Graph Coloring Independent Set and Coloring k -Coloring : 3-Coloring - - PowerPoint PPT Presentation

CS500

Graph Coloring

k-Coloring: Instance: A graph G. Question: Can the vertices of G be colored with k colors such that no two vertices of the same color are adjacent? k-Coloring ≤ (k + 1)-Coloring (Note that k is part of the problem, not an input.) Lemma: 2-Coloring can be solved in linear time. CS500

Independent Set and Coloring

3-Coloring ≤ IndependentSet n vertices 3n vertices, k = n 3-Coloring ⇔ independent set of size n CS500

Hamiltonian Cycle

HamiltonianCycle: Instance: A directed graph G. Question: Does G contain a Hamiltonian cycle (a cycle that visits every vertex exactly once)? CS500

Vertex Cover and Hamiltonian Cycle

VertexCover ≤ HamiltonianCycle u v (u, v, o) (u, v, i) (v, u, i) (v, u, o) There are three possible ways for a Hamiltonian cycle to visit these four vertices.

CS500

Vertex Cover and Hamiltonian Cycle

VertexCover ≤ HamiltonianCycle u v (u, v, o) (u, v, i) (v, u, i) (v, u, o) There are three possible ways for a Hamiltonian cycle to visit these four vertices. CS500

Vertex Cover and Hamiltonian Cycle

VertexCover ≤ HamiltonianCycle u v (u, v, o) (u, v, i) (v, u, i) (v, u, o) There are three possible ways for a Hamiltonian cycle to visit these four vertices. CS500

Vertices of G

u v1 v2 v3 v4 (u, v1, i) (u, v1, o) (u, v2, i) (u, v3, i) (u, v4, i) (u, v2, o) (u, v3, o) (u, v4, o) (u, v2, o) Vertex chain of u CS500

Cover vertices

k cover vertices connected to the ends of all vertex chains. (u, v1, i) (u, v1, o) (u, v2, i) (u, v3, i) (u, v4, i) (u, v2, o) (u, v3, o) (u, v4, o) (u, v2, o) 1 2 3 k = 3

CS500 G has a vertex cover of size k if and only if G′ has a Hamiltonian path. Let u1, u2, . . . , uk be the vertices of the vertex cover C. Then the Hamiltonian path starts in cover vertex 1, visits the vertex chain of u1, goes to cover vertex 2, visits the vertex chain of u2, and so on, until returning to cover vertex 1. u (u, v, o) (u, v, i) (v, u, i) (v, u, o) u ∈ C v ∈ C v CS500 G has a vertex cover of size k if and only if G′ has a Hamiltonian path. Let u1, u2, . . . , uk be the vertices of the vertex cover C. Then the Hamiltonian path starts in cover vertex 1, visits the vertex chain of u1, goes to cover vertex 2, visits the vertex chain of u2, and so on, until returning to cover vertex 1. u (u, v, o) (u, v, i) (v, u, i) (v, u, o) u ∈ C v ∈ C v CS500

Example

u v w x w x u v CS500

Example Cover

u v w x u v w x

CS500

Subset Sum

SubsetSum: Instance: A set X of positive integers and an integer t. Question: Does X have a subset whose elements sum to t? VertexCover ≤ SubsetSum Number edges from 0 to m − 1. Our set X contains bi = 4i for each edge i, and av for each vertex v: av = 4m +

• i∈∆(v)

4i. The target sum t is t = k · 4m +

m−1

• i=0

2 · 4i. CS500

Summary

3-Coloring ≤ IndependentSet VertexCover ≤ HamiltonianCycle VertexCover ≤ SubsetSum IndependentSet ≤ VertexCover VertexCover ≤ SetCover HamiltonianCycle ≤ HamiltonianPath HamiltonianPath ≤ LongestPath 3-Coloring ≤ Planar3Coloring IndependentSet ≤ Clique SubsetSum ≤ Partition