Abstract
We study the Minimum Maximal Matching problem, where we are asked to find in a graph the smallest matching that cannot be extended. We show that this problem is hard to approximate with any constant smaller than 2 even in bipartite graphs, assuming either of two stronger variants of Unique Games Conjecture. The bound also holds for computationally equivalent Minimum Edge Dominating Set. Our lower bound matches the approximation provided by a trivial algorithm.
Our results imply conditional hardness of approximating Maximum Stable Matching with Ties and Incomplete Lists with a constant better than \(\frac{3}{2}\), which also matches the best known approximation algorithm.
Supported by Polish NSC grants 2018/29/B/ST6/02633 and 2015/18/E/ST6/00456.
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Notes
- 1.
To illustrate that imagine a cycle of four edges with weights (1, 1, 100, 100). The lightest EDS weighs 2, but MMM must contain two opposite edges and weighs 101.
- 2.
- 3.
- 4.
- 5.
This is similar to Strong UGC except that Strong UGC deals with vertex expansion whereas SSEH concerns with edge expansion.
- 6.
These results talk about Balanced Biclique which is more natural, but since bi-independent set fits our needs more, we will silently take graph complement when recalling their reductions.
- 7.
The character ‘\(\circ \)’ is a function composition operator. Since a vector can be viewed as a function mapping indices to values, for a permutation \(\pi \) and a vector x, \(\pi \circ x\) is the vector with permuted elements.
- 8.
(A, B) is a bisection of X if \(|A| = |B| = |X|/2\) and \(A \cup B = X\).
- 9.
The vectors \(\mathbf {v}\) and \(\mathbf {w}\) are neighbours in \(G^{\otimes R}\) iff \(v_i\) is connected to \(w_i\) in G for every \(i\in [R]\).
- 10.
Equivalently we will write that \(\pi ^{1,\dots ,k}(\mathbf {u})\) and \(\mathbf {v}\) are neighbours in \(G^{\otimes (R\times k)}\).
- 11.
This may be not unique. The choice of a permutation is arbitrary.
- 12.
\(T_0\) and \(T_1\) are called \(T'_0\) and \(T'_1\) respectively in [20].
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Dudycz, S., Manurangsi, P., Marcinkowski, J. (2021). Tight Inapproximability of Minimum Maximal Matching on Bipartite Graphs and Related Problems. In: Koenemann, J., Peis, B. (eds) Approximation and Online Algorithms. WAOA 2021. Lecture Notes in Computer Science(), vol 12982. Springer, Cham. https://doi.org/10.1007/978-3-030-92702-8_4
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