Gabow's $O(\sqrt{n}m)$ Maximum Cardinality Matching Algorithm, Revisited

Abstract

We revisit Gabow's $O(\sqrt{n} m)$ maximum cardinality matching algorithm (The Weighted Matching Approach to Maximum Cardinality Matching, Fundamenta Informaticae, 2017). It adapts the weighted matching algorithm of Gabow and Tarjan~\cite{GT91} to maximum cardinality matching. Gabow's algorithm works iteratively. In each iteration, it constructs a maximal number of edge-disjoint shortest augmenting paths with respect to the current matching and augments them. It is well-known that $O(\sqrt{n})$ iterations suffice. Each iteration consists of three parts. In the first part, the length of the shortest augmenting path is computed. In the second part, an auxiliary graph $H$ is constructed with the property that shortest augmenting paths in $G$ correspond to augmenting paths in $H$. In the third part, a maximal set of edge-disjoint augmenting paths in $H$ is determined, and the paths are lifted to and augmented to $G$. We give a new algorithm for the first part. Gabow's algorithm for the first part is derived from Edmonds' primal-dual algorithm for weighted matching. We believe that our approach is more direct and will be easier to teach. We have implemented the algorithm; the implementation is available at the companion webpage (https://people.mpi-inf.mpg.de/~mehlhorn/CompanionPageGenMatchingImplementation.html).

Figures (6)

• Figure 1: The vertex $z$ becomes even by the addition of the bridge $xy$. The path from $r$ to $x$ has length $\mathit{lcp}(x)$, the path from $r$ to $y$ has length $\mathit{lcp}(y)$, the path from $r$ to $z$ has length $\mathit{lcp}_{\mathit{odd}}(z)$, the path from $y$ to $z$ has length $\mathit{lcp}(y) - \mathit{lcp}_{\mathit{odd}}(z)$; therefore, $\mathit{lcp}(z) \mathbin{\raisebox{0.05ex}{\rm :}\!\!=} \mathit{lcp}(x) + 1 + \mathit{lcp}(y) - \mathit{lcp}_{\mathit{odd}}(z)$. Moreover, the even length path from $b$ to $z$ has length $\mathit{lcp}(z) - \mathit{lcp}(b)$. Edges are drawn solid, and paths are drawn dash-dotted.
• Figure 2: The execution of part I on the example given in Gabow:GeneralMatching: (a) Shows the input graph. (b) At the end of phase 0, $S$ consists of the two free nodes $a$ and $l$. (c) At the end of phase 2, we have grown $S$ to level 2. (d) After the growth-step in phase 4, $S$ has grown to level 4. There is an even-even edge $ce$ with $\mathit{lcp}(c) + \mathit{lcp}(e) = 2 + 4 = 2\cdot 4 - 2$. (e) After the bridge-steps in phase 4, $d$ is now even with $\mathit{lcp}(d) = 4$. (f) In phase 5, we add the bridge $ik$ with $\mathit{lcp}(i)+ \mathit{lcp}(k) = 4 + 4 = 2\cdot 5 - 2$. The vertices $h$ and $j$ become even with $\mathit{lcp}$-value equal to $6$. (g) In phase 6, we grow the tree and add node $r$ at level 5 and node $q$ at level 6. Then we process the bridge $pq$ with $\mathit{lcp}(p) + \mathit{lcp}(q) = 4 + 6 = 2\cdot6 - 2$, $r$ becomes even with $\mathit{lcp}(r) = 6$. We also add the bridge $eh$ with $\mathit{lcp}(e) + \mathit{lcp}(h) = 4 + 6 = 2\cdot 6 - 2$. Vertices $b$ and $f$ become even with $\mathit{lcp}$-value equal to $10$. (h) Finally, in phase 7, we add the bridge $fn$ with $\mathit{lcp}(f) + \mathit{lcp}(n) = 10 + 2 = 2\cdot 7 - 2$. We have found an augmenting path of length 13, and Part I ends. (j) The contracted graph: Vertices $a$ to $k$ are contracted into the red supernode, and vertices $p$ to $r$ into the green supernode. Edges $mf$ and $pf$ are discarded. (i) also shows the structure inside the supernodes. In (d) we grow out of $c$ using edge $cd$ and set $\mathit{lcp}_{\mathit{odd}}(d) = 3$ and $\mathit{lcp}(e) = 4$. When we consider $ce$, we set $\mathit{lcp}_{\mathit{odd}}(e) = 3$.
• Figure 3: Part I of the Matching Algorithm. Augmenting paths are only found for $\Delta \le n/2$. The remaining phases are needed for the correct construction of the witness of optimality; see Section \ref{['Implementation']}.
• Figure 4: Cascading bridges: (a) $\Delta$ is assumed to be odd. At the beginning of phase $\Delta$ , $a$, $b$, $d$, and $f$ are even, and $c$, $e$, and $g$ are odd. Nodes $a$ and $b$ are guaranteed to have their correct $\mathit{lcp}$-values by the first part of Lemma \ref{['correctness']}. We add the bridge $ab$ with $\mathit{lcp}$-sum $2\Delta - 2$; its skew is zero. As a consequence, $c$ becomes even with $\mathit{lcp}$-value $\Delta + 1$ and the edge $cd$ becomes an even-even edge with $\mathit{lcp}$-sum $2\Delta - 2$ and a skewness of 4. We add $cd$, $e$ becomes even with the $\mathit{lcp}$-value $\Delta + 3$, and $ef$ becomes an even-even edge with $\mathit{lcp}$-sum $2\Delta - 2$ and skew 8. Note that the edges $ab$, $cd$, and $ef$ have increasing skew. (b) $\Delta$ is assumed to be even. In the growth-step in phase $\Delta$ we have added node $h$ at level $\Delta - 1$ and node $i$ at level $\Delta$. The edge $ei$ is a bridge with $\mathit{lcp}$-sum $2\Delta - 2$ and skew 2. The addition of $ei$ makes $h$ even with $\mathit{lcp}(h) = \Delta$ and creates the bridges $hg$ and $hj$ with skew 2. Node $k$ becomes even with $\mathit{lcp}(k) = \Delta + 2$. Then $kl$ has two even endpoints, $\mathit{lcp}$-sum $2\Delta - 2$, and skew 6. So we add $kl$.
• Figure 5: The graph has seven vertices, and a maximum matching is shown. A certificate of optimality gives label one to vertices $b$ and $d$, label two to vertices $e$, $f$, and $g$, and label zero to vertices $a$ and $c$. After phase $\left\lfloor 7/2 \right\rfloor$, the search structure consists of the path from $a$ to $c$, and the two matching edges $de$ and $fg$ still belong to the reservoir of unexplored edges. The labeling algorithm would give label one to $b$ and to one of the vertices in $\{\space d,e,f,g \space\}$ and would give label two to the other vertices in this set. If $d$ is labeled two, the edge $cd$ connects labels zero and two and violates the witness property. After phase seven, all vertices belong to the search structure, and the correct labeling is computed.

Theorems & Definitions (14)

• Lemma 1
• Lemma 2: Even-odd edges go up at most one level
• Lemma 3: Discover bridges in order
• Lemma 4: Bridge-steps generate growth-steps only for later phases
• Lemma 5: Non-horizontal bridges: Skewness increases
• Lemma 6: Growth-steps in phase $\Delta$ generate bridge-steps for phases $\ge \Delta$
• Lemma 7: Correctness
• Lemma 8
• Lemma 9
• Lemma 10