# Maximum Matching via Maximal Matching Queries

Christian Konrad, Kheeran K. Naidu, Arun Steward
2023

#### Abstract

We study approximation algorithms for Maximum Matching that are given access to the input graph solely via an edge-query maximal matching oracle. More specifically, in each round, an algorithm queries a set of potential edges and the oracle returns a maximal matching in the subgraph spanned by the query edges that are also contained in the input graph. This model is more general than the vertex-query model introduced by Binti Khalil and Konrad [FSTTCS’20], where each query consists of a subset of vertices and the oracle returns a maximal matching in the subgraph of the input graph induced by the queried vertices.

In this paper, we give tight bounds for deterministic edge-query algorithms for up to three rounds. In more detail:

• As our main result, we give a deterministic $3$-round edge-query algorithm with approximation factor $0.625$ on bipartite graphs. This result establishes a separation between the edge-query and the vertex-query models since every deterministic 3-round vertex-query algorithm has an approximation factor of at most $0.6$ [Binti Khalil, Konrad, FSTTCS’20], even on bipartite graphs. Our algorithm can also be implemented in the semi-streaming model of computation in a straightforward manner and improves upon the state-of-the-art $3$-pass $0.6111$-approximation algorithm by Feldman and Szarf [APPROX’22] for bipartite graphs.

• We show that the aforementioned algorithm is optimal in that every deterministic $3$-round edge-query algorithm has an approximation factor of at most $0.625$, even on bipartite graphs.

• Last, we also give optimal bounds for one and two query rounds, where the best approximation factors achievable are $1/2$ and $1/2 + \Theta(\frac{1}{n})$, respectively, where $n$ is the number of vertices in the input graph.