Tight Approximation Ratio for Minimum Maximal Matching

Abstract

We study a combinatorial problem called Minimum Maximal Matching, where we are asked to find in a general graph the smallest matching that can not be extended. We show that this problem is hard to approximate with a constant smaller than 2, assuming the Unique Games Conjecture.

As a corollary we show, that Minimum Maximal Matching in bipartite graphs is hard to approximate with constant smaller than $\frac{4}{3}$, with the same assumption. With a stronger variant of the Unique Games Conjecture—that is Small Set Expansion Hypothesis—we are able to improve the hardness result up to the factor of $\frac{3}{2}$.

S. Dudycz—Supported by the Polish National Science Centre grant 2013/11/B/ST6/01748.

M. Lewandowski and J. Marcinkowski—Supported by the Polish National Science Centre grant 2015/18/E/ST6/00456.

A Hardness of Bipartite MMM

In this section we will perform a natural reduction to prove the following theorem.

Theorem 6

Assuming the Unique Games Conjecture, for any $\epsilon > 0$ it is NP-hard to distinguish between balanced bipartite graphs of $2n$ vertices:

• (yes instance) with a Maximal Matching of size smaller than $n \left( \frac{1}{2} + \epsilon \right)$.

• (no instance) with no Maximal Matching of size smaller than $n \left( \frac{2}{3} - \epsilon \right)$.

We will start with the graph $G_{\Phi }^{\rho }$ defined in Sect. 5. The bipartite graph $H_{\Phi }$ has two copies $v^{l}$ and $v^{r}$ of every vertex $v \in G_\Phi ^\rho$. The vertices $u^{l}$ and $v^r$ are connected with an edge if there is an edge $(u, v)$ in $G_\Phi ^\rho$. $n$ is going to be equal to $|V(G_\Phi ^\rho )|$. We will call this construction bipartisation of an undirected graph.

It is easy to see, that if $\Phi$ is a yes instance of the Unique Label Cover problem, we can use the matching from Lemma 8 ($M$ in $G_\Phi ^\rho$) to produce a maximal matching in $H_\Phi$. For every edge $(u, v)\in M$ we will put its two copies, $(u^l, v^r)$ and $(v^l, u^r)$ into the matching. The resulting matching size is thus equal to $2 \cdot |M| < n (\frac{1}{2} + \epsilon )$.

1.1 A.1 Covering with Paths

In order to analyse the no case, we need to look at the bipartite instance and its matchings from another angle. For any matching in $H_{\Phi }$, we will view its edges as directed edges in $G_\Phi ^{\rho }$—the vertices on the left will be viewed as out vertices, and those on the right as in vertices. The graph $G_{\Phi }^{\rho }$ will thus be covered with directed edges. Every vertex will be incident to at most one outgoing and one incoming edge, which means that the edges will form a structure of directed paths and cycles. The set of these paths and cycles will be called $\mathscr {P}(M)$ for a matching $M$.

Observation 7

If $M$ is a maximal matching, every path $P \in \mathscr {P}(M)$ has a length $|P| \geqslant 2$.

Proof

Assume, that for a maximal matching $M$ in $H_\Phi$ there is a length-one path $P = {(u, v)} \in \mathscr {P}(M)$. This means, that the vertices $v^l$ and $u^r$ are unmatched in $M$—yet, they are connected with an edge, that can be added to the matching (that would form a length-2 cycle in $\mathscr {P}(M)$).    $\square$

We will now use this observation to prove the relation between maximal matchings in $H_\Phi$ and vertex covers in $G_\Phi ^\rho$.

Lemma 9

For any maximal matching $M$ in $H_\Phi$, there exists a vertex cover $C$ in $G_\Phi ^{\rho }$ of size $|C| \leqslant \frac{3}{2}|M|$.

Proof

We will construct the vertex cover using paths and cycles of $\mathscr {P}(M)$. For every $P \in \mathscr {P}(M)$ we add all the vertices of $P$ into $C$. When $P$ is a cycle, it contains as many vertices as edges. A path has at most $\frac{3}{2}$ as many vertices as edges, since its length is at least 2.    $\square$

As shown in Lemma 7, when $\Phi$ is a no instance, the Minimum Vertex Cover in $G_\Phi ^\rho$ has at least $n (1 - \epsilon )$ vertices. The Minimum Maximal Matching in $H_{\Phi }$ must in this case have at least $\frac{2}{3}n(1 - \epsilon ) > n(\frac{2}{3} - \epsilon )$ edges.

The hardness coming from Theorem 6 is, that assuming UGC, no polynomial-time algorithm will provide approximation for Minimum Maximal Matching with a factor $\frac{4}{3} - \epsilon$ for any $\epsilon > 0$.