models including dynamic graph streams, and considers a larger family of parameterized problems for which this primitive yields fast, small-space dynamic graph stream algorithms.

### Maximum Matchings in Dynamic Graph Streams and the Simultaneous Communication Model

- Computer Science, MathematicsSODA
- 2016

The space complexity of single-pass turnstile streaming algorithms for approximating matchings by showing that for any e > 0, Θ(n2-3e) space is both sufficient and necessary to compute an ne-approximate matching.

### An optimal space lower bound for approximating MAX-CUT

- Computer ScienceSTOC
- 2019

Any (randomized) single pass streaming algorithm that breaks the 2-approximation barrier requires Ω(n)-space, thus resolving the space complexity of any non-trivial approximations of the MAX-CUT value to within polylogarithmic factors in the single pass Streaming model of computation.

### Densest Subgraph in Dynamic Graph Streams

- Computer ScienceMFCS
- 2015

A single-pass algorithm that returns a \((1+\epsilon )\) approximation of the maximum density with high probability of the dynamic graph stream model and uses space that matches the lower bound of Bahmani et al. (PVLDB 2012).

### Kernelization via Sampling with Applications to Dynamic Graph Streams

- Computer Science, MathematicsArXiv
- 2015

This paper presents a simple but powerful subgraph sampling primitive that is applicable in a variety of computational models including dynamic graph streams, and considers a larger family of parameterized problems for which this primitive yields fast, small-space dynamic graph stream algorithms.

### Sublinear Estimation of Weighted Matchings in Dynamic Data Streams

- Computer Science, MathematicsESA
- 2015

An algorithm for estimating the weight of a maximum weighted matching by augmenting any estimation routine for the size of an unweighted matching is presented and the first constant estimation for the maximum matching size in a dynamic graph stream for planar graphs using \(\tilde{O}(n 4/5})\) space is given.

### Maximum Matching Turnstile

- Computer Science
- 2015

The unweighted bipartite maximum matching problem in the one-pass turnstile streaming model where the input stream consists of edge insertions and deletions is considered and no such result is possible if edge deletions are allowed.

### J un 2 01 5 Densest Subgraph in Dynamic Graph Streams ∗

- Computer Science
- 2021

A single-p ass algorithm is presented that returns a (1 + ǫ) approximation of the maximum density with high probability of the dynamic graph stream model and uses similar space and another algorithm that both pr ocessed each update and maintained a (4+ǫ), which is the current maximum density in polylog(n) time per-update.

### Almost Optimal Streaming Algorithms for Coverage Problems

- Computer Science, MathematicsSPAA
- 2017

This paper presents optimal approximation algorithms for maximum coverage and minimum set cover problems in the streaming model with an (almost) optimal space complexity of Õ(n), and introduces a new general sketching technique for coverage functions.

## References

SHOWING 1-10 OF 53 REFERENCES

### Better bounds for matchings in the streaming model

- Computer ScienceSODA
- 2013

Improved bounds for approximating maximum matchings in bipartite graphs in the streaming model are presented and it is shown that a simple fractional load balancing approach achieves approximation ratio.

### Approximating matching size from random streams

- Computer Science, MathematicsSODA
- 2014

This work gives the first algorithm where both the space and approximation factors are smaller than any polynomial in n, and shows, somewhat surprisingly, that the local algorithm can be implemented in the streaming setting even for k = Ω(log n/log log n).

### Maximum Matching in Turnstile Streams

- Computer ScienceESA
- 2015

In the one-pass turnstile streaming model, in order to compute a O(n e )-approximation, space Ω(n3/2 − 4e) is required for constant error randomized algorithms, and, up to logarithmic factors, space \(\tilde{\mathrm{O}}( n^{2-2\epsilon} )\) is sufficient.

### Improved Approximation Guarantees for Weighted Matching in the Semi-streaming Model

- Computer ScienceSIAM J. Discret. Math.
- 2010

This work improves on the currently best one-pass algorithm due to Zelke by devising a deterministic approach, and provides a lower bound on the competitive ratio of any such deterministic algorithm, and shows that future improvements will have to store in memory a set of edges which is not necessarily a feasible matching.

### Parameterized Streaming: Maximal Matching and Vertex Cover

- Computer Science, MathematicsSODA
- 2015

This paper presents the first graph streaming algorithm that combines linear sketching with sequential operations that depend on the graph at the current time, and shows a tight lower bound of Ω(k2) for the space complexity of any (randomized) streaming algorithms for the parameterized Vertex Cover, even in the insertion-only model.

### Analyzing graph structure via linear measurements

- Computer ScienceSODA
- 2012

The study of graph sketching is initiated, i.e., algorithms that use a limited number of linear measurements of a graph to determine the properties of the graph are studied, including the first dynamic graph semi-streaming algorithms for connectivity, spanning trees, sparsification, and matching problems.

### Kernelization via Sampling with Applications to Finding Matchings and Related Problems in Dynamic Graph Streams

- Computer Science, MathematicsSODA
- 2016

This paper presents a simple but powerful subgraph sampling primitive that is applicable in a variety of computational models including dynamic graph streams, and considers a larger family of parameterized problems for which this primitive yields fast, small-space dynamic graph stream algorithms.

### Streaming Algorithms for Estimating the Matching Size in Planar Graphs and Beyond

- Computer Science, MathematicsSODA
- 2015

The adversarial-order model is circumvented by exploiting several structural properties of planar graphs, and more generally, graphs with bounded arboricity, and a reduction from the Boolean Hidden Matching Problem is designed to show that there is no randomized streaming algorithm that estimates the size of the maximum matching to within a factor better than 3/2 and uses only o(n1/2) bits of space.

### Bipartite Graph Matchings in the Semi-streaming Model

- Computer Science, MathematicsESA
- 2009

This work presents an algorithm for finding a large matching in a bipartite graph in the semi-streaming model, which finds a \(\frac{1}{1+\epsilon}\)-approximation of a maximum-cardinality matching and uses \(O{({(\frac{ 1}{\ep silon})^8}\) passes over the input stream.

### Superlinear Lower Bounds for Multipass Graph Processing

- Computer Science, Mathematics2013 IEEE Conference on Computational Complexity
- 2013

The line of attack requires proving an information cost lower bound for a decision version of the classic pointer chasing problem and a direct sum type theorem for the disjunction of several instances of this problem.