4.7 Multipath Blocking Vertices Property
4.7.1 Non-Blocking Multipaths
In analogy toCorollary 4.11for the Blocking Vertices Property and simple forward paths, we can formulate the existence of non-blocking multipaths as follows:
Lemma 4.14 Multipath Blocking Vertices Property
Let π∗be an optimal linear ordering of a graph G. Then, π∗
sp/land π∗sp/rrespect the Multipath Property.
Proof. Just as the Multipath Blocking Vertices Property is obtained from a combination of the Blocking Vertices Property and the Multipath Property, so is its proof. Let π∗ be an optimal linear ordering of G= (V, A) and suppose, for the sake of contradiction, that
π∗ does not respect the Multipath Blocking Vertices Property.
The Right-Blocking Split Graph and Outgoing Backward Arcs We assume first that πsp/r∗
does not respect the Multipath Property. In this context, let v be a vertex of G, and suppose that the Multipath Property is violated for the set of outgoing backward arcs B+(v) of v. As in the proof of the Multipath Property, the forward paths for the outgoing
and incoming backward arcs of a vertex can be considered separately. The argument for the set of incoming backward arcs B−(v) is symmetric and will be addressed at the end
of the proof.
In case that v itself is right-blocking and therefore split into vland vr, substitute v by vl in the following paragraphs. For the correctness, recall that vlinherits F−(v) and B+(v),
so F−(vl) corresponds to F−(v) and B+(vl) corresponds to B+(v). Furthermore, arcs in
F+(v) cannot be part of forward paths for backward arcs in B+(v).
By assumption, there are no pairwise arc-disjoint forward paths for the backward arcs in B+(v) in the right-blocking split graph with linear ordering π∗
sp/r. Let s, D, X, and Y be defined as in the proof ofLemma 4.10, i. e. , s is the newly introduced source with an outgoing arc to every head of a backward arc in B+(v), D is a minimum cut with
|D| < b+(v) that separates s from v, X = D ∩ F, and Y = B+(v) \ {(v, h) | (s, h) ∈ D} (cf.Figure 4.10(b)).
Because we considered the split graph when computing the minimum cut, X only covers forward paths that are preserved during the splitting of right-blocking vertices. In consequence, there may be uncovered forward paths in the unsplit graph G that contain right-blocking vertices.
To overcome this issue, construct a set Z of right-blocking vertices as follows: For each backward arc in Y , consider all forward paths according to π∗ in the unsplit graph G
4.7 Multipath Blocking Vertices Property 99 Gsp/r F h h0 l h0r v part of minimum cut D or ∈ Z
Figure 4.13:Proof of Multipath Blocking Vertices Property: A forward path h ⇝ v for the
backward arc(v, h) in G (but not in Gsp/r) that contains the head of another outgoing backward arc(v, h′) (respectively (v, h′
r) in Gsp/r) of v. If all forward paths for(v, h′) are covered, so are all forward paths for(v, h) via h′.
that do not contain an arc of the reduced minimum cut X. Place the last right-blocking vertex that occurs on a traversal of the respective cropped forward paths in Z. Because
X is a minimum cut for all forward paths for arcs in Y according to π∗
sp/r in the split graph, every cropped forward path not passing through an arc in X must contain a right-blocking vertex.
The intention now is to additionally exchange the incoming backward arcs for the outgoing forward arcs of the right-blocking vertices in Z in analogy to the proof of the Blocking Vertices Property. Let H = {h ∈ V | (v, h) ∈ Y } be the set of all heads of the backward arcs in the reduced set Y of outgoing backward arcs of v. Suppose that some vertices in Z are at the same time the head of an outgoing backward arc of
v, i. e. , H ∩ Z ̸= ∅. As we only consider cropped forward paths, there must hence be
two backward arcs b= (v, h) , b′ = (v, h′) ∈ Y such that a forward path Pb for b in the unsplit graph G passes h′ and h′ is right-blocking, i. e. , b′ = (v, h′) = (v, h′
r) (see also
Figure 4.13). Note, however, that this also implies that every forward path for b′in G is a subpath of a forward path for b and that destroying all forward paths of b′ also cuts all forward paths of b that contain h′. As we placed the last right-blocking vertex that occurs on a traversal of every cropped forward path for b in Z and every subpath h′ ⇝ v either contains a right-blocking vertex or an arc of X, h′̸∈ Z. Subsequently, Z ∩ H = ∅, and, as all forward paths end at v, v ̸∈ Z either.
Let now
B′ = B \ B−[Z] ∪ F+[Z]
and
ByLemma 4.7, B′is feasible and |B′| = |B|. Furthermore, every path that consists only of arcs in F′ and contains a vertex in Z must end within Z. Next, consider
B′′= B′\ Y ∪ X = B \ (B−[Z] ∪ Y ) ∪ F+[Z] ∪ X
and
F′′= A \ B′′= F′\ X ∪ Y = F \ (F+[Z] ∪ X) ∪ B−[Z] ∪ Y.
Note that Z∩H = ∅ immediately implies that B−[Z]∩Y = ∅. Furthermore, F+[Z]∩X = ∅
because X covers exactly all forward paths in π∗
sp/r and no forward path in πsp/r∗ can contain a vertex of Z. Hence, if the tail t of an arc in X were in Z, then t must be the head of an outgoing backward arc of v, but Z ∩ H = ∅. As shown in the proof ofLemma 4.10, |X| < |Y |. Thus, |B′′| < |B′| = |B|.
It remains to show that B′′is feasible, i. e. , G⏐
⏐F′′ is acyclic. Suppose, for contradiction,
that G⏐
⏐F′′contains a simple cycle C, i. e. , all arcs of C are in F′′. Due to B′ being feasible,
Cmust contain exactly one arc(v, h) ∈ Y . Observe that if C contained more than one arc of Y , it would contain v at least twice and would therefore not be simple. Then, the remaining arcs of the cycle must constitute a path P= h ⇝ v and consist solely of arcs in F′\ X. As every path in F′that contains a vertex of Z must also end in Z and v ̸∈ Z by construction, P cannot contain a vertex of Z. Thus, P is a forward path for(v, h) in
G⏐
⏐F and contains neither an arc of X nor a vertex of Z, a contradiction. Subsequently,
Ccannot exist, so B′′ is feasible. Because |B′′| < |B|, this, however, contradicts the optimality of π∗.
The Left-Blocking Split Graph and Outgoing Backward Arcs Let us now suppose that
π∗
sp/ldoes not respect the Multipath Property and consider again the outgoing backward arcs B+(v) of a vertex v.
The proof follows largely that for the right-blocking split graph, i. e. , we add a source vertex s with an outgoing arc to every head of a backward arc in B+(v) and consider the
minimum s − v cut D. If the Multipath Property is violated, then |D| < b+(v) and we
define the sets X and Y as in the right-blocking case.
There is a difference in the construction of the set Z in that it contains left- instead of right-blocking vertices: For each backward arc in Y , consider again all forward paths according to π∗in the unsplit graph G that are not covered by an arc of the reduced cut
X, but now place the last left-blocking vertex that occurs on a traversal of the respective
4.7 Multipath Blocking Vertices Property 101 As we are dealing with left-blocking vertices, we are interested in the arc sets F−[Z]
as well as B+[Z] here. By applying againLemma 4.7, we obtain that
B′ = B \ B+[Z] ∪ F−[Z]
is feasible, |B′
| = |B| and that every path that consists only of arcs in F′ = A \ B′ = F \ F−[Z] ∪ B+[Z]
and contains a vertex in Z must start in Z.
With the same argument as above, we obtain that H ∩ Z = ∅, which implies that
Y ∩ B+[Z] = ∅. Furthermore, X ∩ F−[Z] = ∅, because all vertices in Z are left-blocking
and therefore split vertically in πsp/l. This implies that all forward paths using an arc in F−[Z] must end there, but X is a minimum cut for forward paths ending at v. We obtain
the improved set of backward and forward arcs as
B′′= B′\ Y ∪ X = B \ (B+[Z] ∪ Y ) ∪ F−[Z] ∪ X
and
F′′= A \ B′′ = F′\ X ∪ Y = F \ (X ∪ F−[Z]) ∪ B+[Z] ∪ Y.
It remains again to show that B′′is feasible, i. e. , G⏐
⏐F′′ is acyclic. Hence, suppose that G⏐
⏐F′′contains a simple cycle C. As in the right-blocking case, C must then contain exactly
one arc(v, h) ∈ Y and all other arcs of C form a path P = h ⇝ v that uses only arcs in F′\ X. Here, every path in F′that contains a vertex of Z must also start in Z, but h ∈ H and H ∩ Z = ∅, so P again cannot contain a vertex of Z. Subsequently, P is a forward path for(v, h) in G⏐
⏐F that is covered neither by X nor by Z, a contradiction. Hence, B′′
is feasible with |B′′| < |B| a contradiction to π∗ being optimal.
Incoming Backward Arcs The respective proofs for the incoming backward arcs in the
left-blocking as well as the right-blocking case follow by considering the reverse graph
GRalong with the reverse linear ordering π∗R. Note that the combination of incoming backward arcs and the left-blocking case corresponds to the outgoing backward arcs and the right-blocking case in the reverse graph and linear ordering, and the combination of incoming backward arcs and the right-blocking case corresponds to the outgoing backward arcs and the left-blocking case.
Figure 4.14continues the example given in Figure 4.7 and shows how this linear ordering instance is improved by enforcing the Multipath Blocking Vertices Property for the outgoing backward arcs of vertex v: The first subfigure,Figure 4.14(a), depicts
u h0 x h1 yl yr h2 z v w s u h0 x h1 yl yr h2 z v w u h0 x v w h1 y h2 z (a) (b) (c)
Figure 4.14:Enforcing the Multipath Blocking Vertices Property on the instance intro-
duced inFigure 4.7for the outgoing backward arcs of vertex v: Find a minimum s − v cut(a), identify the backward and forward arcs to exchange with each other(b), and obtain the improved linear ordering(c).
4.7 Multipath Blocking Vertices Property 103 the acyclic subgraph of the right-blocking split graph with an additional source s and arcs to every head of an outgoing backward arc of v. The dashed red arcs highlight a minimum s − v cut, which consists of the two arcs (s, h0) and (z, v). As (s, h0) is incident to s, the reduced minimum cut X contains only (z, v), and the subset Y ⊆ B+(v) of
exchangeable backward arcs equals {(v, h1) , (v, h2)}. Additionally, we construct the set of right-blocking vertices Z with Z = {y}, because y is the last right-blocking vertex encountered on a traversal of the cropped forward path ⟨y⟩ of ⟨h1, y, v⟩ in the unsplit acyclic subgraph G⏐
⏐F. We thus have identified two sets of arcs that switch their roles from
backward to forward and vice versa: The arcs in Y ∪ B−(y) = {(v, h1) , (v, h2) , (w, y)}
will become forward arcs, whereas the arcs in X ∪ F+(y) = {(z, v) , (y, v)} will become
backward arcs. InFigure 4.14(b)these two arc sets are highlighted. As the right-blocking split graph is shown, however,(w, y) must be translated to (w, yr) and (y, v) to (yr, v).
Finally,Figure 4.14(c)depicts a linear ordering obtained by topologically sorting the improved acyclic subgraph.
We introduce a predicate MNoBlock(π) that expresses whether a linear ordering π respects the Multipath Blocking Vertices Property and derive fromLemma 4.14:
Corollary 4.17
For every linear ordering π holds: Opt(π) ⇒ MNoBlock(π).
The definition of the Multipath Blocking Vertices Property immediately implies that ∀π : MNoBlock(π) ⇒ NoBlock(π) ∧ MPath(π). WithCorollary 4.14, we thus obtain:
Corollary 4.18
For every linear ordering π holds:
MNoBlock(π) ⇒ Nest(π) ∧ Path(π) ∧ NoBlock(π) ∧ MPath(π).