A Simple LP-Based Approximation Algorithm for the Matching Augmentation Problem
The Matching Augmentation Problem (MAP) has recently received significant attention as an important step towards better approximation algorithms for finding cheap 2-edge connected subgraphs. This has culminated in a 53 $$\frac{5}{3}$$ -approximation algorithm. However, the algorithm and its analysis...
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| Published in | Integer Programming and Combinatorial Optimization Vol. 13265; pp. 57 - 69 |
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| Main Authors | , , |
| Format | Book Chapter |
| Language | English |
| Published |
Switzerland
Springer International Publishing AG
2022
Springer International Publishing |
| Series | Lecture Notes in Computer Science |
| Online Access | Get full text |
| ISBN | 3031069005 9783031069000 |
| ISSN | 0302-9743 1611-3349 |
| DOI | 10.1007/978-3-031-06901-7_5 |
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| Summary: | The Matching Augmentation Problem (MAP) has recently received significant attention as an important step towards better approximation algorithms for finding cheap 2-edge connected subgraphs. This has culminated in a 53 $$\frac{5}{3}$$ -approximation algorithm. However, the algorithm and its analysis are fairly involved and do not compare against the problem’s well-known LP relaxation called the cut LP.
In this paper, we propose a simple algorithm that, guided by an optimal solution to the cut LP, first selects a DFS tree and then finds a solution to MAP by computing an optimum augmentation of this tree. Using properties of extreme point solutions, we show that our algorithm always returns (in polynomial time) a better than 2-approximation when compared to the cut LP. We thereby also obtain an improved upper bound on the integrality gap of this natural relaxation. |
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| Bibliography: | Original Abstract: The Matching Augmentation Problem (MAP) has recently received significant attention as an important step towards better approximation algorithms for finding cheap 2-edge connected subgraphs. This has culminated in a 53\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{5}{3}$$\end{document}-approximation algorithm. However, the algorithm and its analysis are fairly involved and do not compare against the problem’s well-known LP relaxation called the cut LP. In this paper, we propose a simple algorithm that, guided by an optimal solution to the cut LP, first selects a DFS tree and then finds a solution to MAP by computing an optimum augmentation of this tree. Using properties of extreme point solutions, we show that our algorithm always returns (in polynomial time) a better than 2-approximation when compared to the cut LP. We thereby also obtain an improved upper bound on the integrality gap of this natural relaxation. |
| ISBN: | 3031069005 9783031069000 |
| ISSN: | 0302-9743 1611-3349 |
| DOI: | 10.1007/978-3-031-06901-7_5 |