Rooted Routing Using Structural
Decompositions
Jiao Tong University
Shanghai, China
June 17, 2013
The Bridges of Konigsberg
Euler, 1736
A Prescient Observation by Euler
Euler said: “the problem… can be solved by making
an exhaustive list of all possible routes … because of
the number of possibilities this method would be
too difficult and laborious and in problems with
more bridges it would be impossible.
Much of the development in modern graph theory
is tied to efficient computer algorithms which
solves routing problems modeled using graphs
A Prescient Observation by Euler
Euler said: “the problem… can be solved by making
an exhaustive list of all possible routes … because of
the number of possibilities this method would be
too difficult and laborious and in problems with
more bridges it would be impossible.
Much of the development in modern graph theory
is tied to efficient computer algorithms which
solves routing problems modeled using graphs
Routing in The Modern Era
A Long Tour
•
GRAPHS
A Graph is a set of vertices
unordered pair of vertices
K5
and a set of edges, each of which is an
K3,3
Modelling Euler’s Problem Using
Graphs
Paths and Connectivity
• A path is a (finite) sequence of distinct
vertices, consecutive elements of which are
joined by an edge.
Paths and Connectivity
• A path is a (finite) sequence of distinct
vertices, consecutive elements of which are
joined by an edge.
• The first and last vertices of a path are its
endpoints.
Paths and Connectivity
• A path is a (finite) sequence of distinct
vertices, consecutive elements of which are
joined by an edge.
• The first and last vertices of a path are its
endpoints.
• A graph is connected if every pair of vertices
are joined by a path.
Degrees of Connectivity & Cutsets
Degrees of Connectivity & Cutsets
A cutset is a set of vertices whose deletion
leaves a disconnected graph.
Determining Point to Point
Connectivity
• Given a graph G and vertices x,y  V,
determine if there are k paths from x to y
pairwise disjoint except at their endpoints.
Determining Point to Point
Connectivity
• Given a graph G and vertices x,y  V,
determine if there are k paths from x to y
pairwise disjoint except at their endpoints.
• Given a graph G and sets S and T determine if
there are k pairwise disjoint paths from S to
T.
Menger’s Theorem
There exists
exactly one of
k vertex disjoint S - T paths
a subset X of G with |X| < k, that hits all S - T paths.
An Algorithm and Some Applications
• There is an efficient (O(k|E(G)|)-time)
algorithm to find either k vertex disjoint S-T
paths in G or a set of fewer than k vertices
hitting all S-T paths of G.
An Algorithm and Some Applications
• There is an efficient (O(k|E(G)|)-time)
algorithm to find either k vertex disjoint S-T
paths in G or a set of fewer than k vertices
hitting all S-T paths of G.
• This algorithm has a myriad of applications in
routing
An Algorithm and Some Applications
• There is an efficient (O(k|E(G)|)-time)
algorithm to find either k vertex disjoint S-T
paths in G or a set of fewer than k vertices
hitting all S-T paths of G.
• This algorithm has a myriad of applications in
routing,scheduling, resource allocation, game
theory, algebra,…..
k Disjoint Rooted Paths
• Given subsets S={s1,...,sk} and T={t1,…,tk} of V
determine if there are k vertex disjoint paths
P1,…,Pk such that Pi contains si and ti
k Disjoint Rooted Paths
• Given subsets S={s1,...,sk} and T={t1,…,tk} of V
determine if there are k vertex disjoint paths
P1,…,Pk such that Pi contains si and ti
• Robertson and Seymour developed an
efficient algorithm to solve this problem, and
in so doing produced some of the depest and
most important results in graph theory.
An Instructive Example
s1
t2
s2
t1
A Second Instructive Example
Clique C and 2k vertex-disjoint S∪T to C paths => P1,... , Pk exist.
An Illuminating Observation
• If G contains a clique C with 2k+1 vertices and
Q is a maximal set of vertex disjoint paths
from S  T to V(C) then any vertex of C not on
any of the elements of Q is irrelevant (i.e. the
desired P1,…,Pk exist in G precisely if they exist
in G-v).
An Illuminating Observation
• If G contains a clique C with 2k+1 vertices and
Q is a maximal set of vertex disjoint paths
from S  T to V(C) then any vertex of C not on
any of the elements of Q is irrelevant (i.e. the
desired P1,…,Pk exist in G precisely if they exist
in G-v).
Clique Models
• A Kl model in G consists of l vertex disjoint
connected subgraphs of G every two of which
are joined by an edge
Using Clique Models for k-DRP
• If G contains a K8k+5 model then we can quickly
find an irrelevant vertex of G.
Our Algorithm for k-DRP
• If G contains a K8k+5 model then we can quickly
find an irrelevant vertex v of G and recurse on
G-v.
• Otherwise we solve the problem by exploiting
the structure that the exclusion of a clique
model yields.
Graphs Without Clique Minors I:
Forests
• A connected graph G has no K3 model precisely if
it has no cycle. I.e. if each of its connected
components is a tree.
• Thus if a graph with no K3 model does not have a
cutvertex it is an edge.
Graphs Without Clique Minors II:
Forbidding K5
• If a graph with no K5 model does not have a
cutset of size 3 then it is planar or a special 8
vertex graph L.
Wagner, 1946.
Tree Decompositions
A tree decomposition decomposes G into pieces
corresponding to the nodes of a tree using
cutsets corresponding to its arc.
Tree Decompositions
• A tree decomposition for
G consists of a tree T and
a subtree Sv for each
vertex v of G s.t. if uv is
an edge of G then Su
intersects Sv.
• We set Wt ={v| t Sv} and
Ht to be the graph with
vertex set Wt s.t. uv εE(Ht)
iff. uvεE(G) or u,v ε Ws 
Wt for some st εE(T).
• Every G has a one node
tree decomposition.
• If uv is a nonedge of G
then G has a tree
decomposition using tree
st where st corresponds
to the cutset V-u-v.
• The k by k grid has a tree
decomposition where T is
a path on k^2 nodes each
internal node of which
corresponds to a cutset of
the grid of size k
Tree Decompositions
• A tree decomposition for
G consists of a tree T and
a subtree Sv for each
vertex v of G s.t. if uv is an
edge of G then Su
intersects Sv.
• We set Wt ={v| t Sv} and
Ht to be the graph with
vertex set Wt s.t. uv εE(Ht)
iff. uvεE(G) or u,v ε Ws 
Wt for some st εE(T).
• Every G has a one node
tree decomposition.
• If uv is a nonedge of G
then G has a tree
decomposition using tree
st where st corresponds
to the cutset V-u-v.
• The k by k grid has a tree
decomposition where T is
a path on k^2 nodes each
internal node of which
corresponds to a cutset of
the grid of size k
Tree Decompositions
• A tree decomposition for
G consists of a tree T and
a subtree Sv for each
vertex v of G s.t. if uv is an
edge of G then Su
intersects Sv.
• We set Wt ={v| t Sv} and
Ht to be the graph with
vertex set Wt s.t. uv εE(Ht)
iff. uvεE(G) or u,v ε Ws 
Wt for some st εE(T).
• Every G has a one node
tree decomposition.
• If uv is a nonedge of G
then G has a tree
decomposition using tree
st where st corresponds
to the cutset V-u-v.
• The k by k grid has a tree
decomposition where T is
a path on k^2 nodes each
internal node of which
corresponds to a cutset of
the grid of size k
Tree Decompositions
• A tree decomposition for
G consists of a tree T and
a subtree Sv for each
vertex v of G s.t. if uv is an
edge of G then Su
intersects Sv.
• We set Wt ={v| t Sv} and
Ht to be the graph with
vertex set Wt s.t. uv εE(Ht)
iff. uvεE(G) or u,v ε Ws 
Wt for some st εE(T).
• Every G has a one node
tree decomposition.
• If uv is a nonedge of G
then G has a tree
decomposition using tree
st where st corresponds
to the cutset V-u-v.
• The k by k grid has a tree
decomposition where T is
a path on k2+k nodes
each arc of which
corresponds to a cutset of
the grid of size k
Tree Decompositions
• A tree decomposition for
G consists of a tree T and
a subtree Sv for each
vertex v of G s.t. if uv is an
edge of G then Su
intersects Sv.
• We set Wt ={v| t Sv} and
Ht to be the graph with
vertex set Wt s.t. uv εE(Ht)
iff. uvεE(G) or u,v ε Ws 
Wt for some st εE(T).
• Every G has a one node
tree decomposition.
• If uv is a nonedge of G
then G has a tree
decomposition using tree
st where st corresponds
to the cutset V-u-v.
• The k by k grid has a tree
decomposition where T is
a path on k2+k nodes
each arc of which
corresponds to a cutset of
the grid of size k
Adhesion and Width
• The Adhesion of a tree decomposition is the
maximum over all of its arcs st of |Ws ∩ Wt|.
• The width of a tree decomposition is the
maximum over all of its nodes s of |Ws |.
Adhesion and Width
• The Adhesion of a tree decomposition is the
maximum over all of its arcs st of |Ws ∩ Wt|.
• The width of a tree decomposition is the
maximum over all of its nodes s of |Ws |.
Graphs Without Clique Minors III:
Forbidding K5 Revisited
• A graph has no K5 model precisely if it has a
tree decomposition of adhesion at most 3
such that each Ht is planar or a special 8
vertex graph L.
Wagner, 1946.
Graphs Without Clique Minors IV:
Forbidding Kl
• If G contains no Kl model then it has a tree
decomposition of bounded adhesion such that
each Ht is a graph which can be “almost
embedded” in a surface in which Kl cannot be
embedded.
Robertson and Seymour, 1990.
The k-DRP Algorithm
• There is a linear time algorithm which finds either
(i) a Kl model in G, or (ii) a tree decomposition of
G of bounded adhesion such that each Ht is
almost embeddable in a surface in which Kl
cannot be embedded.
(Li, Kawarabayashi, R., 2009).
• We can solve k-DRP in G given such a tree
decomposition in linear time.
Bounded Extension:
An “Almost Embeddability” Ingredient
• If G-v is planar for some v, then G contains no
K6 model
• If G is obtained from a graph H embeddable in
a surface Σ by adding a bounded number of
vertices then G is almost embeddable in Σ.
• For a tree decomposition of bounded width,
every Ht is almost embeddable in the plane.
Bounded Extension:
An “Almost Embeddability” Ingredient
• If G-v is planar for some v, then G contains no
K6 model
• If G is obtained from a graph H embeddable in
a surface Σ by adding a bounded number of
vertices then G is almost embeddable in Σ.
• For a tree decomposition of bounded width,
every Ht is almost embeddable in the plane.
Bounded Extension:
An “Almost Embeddability” Ingredient
• If G-v is planar for some v, then G contains no
K6 model
• If G is obtained from a graph H embeddable in
a surface Σ by adding a bounded number of
vertices then G is almost embeddable in Σ.
• For a tree decomposition of bounded width,
every Ht is almost embeddable in the plane.
Brambles
• A bramble is a set of
connected subgraphs
every two of which
intersect or are joined
by an edge.
• The order of a bramble
is the minimun size of a
set of vertices which
intersects all its
elements.
• The trees of a clique
model.
• For any set S of vertices,
βS is the set of
connected subgraphs
containing more than
half the vertices of S.
• The set of subgraphs of
a grid formed by the
union of a row and a
column.
Brambles
• A bramble is a set of
connected subgraphs
every two of which
intersect or are joined
by an edge.
• The order of a bramble
is the minimun size of a
set of vertices which
intersects all its
elements.
• The trees of a clique
model.
• For any set S of vertices,
βS is the set of
connected subgraphs
containing more than
half the vertices of S.
• The set of subgraphs of
a grid formed by the
union of a row and a
column.
Brambles
• A bramble is a set of
connected subgraphs
every two of which
intersect or are joined
by an edge.
• The order of a bramble
is the minimun size of a
set of vertices which
intersects all its
elements.
• The trees of a clique
model.
• For any set S of vertices,
βS is the set of
connected subgraphs
containing more than
half the vertices of S.
• The set of subgraphs of
a grid formed by the
union of a row and a
column.
Brambles
• A bramble is a set of
connected subgraphs
every two of which
intersect or are joined
by an edge.
• The order of a bramble
is the minimun size of a
set of vertices which
intersects all its
elements.
• The trees of a clique
model.
• For any set S of vertices,
βS is the set of
connected subgraphs
containing more than
half the vertices of S.
• The set of subgraphs of
a grid formed by the
union of a row and a
column.
Brambles
• A bramble is a set of
connected subgraphs
every two of which
intersect or are joined
by an edge.
• The order of a bramble
β denoted ord(β) is the
minimun size of a set of
vertices which
intersects all its
elements.
• The trees of a clique
model.
• For any set S of vertices,
βS is the set of
connected subgraphs
containing more than
half the vertices of S.
• The set of subgraphs of
a grid formed by the
union of a row and a
column.
Brambles and Tree Width
• A graph has a tree decomposition of width at
most w precisely if it has no bramble of order
w+2.
Robertson,Seymour,Thomas,1986
• We can find such a tree decomposition if it
exists in linear time Bodlaender
Brambles and Tree Width
• A graph has a tree decomposition of width at
most w precisely if it has no bramble of order
w+2.
Robertson,Seymour,Thomas,1986
• We can find such a tree decomposition if it
exists in linear time Bodlaender
Distinguishing Brambles
• For any bramble β and set X of less than
ord(β) vertices, there is a unique component
f(β,X) of G-X containing an element of β.
• X is a distinguisher of β1 and β2 if
|X|<min (ord(β1),ord(β2)) and
f(β1,X) ≠f(β2,X).
Distinguishing Brambles
• For any bramble β and set X of less than
ord(β) vertices, there is a unique component
f(β,X) of G-X containing an element of β.
• X distinguishes β1 and β2 if
|X|<min (ord(β1),ord(β2)) and
f(β1,X) ≠f(β2,X).
A Canonical Tree Decomposition
• There is a tree decomposition of G in which
the nodes correspond to the “maximal
brambles” of G and the arcs correspond to the
cutsets of minimum order distinguishing
them.
• If G contains no Kl model we can massage this
to obtain the decomposition with almost
embeddable Ht
A Canonical Tree Decomposition
• There is a tree decomposition of G in which
the nodes correspond to the “maximal
brambles” of G and the arcs correspond to the
cutsets of minimum order distinguishing
them.
• If G contains no Kl model we can massage this
to obtain the decomposition with almost
embeddable Ht
Building The Tree Decomposition
• For every large order bramble corresponding
to a node t of the canonical tree
decomposition there is a high wall W which is
a subgraph of Ht.
• If the attachments between this wall and the
rest of the graph are sufficiently non-planar
we find a Kl model. Otherwise we obtain a
near embedding.
Building The Tree Decomposition
• For every large order bramble corresponding
to a node t of the canonical tree
decomposition there is a high wall W which is
a subgraph of Ht.
• If the attachments between this wall and the
rest of the graph are sufficiently non-planar
we find a Kl model. Otherwise we obtain a
near embedding of Ht.
Details to Follow
• Lecture II: Sketch of (i) why excluding clique
models yields an RS tree decomposition, (ii) why
excluding brambles yields tree decompositions of
bounded width, and (iii) why excluding walls
excludes brambles.
• Lecture III: Discussion of how to optimize in
graphs with well-behaved tree decompositions
• Lecture IV: linear algorithms for constructing
well-behaved tree decompositions.
Details to Follow
• Lecture II: Sketch of (i) why excluding clique
models yields an RS tree decomposition, (ii) why
excluding brambles yields tree decompositions of
bounded width, and (iii) why excluding walls
excludes brambles.
• Lecture III: Discussion of how to optimize in
graphs with well-behaved tree decompositions
• Lecture IV: linear algorithms for constructing
well-behaved tree decompositions.
Details to Follow
• Lecture II: Sketch of (i) why excluding clique
models yields an RS tree decomposition, (ii) why
excluding brambles yields tree decompositions of
bounded width, and (iii) why excluding walls
excludes brambles.
• Lecture III: Discussion of how to optimize in
graphs with well-behaved tree decompositions
• Lecture IV: linear algorithms for constructing
well-behaved tree decompositions.
Minors and Models
• A model of H in G consists of a set of disjoint
trees {Tv | v in V(H)} of G, such that for every
edge uv of H there is an edge xy of G with x in
Tu and y in Tv.
• H is a minor of G if there is a model of H in G.
• The order of a bramble is the minimum size of
a subset of V intersecting each of its elements.
Minors and Models
• A model of H in G consists of a set of disjoint
trees {Tv | v in V(H)} of G, such that for every
edge uv of H there is an edge xy of G with x in
Tu and y in Tv.
• H is a minor of G if there is a model of H in G.
• The order of a bramble is the minimum size of
a subset of V intersecting each of its elements.
Three Consequences of The Excluded
Minor Structure Theorem
1) There is an efficient algorithm to determine if
H is a minor of G.
2) In any infinite sequence G1,G2,… there is an i<j
such that Gi is a minor of Gj.
3) There is an efficient algorithm to test
memebrship in any minor closed family.
Three Consequences of The Excluded
Minor Structure Theorem
1) There is an efficient algorithm to determine if
H is a minor of G.
2) In any infinite sequence G1,G2,… there is an i<j
such that Gi is a minor of Gj.
3) There is an efficient algorithm to test
memebrship in any minor closed family.
Three Consequences of The Excluded
Minor Structure Theorem
1) There is an efficient algorithm to determine if
H is a minor of G.
2) In any infinite sequence G1,G2,… there is an i<j
such that Gi is a minor of Gj.
3) There is an efficient algorithm to test
membership in any minor closed family.
Thanks for your attention.