Output Sensitive Fault Tolerant Maximum Matching

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,

• 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.

• 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.

N. Banerjee—Supported by JSPS KAKENHI Grant No. JP20H05967.