Maximum Cardinality $f$-Matching in Time $O(n^{2/3}m)$

Consider an $f$-matching $M$. Let $S$ be an augmenting trail for $M$. $|S|$ denotes the length of $S$. $S$ is a shortest augmenting trail if it has the minimum length possible. We call $S$ a sat.

Hopcroft and Karp [16] and Karzanov [18] proved the fundamental result that for 1-matching the length of a sat is nondecreasing. The proof immediately extends to $f$-matching. For completeness we sketch it.

For $i=1,2$ let trail $S_{i}$ have ends $v_{i}$ and $w_{i}$. (These four ends need not be distinct, in fact all four may be the same vertex.) Enlarge $M$ to an $f$-matching $M^{\prime}$ by adding artificial edges $v_{i}w_{i}$, $i=1,2$. Every vertex $u\in V$ has $d(u,N)=d(u,M^{\prime})$. So $N\oplus M^{\prime}$ is a collection of edge-disjoint alternating circuits. Removing the two artificial edges creates two alternating trails that begin and end with edges of $N-M$ and hence are augmenting for $M$. (In detail let $C_{i}$ be the circuit containing $v_{i}w_{i}$, $i=1,2$. If $C_{1}\neq C_{2}$ then the desired augmenting trails are $T_{i}=C_{i}-v_{i}w_{i}$. If $C_{1}=C_{2}$ then $C_{1}-v_{1}w_{1}-v_{2}w_{2}$ is the disjoint union of trails $T_{1}$ and $T_{2}$.) $\Box$

The lemma leads to the following high-level algorithm for finding a maximum cardinality $f$-matching. Define a blocking set to be a maximal size collection of edge-disjoint shortest augmenting trails. The algorithm repeatedly finds a blocking set and augments each of its trails. When no augmenting trail exists the matching is maximum.

Our goal is to show a maximum cardinality $f$-matching on a general graph can be found in time (1.1). Section 4 shows a blocking set can be found in time $O(m)$. So the goal is achieved by showing the high-level algorithm finds $O(\min\{\sqrt{f(V)},n\}$ blocking sets on multigraphs and $O(n^{2/3})$ blocking sets on simple graphs. The arguments are as follows.

Let $M$ be the $f$-matching at some point in the high-level algorithm. Let $s$ be the length of a sat for $M$. We start with the simple analysis for multigraphs.

The proof of $(*)$ requires some basic properties of $f$-matching blossoms. We present them here. Readers familiar with [12] need only skim this section to review the notation. Fig. 2 illustrates the discussion.

We begin with more terminology for matchings.

Contracting a subgraph contracted vertex (all internal edges including internal loops disappear). We will even contract subgraphs that just contain a loop. We use the following notation for contractions. Let $\overline{G}$ be a graph derived from $G$ by contracting a number of vertex-disjoint subgraphs. $V$ ($\overline{V}$) denotes the vertex set of $G$ ($\overline{G}$), respectively. A vertex of $\overline{G}$ that belongs to $V$ (i.e., it is not in a contracted subgraph) is an atom. We identify an edge of $\overline{G}$ with its corresponding edge in $G$. Thus an edge of $\overline{G}$ incident to a contracted vertex is denoted as $xy$, where $x$ and $y$ are $V$-vertices in distinct $\overline{V}$-vertices, and $xy\in E$. Let $H$ be a subgraph of $\overline{G}$. The preimage of $H$ is a subgraph of $G$ consisting of the edges of $H$, plus the subgraphs whose contractions are vertices of $H$, plus the atoms of $H$. $\overline{V}(H)$ ($V(H)$) denotes the vertex set of $H$ (the preimage of $H$), respectively. Similarly $\overline{E}(H)$ ($E(H)$) denotes the edge set of $H$ (the preimage of $H$), respectively.

Figures in this paper use the following conventions. Matched edges are drawn heavy, unmatched edges light. Free vertices are drawn as rectangles. Edge weights are always given by (2.1). Trails are indicated by arrowheads on their edges.

When discussing a matching $M$, the M-type of an edge $e$ is $M$ or $\overline{M}$ depending on whether $e$ is matched or unmatched, respectively. We usually denote an arbitrary M-type as $\mu$, and $\mu(e)$ denotes the M-type of edge $e$.

Let $G$ be a graph with an $f$-matching. A trail is closed if it starts and ends at the same vertex. A trail is alternating if every 2 consecutive edges have opposite M-types. The first and last edges of a closed trail are not consecutive.

Blossoms are defined recursively as follows. Let $\overline{G}$ be a graph derived from $G$ by contracting a family ${\cal A}$ of zero or more vertex-disjoint blossoms. Let $C$ be a closed trail in $\overline{G}$ that starts and ends at a vertex $\alpha\in V(\overline{G})$. The preimage of $C$ is a blossom $B$ with base vertex $\beta(B)$ if $C$ has the following properties:

If $\alpha$ is an atom then $C$ starts and ends with edges of the same $M$-type. $B$ has this $M$-type and $\beta(B)=\alpha$.

If $\alpha\in{{\cal A}}$ then $B$ has the same $M$-type as $\alpha$ and $\beta(B)=\beta(\alpha)$.

If $v$ is an atom of $C$ then every 2 consecutive edges of $\delta(v,C)$ alternate.

If $v\in{{\cal A}}\cap C$ then $d(v,C)=2$. Furthermore if $v\neq\alpha$ then $\delta(\beta(v),C)$ contains an edge of opposite $M$-type from $v$.

Note that as a subgraph of $G$, a blossom $B$ has vertex and edge sets $V(B)$ and $E(B)$. Also note in Fig.2 the blossom of part (b) can be viewed as being formed from the blossom of part (c) or the blossom of part (d). (This nonuniqueness is usually irrelevant.)

In addition we define the base edge of a blossom $A$, denoted $\eta(A)$. $\eta(A)$ is an edge that leaves $V(A)$ from vertex $\beta(A)$, and it has opposite $M$-type from $A$. Every blossom with base $\beta(A)$ has the same base edge $\eta(A)$. So now assume $A$ is the maximal blossom with base vertex $\beta(A)$. If $A$ is in the closed trail $C$ of some blossom, $\eta(A)$ is the edge required in the above definition. (In case both edges of $\delta(\beta(v),C)$ qualify, the choice for $\eta(A)$ is arbitrary.) If $A$ is a maximal blossom (i.e., not in such a closed trail) $\eta(A)$ is some chosen edge (of opposite $M$-type from $A$); it will be clear from context.

Blossoms where $\beta(A)$ is a free vertex are a special case. We say the blossom is free. A free blossom has $M$-type $\overline{M}$ (since an augmenting trail always begins with an unmatched edge). Also the vertex $\beta(A)$ has deficiency 1. (Otherwise the edges of $A$ contain an augmenting trail, and the blossom structure is irrelevant.) For a free blossom $\eta(A)$ is a matched edge from an artificial vertex $\varepsilon$ to $\beta(A)$.

For any blossom $B$, any edge leaving $B$ other than $\eta(B)$ is a petal of $B$. Blossom $B$ is heavy (light) if its M-type is $M$ ($\overline{M}$), respectively.

In a blossom $B$ with base vertex $\beta$, each vertex $v\in V(B)$ has 2 associated $v\beta$-trails in $E(B)$, $P_{i}(v,\beta)$, $i=0,1$. Each $P_{i}$ is alternating and ends with an edge whose $M$-type is that of $B$. The starting edge for $P_{1}(v,\beta)$ has the same $M$-type as $B$; it has the opposite $M$-type for $P_{0}(v,\beta)$. There is one exceptional trail: $P_{0}(\beta,\beta)$ is the length 0 trail $\beta$.

Next we describe the linear program dual variables for weighted matching, as they are maintained in [12] (extending [8]). Assume every edge of $G$ has a numerical weight $w(e)$. We define a subset of the edges incident to a blossom $B$ by

(Here $M$ is the matching and $I$ stands for “incident”.) There are two dual functions $y:V\to\mathbb{R}$, $z:2^{V}\to\mathbb{R_{+}}$. Define $\widehat{yz}:E\to\mathbb{R}$ by

The duals satisfy the LP feasibility conditions, which (using CS for blossoms) are:

We say $e$ is tight if equality holds, otherwise loose. The terms strictly dominated and strictly underrated refer to the possibilities $>w(e)$ and $<w(e)$ respectively.

A positive blossom has positive $z$-value. For any positive blossom $B$ every edge of $E(B)+\eta(B)$ is tight. (A zero blossom has $z(B)=0$. Note that the subgraph of a zero blossom can be part of the subgraph of a positive blossom. For example the blossoms of Fig.2(b)–(d) are zero blossoms. We can make (b) a positive blossom (lower $y(\phi)$ to $-4$) but (c) or (d) remains a zero blossom.)

The last property of duals is that all free vertices $\phi$ have the same $y$-value $y(\phi)$. A structured matching consists of a matching, duals $y,z$, and positive blossoms wherein every unmatched (matched) edge is dominated (underrated) respectively; every edge in a positive blossom subgraph is tight; and every free vertex has the same $y$-value.

Edmonds’ algorithm and its extension to $f$-matching both work by constructing a “search structure” $\cal S$. The structure consists of atoms and blossoms, and $\overline{\cal S}$ denotes $\cal S$ with all blossoms contracted. $\overline{\cal S}$ is a forest, and we call the elements of $V({\overline{\cal S}})$ “nodes”. (Recall our tree terminology from Section 1.) Every edge of $\cal S$ is tight. The roots of $\overline{\cal S}$ are the free vertices/blossoms. If $v$ is an atom of $\overline{\cal S}$ then $\tau(v)$ alternates with each child edge of $v$. (For a free vertex $v$ take $\tau(v)=\varepsilon v$.) The base edge of a blossom $B$ in $\cal S$ is its parent edge $\tau(B)$.

Every node of $\overline{\cal S}$ has a corresponding path to the root of its search tree. These paths expand to trails in $G$ for every vertex of $\cal S$. We denote the trails as $P_{j}(v)$, defined as follows. Call an atom $v$ of $\overline{\cal S}$ inner if $\tau(v)$ is unmatched and outer if $\tau(v)$ is matched. Correspondingly an io-type is $i$ or $o$, for inner and outer respectively. A vertex $v$ in a blossom has both io-types. Any vertex of $\cal S$ has a trail $P_{j}(v)$. Here $j$ is the io-type of $v$. $P_{j}(v)$ starts with an edge from $v$ that is unmatched (matched) for $j=i$ ($j=o$). It ends on an unmatched edge to a free vertex. $P_{j}(v)$ starts with $\tau(v)$ if $v$ is atomic. If $v$ is in a blossom $P_{j}(v)$ starts with the trail $P_{i}(v)$, where $i$ is chosen to correspond with $j$. A node on a path to the root in $\overline{\cal S}$ expands in a $P_{\cdot}$ trail in the obvious way. Every trail $P_{j}(v)$ is alternating.

For a blossom $B$, two edges of $\delta(B)$ alternate if one of the edges is $\eta(B)$. The motivation for this definition is that the two edges can be combined with the appropriate $P_{i}$ trail to form an alternating trail in $G$. Note that with this definition, ${\overline{\cal S}}$ is an alternating forest, i.e., its edges alternate at every node.

$P_{j}(v)$ is a shortest alternating trail that starts with an unmatched (matched) edge for $j=i$ ($j=0$) and ends with an unmatched edge to a free vertex. (This can be proved following the development in [11] for 1-matching, or using the tracking algorithm of Section 3. But we do not use this fact.) We denote the length of $P_{j}(v)$ as $l_{j}(v)$. $l_{j}(v)$ is the basic level of a vertex in the level graph. But unlike level graphs of previous papers, vertices have other levels. We call these other levels petalevels, since they are caused by petals in shortest trails.²²2In more detail ordinary levels are formed by trails that enter every positive blossom on its base edge. So they are “basic levels”. In contrast petalevels enter at least one positive blossom on a petal. Fig. 2(c)-(d) show a vertex with ordinary level $l_{o}(x)=2$ and petalevel 4.

This section provides a central motivation for our proof of $(*)$, an example graph with too many petalevels to include in the desired level graph. Strictly speaking the example is not needed to prove $(*)$. But we include it here to provide an in-depth look at the search structure $\cal S$, to illustrate how multiple petalevels arise, and to provide details about how the example graph’s search structure can actually occur in our algorithm Fig. 1. (The remaining details for this are given in Appendix A.)

The example graph $EG(b)$ has $b$ blossoms, for any even $b\geq 2$. $EG(b)$ has $\Theta(b)$ vertices and edges. It has sat length $4b+1$ and importantly, $\Theta(b^{2})$ petalevels. Fig.3 gives a simple example, with $b=4$ and sat length 17. Its table shows there are 6 petalevels (that are not ordinary levels).

As usual vertical level in the figure corresponds to level, i.e., shortest distance from a free vertex. So the two vertices with $y$-value 0 are at level 8. Edges drawn vertically are in the search structure $\cal S$.

The blossoms $B_{i}$ are alternately heavy and light. The triangle blossom $B_{1}$ has a slightly different structure from the others. Changing it to have the same structure gives a similar construction, with multiple petalevels. We do not present it because that structure cannot be grown using the algorithm of Fig.1. Fig.5 shows how the algorithm grows the search structure of Fig. 3.

Note also that a similar example uses only light blossoms. So our construction can be modified for ordinary matching. The given construction has the advantage of being smaller.

Fig.4 illustrates the general structure of the example graph $EG$. In addition we assume $f(p_{0})=f(\beta_{b})=1$. The remaining vertices are saturated, as implied by the figure, and their $f$-value is 1 or 2. The figure implies the following expressions that combine to define the augmenting trail of a cross edge $\beta_{i}p_{1+3(b-i)}$.

Let $j$ be the io-type $o$ ($i$) for $i$ odd (even). Let $S_{i}=s_{i1}s_{i2}s_{i3}s_{i4}$. For every $i\geq 1$,

To calculate the length of $\beta_{i}$’s trail first note $|S|=3(b-i)$. So $|l_{j}(\beta_{i})|=(b-i)+1+3(i-1)+1+1+(i-1)=b+3i-1$.

The augmenting trail for $\beta_{i}$ is

Its length is $(b+3i-1)+1+(1+3(b-i))=4b+1$ as desired.

It is clear that the graph has size $\Theta(b)$, and (2.4) implies there are $\Theta(b^{2})$ petalevels. It only remains to prove the augmenting trails of (2.4) are all sats, i.e., $s=4b+1$. Appendix A shows this by exhibiting the optimum dual values. Here we present a path-tracing argument for greater intuition.

Let $T$ be a sat. The degree constraints imply $T$ contains both free vertices $p_{0}$ and $\beta_{b}$. So $T$ contains a cross edge $p_{1+3(b-i)}\beta_{i}$. The alternation of $T$ implies similar properties on both sides of the cross edge:

The arguments for (b)–(c) are similar to (a): Arriving at a vertex $v$ on a directed edge $uv$, alternation implies there is only one possibility for the next edge.

(b) Examining Fig. 4(b), we use the nonalteration of directed edge $s_{24}s_{23}$.

(c) Examining Fig. 4(a)-(b), we successively use nonalteration of directed edges $s_{j4}\beta_{j-1}$, $s_{24}\beta_{1}$, $\beta_{j}s_{j+1,4}$. $\Box$

Part (a) of the lemma shows $T$ consists of the trail $p_{0}\ldots p_{1+3(b-i)}\beta_{i}$ and a trail from $\beta_{i}$ to $\beta_{b}$, that begins with an edge of opposite M-type from the cross edge $p_{1+3(b-i)}\beta_{i}$. We claim that trail must be $l_{j}(\beta_{i})$. (2.6) then shows $s=4b+1$ as desired.

Before proving the claim we show another simple property (self-evident from Fig. 4(a)-(b)).

For $i=1$ a shortest path from $\beta_{1}$ to $\beta_{b}$ is $\beta_{1}\ldots\beta_{b}$. For every $i>1$, a shortest path from $s_{i1}$ to $\beta_{b}$ is $s_{i1}\beta_{i}\ldots\beta_{b}$.

$T$ enters blossom $B_{i}$ on cross edge $p_{1+3(b-i)}\beta_{i}$. To reach $\beta_{b}$ it leaves $B_{i}$ either on edge $\beta_{i}\beta_{i+1}$ or $s_{i+1,4}s_{i+1,3}$. Observe that in both cases $T$ contains an alternating trail from $\beta_{i}$ to the edge $s_{i1}s_{i2}$: To leave on $\beta_{i}\beta_{i+1}$ alteration implies $T$ uses both edges descending from the cross edge, and alteration at $s_{i1}$ implies $T$ uses $s_{i1}s_{i2}$ as claimed. To leave on $s_{i+1,4}s_{i+1,3}$, alteration at $s_{i+1,4}=s_{i1}$ also implies $T$ uses $s_{i1}s_{i2}$.

Now Proposition 2.2(c) shows $T$ contains

The rest of $T$ joins $s_{i1}$ to $\beta_{b}$. Proposition 2.3 implies a shortest such path path is

$T$ can use this path, since it alternates with $s_{i1}s_{i2}$. Since $T$ is a sat we can assume $T$ uses it. Now (2.4) shows the claim holds.