Document

# Sensitivity Analysis of the Maximum Matching Problem

## File

LIPIcs.ITCS.2021.58.pdf
• Filesize: 0.59 MB
• 20 pages

## Acknowledgements

We would like to thank anonymous ITCS reviewers for helpful comments on an earlier version of the paper.

## Cite As

Yuichi Yoshida and Samson Zhou. Sensitivity Analysis of the Maximum Matching Problem. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 58:1-58:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ITCS.2021.58

## Abstract

We consider the sensitivity of algorithms for the maximum matching problem against edge and vertex modifications. When an algorithm A for the maximum matching problem is deterministic, the sensitivity of A on G is defined as max_{e ∈ E(G)}|A(G) △ A(G - e)|, where G-e is the graph obtained from G by removing an edge e ∈ E(G) and △ denotes the symmetric difference. When A is randomized, the sensitivity is defined as max_{e ∈ E(G)}d_{EM}(A(G),A(G-e)), where d_{EM}(⋅,⋅) denotes the earth mover’s distance between two distributions. Thus the sensitivity measures the difference between the output of an algorithm after the input is slightly perturbed. Algorithms with low sensitivity, or stable algorithms are desirable because they are robust to edge failure or attack. In this work, we show a randomized (1-ε)-approximation algorithm with worst-case sensitivity O_ε(1), which substantially improves upon the (1-ε)-approximation algorithm of Varma and Yoshida (SODA'21) that obtains average sensitivity n^O(1/(1+ε²)) sensitivity algorithm, and show a deterministic 1/2-approximation algorithm with sensitivity exp(O(log^*n)) for bounded-degree graphs. We then show that any deterministic constant-factor approximation algorithm must have sensitivity Ω(log^* n). Our results imply that randomized algorithms are strictly more powerful than deterministic ones in that the former can achieve sensitivity independent of n whereas the latter cannot. We also show analogous results for vertex sensitivity, where we remove a vertex instead of an edge. Finally, we introduce the notion of normalized weighted sensitivity, a natural generalization of sensitivity that accounts for the weights of deleted edges. For a graph with weight function w, the normalized weighted sensitivity is defined to be the sum of the weighted edges in the symmetric difference of the algorithm normalized by the altered edge, i.e., max_{e ∈ E(G)}1/(w(e))w (A(G) △ A(G - e)). Hence the normalized weighted sensitivity measures the weighted difference between the output of an algorithm after the input is slightly perturbed, normalized by the weight of the perturbation. We show that if all edges in a graph have polynomially bounded weight, then given a trade-off parameter α > 2, there exists an algorithm that outputs a 1/(4α)-approximation to the maximum weighted matching in O(m log_α n) time, with normalized weighted sensitivity O(1).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Graph algorithms analysis
• Theory of computation → Approximation algorithms analysis
• Theory of computation → Models of computation
##### Keywords
• Sensitivity analysis
• maximum matching
• graph algorithms

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Marc Bury, Elena Grigorescu, Andrew McGregor, Morteza Monemizadeh, Chris Schwiegelshohn, Sofya Vorotnikova, and Samson Zhou. Structural results on matching estimation with applications to streaming. Algorithmica, 81(1):367-392, 2019.
2. Keren Censor-Hillel, Elad Haramaty, and Zohar Karnin. Optimal dynamic distributed mis. In Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing (PODC), pages 217-226, 2016.
3. Vincent Cohen-Addad, Niklas Hjuler, Nikos Parotsidis, David Saulpic, and Chris Schwiegelshohn. Fully dynamic consistent facility location. In Advances in Neural Information Processing Systems (NeurIPS), pages 3250-3260, 2019.
4. Richard Cole and Uzi Vishkin. Deterministic coin tossing with applications to optimal parallel list ranking. Information and Control, 70(1):32-53, 1986.
5. Michael Crouch and Daniel S. Stubbs. Improved streaming algorithms for weighted matching, via unweighted matching. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), volume 28, pages 96-104, 2014.
6. P. Erdós and R. Rado. Combinatorial theorems on classifications of subsets of a given set. Proceedings of the London Mathematical Society, s3-2(1):417-439, 1952.
7. Guy Even, Moti Medina, and Dana Ron. Distributed maximum matching in bounded degree graphs. In Proceedings of the International Conference on Distributed Computing and Networking (ICDCN), pages 18:1-18:10, 2015.
8. Silvio Lattanzi and Sergei Vassilvitskii. Consistent k-clustering. In Proceedings of the 34th International Conference on Machine Learning (ICML), pages 1975-1984, 2017.
9. Andrew McGregor. Finding graph matchings in data streams. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, pages 170-181, 2005.
10. Michal Parnas and Dana Ron. Approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms. Theoretical Computer Science, 381(1-3):183-196, 2007.
11. Pan Peng and Yuichi Yoshida. Average sensitivity of spectral clustering. In Proceedings of the 26th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD), pages 1132-1140, 2020.
12. Nithin Varma and Yuichi Yoshida. Average sensitivity of graph algorithms. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), 2021. to appear.