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

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GRAPHS A Graph is a set of vertices V and a set E of edges, each of which is an unordered pair of vertices

K5 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 TheoremThere 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 Examples1

t1

t2 s2

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 K l 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.