Abstract
In a weighted undirected graph, a matching is said to be α-robust if for all p, the total weight of its heaviest p edges is at least α times the maximum weight of a p-matching in the graph. Here a p-matching is a matching with at most p edges. In 2002, Hassin and Rubinstein [4] showed that every graph has a \(\frac{1}{\sqrt 2}\)-robust matching and it can be found by k-th power algorithm in polynomial time.
In this paper, we show that it can be extended to the matroid intersection problem, i.e., there always exists a \(\frac{1}{\sqrt 2}\)-robust matroid intersection, which is polynomially computable. We also study the time complexity of the robust matching problem. We show that a 1-robust matching can be computed in polynomial time (if exists), and for any fixed number α with \(\frac{1}{\sqrt 2} < \alpha < 1\), the problem to determine whether a given weighted graph has an α-robust matching is NP-complete. These together with the positive result for \(\alpha =\frac{1}{\sqrt 2}\) in [4] give us a sharp border for the complexity for the robust matching problem. Moreover, we show that the problem is strongly NP-complete when α is a part of the input. Finally, we show the limitations of k-th power algorithm for robust matchings, i.e., for any ε> 0, there exists a weighted graph such that no k-th power algorithm outputs a \(\left( \frac{1}{\sqrt 2} + \epsilon \right)\)-approximation for computing the most robust matching.
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Fujita, R., Kobayashi, Y., Makino, K. (2010). Robust Matchings and Matroid Intersections. In: de Berg, M., Meyer, U. (eds) Algorithms – ESA 2010. ESA 2010. Lecture Notes in Computer Science, vol 6347. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15781-3_11
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DOI: https://doi.org/10.1007/978-3-642-15781-3_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15780-6
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