Abstract
We consider the problem of maintaining the size of a Maximum Matching in the presence of failures of vertices and edges. For a graph G, we use \(\mu (G)\) to denote the size of a maximum matching of G. A subgraph H of G is an \((\alpha ,f)\) -Fault Tolerant Matching Subgraph (\((\alpha ,f)\) -FTMS) if it has the following property: For any set F of at most f vertices or edges in G, \(\alpha \cdot \mu (G-F) \le \mu (H-F)\). Assadi and Bernstein [SOSA 2019] showed that for any \(\epsilon > 0\), there exists a \((\frac{2}{3}-\epsilon ,f)\)-FTMS of size \(\mathcal {O}(n+f)\). In this paper we initiate a study of (1, f)-FTMS or f-FTMS in short.
In particular we obtain the following results,
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On bipartite graphs, there exists 1-FTMS, for one edge fault with \(\mathcal {O}(\mu (G))\) vertices and edges. We complement this upper bound with the matching lower bound of \(\varOmega (\mu (G))\) on 1-FTMS for one edge fault.
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On general graphs, there exists f-FTMS for at most f edge faults with \(\mathcal {O}(\mu (G)^2+ \mu (G)f)\) edges and \(\mathcal {O}(\mu (G)\cdot f)\) vertices. We also provide a matching lower bound of \(\varOmega (\max \{\mu (G)^2,\mu (G)f\})\) edges and \(\varOmega (\mu (G)f)\) vertices for f-FTMS, \(f \ge 2\) for at most f edge faults.
The same construction works for vertex faults, and they result in even tighter bounds for \(f=1\). Our algorithmic results exploit the structural properties of matchings and use tools from Parameterized Algorithms, such as Expansion Lemma. We leave open the question of existence of 1-FTMS for one edge fault, of linear size (in terms of \(\mu (G)\)) on general graphs.
Keywords
N. Banerjee—Supported by JSPS KAKENHI Grant No. JP20H05967.
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Results marked with \(\star \) are deferred to the full version.
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Banerjee, N., Gupta, M., Raman, V., Saurabh, S. (2022). Output Sensitive Fault Tolerant Maximum Matching. In: Kulikov, A.S., Raskhodnikova, S. (eds) Computer Science – Theory and Applications. CSR 2022. Lecture Notes in Computer Science, vol 13296. Springer, Cham. https://doi.org/10.1007/978-3-031-09574-0_8
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