At the end of the previous section, we introduced the Extended Multipath Property, which is stronger than the Multipath Property, but not efficiently enforceable anymore. A similar extension is possible for the Multipath Blocking Vertices Property, which uses the following statement:
Lemma 4.16
Let B be the set of backward arcs induced by a linear ordering π of a graph G such that π respects the Multipath Blocking Vertices Property and b= (u, w) ∈ B. Let v be a vertex on a forward path Pbfor b that contains no left-blocking vertex and let Y ⊆ B+(v) or Y ⊆ B−(v) be a set of either incoming or outgoing backward
arcs of v. Furthermore, let Z ⊆ V be a set of left-blocking vertices that contains neither w nor, if Y ⊆ B+(v), the head of any arc in Y . If there is a set X of
forward arcs induced by πsp/lthat covers every forward path of a backward arc in Y ∪ [b]∥in πsp/l, Z covers every forward path of a backward arc in Y ∪ [b]∥in π that contains a left-blocking vertex, and |X| ≤ |Y |, then B \(
B+[Z] ∪ Y ∪ [b]∥)∪ F−[Z] ∪ X is feasible.
Likewise, let v be a vertex on a forward path Pb for b that contains no right- blocking vertex and let Y ⊆ B+(v) or Y ⊆ B−(v) be a set of either incoming or
outgoing backward arcs of v. Furthermore, let Z ⊆ V be a set of right-blocking vertices that contains neither u nor, if Y ⊆ B−(v), the tail of any arc in Y . If there
is a set X of forward arcs induced by πsp/rthat covers every forward path of a backward arc in Y ∪ [b]∥ in πsp/r, Z covers every forward path of a backward arc in Y ∪ [b]∥ in π that contains a right-blocking vertex, and |X| ≤ |Y |, then B \(B−[Z] ∪ Y ∪ [b]∥)∪ F+[Z] ∪ X is feasible.
Proof. Let G = (V, A) and denote by F = A \ B the set of forward arcs induced by π. We consider first the left-blocking case. As v is on the forward path Pb = w ⇝ u in
πsp/l, Pb consists of two subpaths P′b = w ⇝ v and P′′b = v ⇝ u. Furthermore, v is not left-blocking by the lemma’s preconditions.
As π respects the Multipath Blocking Vertices Property, πsp/lrespects the Multipath Property byLemma 4.14. Consequently, there must be pairwise arc-disjoint forward paths for the backward arcs in B(v) = B+(v) ∪ B−(v) in π
sp/l, which in particular also holds for those in Y . Hence, every set of forward arcs that covers all forward paths for the backward arcs in Y must have cardinality at least |Y |. As |X| ≤ |Y |, we can conclude that |X| = |Y | and that every arc in X is part of at least one forward path for a backward arc in Y in πsp/l.
Next, consider the set of left-blocking vertices Z. ByLemma 4.7, B′ = B \ B+[Z] ∪ F−[Z]
4.7 Multipath Blocking Vertices Property 109 is feasible, |B| = |B′| and every path consisting only of arcs in
F′ = A \ B′ = F \ F−[Z] ∪ B+[Z]
that contains a vertex in Z must also start at a vertex in Z. Let(t, h) ∈ Y ∪ [b]∥ and suppose there is a path P = h ⇝ t that uses only arcs in F′. If P contains a vertex in
Z, then, due toLemma 4.7, h ∈ Z, a contradiction to our precondition that Z contains
neither w nor the head of any arc in Y if(t, h) ∈ B+(v) and to v being not left-blocking if
(t, h) ∈ B−(v), i. e. , h = v. Consequently, P contains only arcs in F and no vertex in Z,
which in turn implies that P is covered by X. Let now
B′′= B′\(Y ∪ [b]∥)∪ X
and
F′′= A \ B′′= F′\ X ∪ Y ∪ [b]∥.
Suppose there is a cycle C in G⏐
⏐F′′. W. l. o. g., we assume that C is simple. As B′is feasible, G⏐
⏐F′ is acyclic, so C cannot only consist of arcs in F′\ X. Hence, C must contain at least
two arcs from Y ∪ [b]∥: If C contained just one arc(t, h) ∈ [b]∥∪ Y , then there must be a path h ⇝ t in F′
\ X, a contradiction to our conclusion in the previous paragraph. Furthermore, if C contains two or more parallel arcs in[b]∥, it is not simple. Hence, C must use at least one arc of Y . Suppose C uses neither b nor one of its parallel arcs. In case that
Y ⊆ B+(v), this implies that C consists of a set of arcs Y′ = {(v, hi) | 0 ≤ i < k} ⊆ Y as well as a set of pathsP = {hi⇝ v| 0 ≤ i < k}, where k ≥ 2. Otherwise, if Y ⊆ B−(v),
Canalogously consists of a set of k ≥ 2 arcs Y′ = {(ti, v) | 0 ≤ i < k} ⊆ Y as well as
a set of pathsP = {v ⇝ ti | 0 ≤ i < k}. In both cases, the paths in P use only arcs in F′\ X, which once more contradicts our conclusion in the previous paragraph.
Hence, exactly one arc of[b]∥ and k ≥ 1 arcs of Y must be part of C. Consider the case that Y ⊆ B+(v) and let Y′ = {(v, hi) | 0 ≤ i < k} ⊆ Y be the subset of arcs in Y that is used in C. Then, C must also contain subpaths hj ⇝ u for some j∈ {0, . . . , k − 1} and, more importantly, a subpath P′ = w ⇝ v, such that all use only arcs in F′
\ X. Recall that every arc in X is also part of a forward path for a backward arc in Y in πsp/land all of these forward paths end at v. Hence, P′′
b = v ⇝ u uses neither an arc arc in X nor does it contain a vertex in Z. The same applies for P′, which yields that they together form a path w ⇝ v ⇝ u that contains neither a left-blocking vertex nor an arc in X and is a forward path for b in π, a contradiction. For the alternative case that Y ⊆ B−(v), let
must then contain a subpath w ⇝ tj, for some j ∈ {0, . . . , k − 1}, as well as a subpath P′ = v ⇝ u, and both use only arcs in F′\ X. Every arc in X must again also be part of a forward path for a backward arc in Y in πsp/l, all of which start at v. Thus, P′b = w ⇝ v can neither contain an arc in X nor a vertex in Z. Hence, the combination of P′
b and P′ results in a path w ⇝ v ⇝ u, which is a forward path for b in π that is not covered by X and does not pass through a left-blocking vertex, a contradiction.
Consequently, G⏐
⏐F′′is acyclic, which yields that
B′′= B′\(Y ∪ [b]∥)∪ X =( B \ B+[Z] ∪ F−[Z])\(Y ∪ [b]∥)∪ X = B \( B+[Z] ∪ Y ∪ [b]∥)∪ F−[Z] ∪ X is feasible.
The analogous statement using right-blocking vertices follows once more by consider- ing the reverse graph GRalong with the reverse linear ordering πR.
Note that for an application ofLemma 4.12in the proof ofLemma 4.16, we would have needed a linear ordering that induces the intermediate set of backward arcs B′ and, most notably, respects the Multipath Property, which we cannot guarantee in general.
UsingLemma 4.16, we can strengthen the Multipath Blocking Vertices Property as follows:
Lemma 4.17 Extended Multipath Blocking Vertices Property
Let π∗be an optimal linear ordering of a graph G and let b= (u, w) be a backward arc induced by π∗. If v is a vertex on a forward path Pbfor b in π∗
sp/l(π ∗
sp/r) and
uand w are not themselves left-blocking (right-blocking), then there is a set of
pairwise arc-disjoint forward paths for the backward arcs in B+(v) ∪ [b]∥as well
as one for the backward arcs in B−(v) ∪ [b]∥in π∗
sp/l(πsp/r∗ ).
Proof. Let G = (V, A) and denote by B and F the set of backward and forward arcs induced by π∗, respectively. Note that if b ∈ B(v), then the pairwise arc-disjoint for- ward paths for B+(v) ∪ [b]∥ = B+(v) and B−(v) ∪ [b]∥ = B−(v) are already implied by
Lemma 4.14. Hence, assume in the following that b ̸∈ B(v).
As π∗is optimal and hence respects the Multipath Blocking Vertices Property, both
π∗
4.7 Multipath Blocking Vertices Property 111 arcs before and after a vertical split, π∗
sp/land πsp/r∗ induce the same sets of backward and forward arcs as π∗.
Suppose there is a backward arc b= (u, w) ∈ B and a vertex v on a forward path Pb of b in πsp/l such that u and w are not left-blocking and no set of pairwise arc-disjoint forward paths for the backward arcs in B+(v)∪[b]∥or for the backward arcs in B−(v)∪[b]∥
exists. Then, there must be a set Y ⊆ B+(v) or Y ⊆ B−(v), respectively, as well as a
directed multicut X ⊆ F such that |X| ≤ |Y | +⏐⏐ ⏐[b]∥
⏐ ⏐
⏐, and X covers all forward paths for
the backward arcs in Y ∪ [b]∥in π∗ sp/l.
Identify a set Z of left-blocking vertices by traversing all cropped forward paths for the backward arcs in Y that do not contain an arc of X and placing the first left-blocking vertex that occurs in Z. If Y ⊆ B+(v), we traverse all forward paths backwards, i. e. ,
starting at v, whereas if Y ⊆ B−(v), we use the usual direction (and thereby also start at
v). Proceed likewise in both cases for all cropped forward paths for b that do not contain
an arc in X and add the left-blocking vertices encountered first to Z. As u, v, and w are not left-blocking, they cannot be in Z. Assume that Y ⊆ B+(v). Then, there also is no
head of a backward arc in Y contained in Z (cf. also the proof ofLemma 4.14): In this case, there would have been backward arcs(v, h) and (v, h′) such that, w. l. o. g., h′is left-blocking and part of a cropped forward path P for(v, h). Then, however, P has a subpath which is a cropped forward path for(v, h′), which either contains an arc of X or a left-blocking vertex z ∈ Z. In the former case, P also contains an arc of X and is therefore not considered, whereas in the latter, z must have been encountered before
h′ on a backward traversal of P and we only collect the first left-blocking vertex that
occurs to create Z. The analogous argument yields that if Y ⊆ B−(v), then no tail of a
backward arc in Y can be contained in Z.
As |X| ≤ |Y |, we can applyLemma 4.16both if Y ⊆ B+(v) and if Y ⊆ B−(v), which
yields that
B′ = B \(B+[Z] ∪ Y ∪ [b]∥)∪ F−[Z] ∪ X
is feasible. As | [b]∥| ≥ 1, b+[Z] = f−[Z] byProposition 4.2, and all sets are pairwise
disjoint, |B+[Z] ∪ Y ∪ [b]∥| > |F−[Z] ∪ X|, which implies that |B′| < |B|, a contradiction to π∗being optimal.
The analogous statement for right-blocking vertices follows immediately from the left-blocking case by considering the reverse graph GR along with the reverse linear ordering πRinstead.
For the same reason as for the Extended Multipath Property, the proof of the Extended Multipath Blocking Vertices Property does not suggest an efficient algorithm to check or enforce it, due to the N P-hardness of the DIRECTED MULTICUTproblem (cf. Sec-
tion 4.6.4). In particular their preparatory lemmas,Lemma 4.12andLemma 4.16, may however be useful in case that a directed minimum cut has already been obtained by some other procedure. In this case, weaker versions of the Extended Multipath Prop- erty and the Extended Multipath Blocking Vertices Property may be implemented in polynomial time. In fact, two such properties are employed inSection 5.4.1.