Discrete Optimization Lecture 2 – Part 2

Discrete OptimizationLecture 2 – Part 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online http://cvn.ecp.fr/personnel/pawan/

Outline • Preliminaries • Menger’s Theorem for Disjoint Paths • Path Packing

Directed Graphs (Digraphs) v0 D = (V, A) v1 v4 v2 v5 v3 v6 ‘n’ vertices or nodes V ‘m’ arcs A: ordered pairs from V

Indegree of a Vertex v0 D = (V, A) v1 v4 v2 v5 v3 v6 Number of arcs entering the vertex. indeg(v0) = 1, indeg(v1) = 1, indeg(v4) = 2, …

Indegree of a Subset of Vertices v0 D = (V, A) v1 v4 v2 v5 v3 v6 Number of arcs entering the subset. indeg({v0,v1}) = 1, indeg({v1,v4}) = 3, …

Outdegree of a Vertex v0 D = (V, A) v1 v4 v2 v5 v3 v6 Number of arcs leaving the vertex. outdeg(v0) = 1, outdeg(v1) = 1, outdeg(v2) = 2, …

Outdegree of a Subset of Vertices v0 D = (V, A) v1 v4 v2 v5 v3 v6 Number of arcs leaving the subset. outdeg({v0,v1}) = 1, outdeg({v1,v4}) = 2, …

s-t Path v0 D = (V, A) v1 v4 v2 v5 v3 v6 Sequence P = (s=v0,a1,v1,…,ak,t=vk), ai = (vi-1,vi) Vertices s=v0,v1,…,t=vk are distinct

S-T Path S v0 D = (V, A) T v1 v4 v2 v5 v3 v6 S and T are subsets of V Any st-path where s  S and t T

Outline • Preliminaries • Menger’s Theorem for Disjoint Paths • Path Packing

Vertex Disjoint S-T Paths v0 v8 S v1 v2 v3 v4 v5 v6 T v7 v9 Set of S-T Paths with no common vertex

Vertex Disjoint S-T Paths v0 v8 S v1 v2 v3 ✓ v4 v5 v6 T v7 v9 Set of S-T Paths with no common vertex

Vertex Disjoint S-T Paths v0 v8 S v1 v2 v3 Common Vertex v7 ✗ v4 v5 v6 T v7 v9 Set of S-T Paths with no common vertex

Vertex Disjoint S-T Paths v0 v8 S v1 v2 v3 Common Vertex v0 ✗ v4 v5 v6 T v7 v9 Set of S-T Paths with no common vertex

Internally Vertex Disjoint s-t Paths v0 v8 s v1 v2 v3 v4 v5 v6 t v7 v9 Set of s-t Paths with no common internal vertex

Internally Vertex Disjoint s-t Paths v0 v8 s v1 v2 v3 ✓ v4 v5 v6 t v7 v9 Set of s-t Paths with no common internal vertex

Internally Vertex Disjoint s-t Paths v0 v8 s v1 v2 v3 Common Vertex v5 ✗ v4 v5 v6 t v7 v9 Set of s-t Paths with no common internal vertex

Arc Disjoint s-t Paths v0 v8 s v1 v2 v3 v4 v5 v6 t v7 v9 Set of s-t Paths with no common arcs

Arc Disjoint s-t Paths v0 v8 s v1 v2 v3 ✓ v4 v5 v6 t v7 v9 Set of s-t Paths with no common arcs

Outline • Preliminaries • Menger’s Theorem for Disjoint Paths • Vertex Disjoint S-T Paths • Internally Vertex Disjoint s-t Paths • Arc Disjoint s-t Paths • Path Packing

Vertex Disjoint S-T Paths v0 v8 S Maximum number of disjoint paths? v1 v2 v3 Minimum size of S-T disconnecting vertex set !! v4 v5 v6 T v7 v9 Set of S-T Paths with no common vertex

S-T Disconnecting Vertex Set v0 v8 S v1 v2 v3 ✗ v4 v5 v6 T v7 v9 Subset U of V which intersects with all S-T Paths

S-T Disconnecting Vertex Set v0 v8 S v1 v2 v3 ✓ v4 v5 v6 T v7 v9 Subset U of V which intersects with all S-T Paths

Connection v0 v8 S Maximum number of disjoint paths v1 v2 v3 ≤ Minimum size of S-T disconnecting vertex set !! v4 v5 v6 T v7 v9

Menger’s Theorem v0 v8 S Maximum number of disjoint paths v1 v2 v3 = Minimum size of S-T disconnecting vertex set !! v4 v5 v6 T v7 v9 Proof ? Mathematical Induction on |A|

Menger’s Theorem v0 v8 S True for |A| = 0 v1 v2 v3 Assume it is true for |A| < m v4 v5 v6 T v7 v9 Mathematical Induction on |A|

Menger’s Theorem v0 v8 S Consider |A| = m v1 v2 v3 Let minimum size disconnecting vertex set = k v4 v5 v6 T v7 v9 Remove an arc (u,v) Mathematical Induction on |A|

Menger’s Theorem v0 v8 S Minimum disconnecting vertex set = U. |U| = k v1 v2 v3 Proved by induction v4 v5 v6 T v7 v9 Mathematical Induction on |A|

Menger’s Theorem v0 v8 S v1 v2 v3 v4 v5 v6 T v7 v9 Remove an arc (u,v) Mathematical Induction on |A|

Menger’s Theorem v0 v8 S Minimum disconnecting vertex set U. |U| = k-1 v1 v2 v3 U’ = U union {u} U’ v4 v5 v6 U’’ U’’ = U union {v} T v7 v9 Mathematical Induction on |A|

Menger’s Theorem U’ is an S-T disconnecting vertex set of size k in D v0 v8 S v1 v2 v3 Any S-U’ disconnecting vertex set has size ≥ k U’ v4 v5 v6 Graph contains k disjoint S-U’ paths (induction) T v7 v9 Mathematical Induction on |A|

Menger’s Theorem U’’ is an S-T disconnecting vertex set of size k in D v0 v8 S v1 v2 v3 Any U’’-T disconnecting vertex set has size ≥ k v4 v5 v6 U’’ Graph contains k disjoint U’’-T paths (induction) T v7 v9 Mathematical Induction on |A|

Menger’s Theorem Add arc (u,v) back v0 v8 S v1 v2 v3 v4 v5 v6 T v7 v9 Mathematical Induction on |A|

Menger’s Theorem Add arc (u,v) back v0 v8 S (k-1) disjoint pairs of paths intersecting in U v1 v2 v3 v4 v5 v6 1 disjoint pair of paths now connected by (u,v) T v7 v9 Mathematical Induction on |A|

Menger’s Theorem Total k disjoint paths v0 v8 S Hence proved v1 v2 v3 v4 v5 v6 T v7 v9 Mathematical Induction on |A|

Outline • Preliminaries • Menger’s Theorem for Disjoint Paths • Vertex Disjoint S-T Paths • Internally Vertex Disjoint s-t Paths • Arc Disjoint s-t Paths • Path Packing

Internally Vertex Disjoint s-t Paths v0 v8 s Maximum number of disjoint paths? v1 v2 v3 Minimum size of s-t vertex-cut !! v4 v5 v6 t v7 v9 Set of s-t Paths with no common internal vertex

s-t Vertex-Cut v0 v8 s v1 v2 v3 v4 v5 v6 t v7 v9 t not in U s not in U Subset U of V which intersects with all s-t Paths

s-t Vertex-Cut v0 v8 s v1 v2 v3 ✗ v4 v5 v6 t v7 v9 t not in U s not in U Subset U of V which intersects with all s-t Paths

s-t Vertex-Cut v0 v8 s v1 v2 v3 ✓ v4 v5 v6 t v7 v9 t not in U s not in U Subset U of V which intersects with all s-t Paths

Connection v0 v8 s Maximum number of disjoint paths? v1 v2 v3 ≤ Minimum size of s-t vertex-cut !! v4 v5 v6 t v7 v9

Menger’s Theorem v0 v8 s Maximum number of disjoint paths? v1 v2 v3 = Minimum size of s-t vertex-cut !! v4 v5 v6 t v7 v9 Proof ? Menger’s Theorem for vertex disjoint S-T paths

Menger’s Theorem S = outneighbors of s v0 v8 s S T = inneighbors of t v1 v2 v3 T v4 v5 v6 Remove s and t t v7 v9 Menger’s Theorem for vertex disjoint S-T paths

Discrete Optimization Lecture 2 – Part 2