Algorithms to Approximately Count and Sample Conforming Colorings of Graphs

Sarah Miracle and Dana RandallGeorgia Institute of Technology

(B,B)

(R,B)

(G,R)

(R,G

)

(R,B

)

Given:• a (multi) graph G = (V,E)• a set of colors [k] = {1, . . .,k} • a set of edge constraints F: E [k] x [k]

A coloring C of the vertices of G is conforming to F if for every edge e=(u,v),

F(e) ≠ ( C(v), C(u) ).

“Conforming Colorings”

A coloring C of the vertices of G is conforming to F if for every edge e=(u,v), F(e)≠(C(v),C(u)).

Conforming Colorings

(B,B)

(R,B)

(G,R)

(R,G

)

(R,B

)

(B,B)

(R,B)

(G,R)

(R,G

)

(R,B

)(B,B)

(R,B)

(G,R)

(R,G

)

(R,B

)

✔ ✗✔

1. Resource Allocation– Colors are Resources– Vertices are Jobs– Edge Constraints are Incompatible Scheduling

Assignments

2. Generalizes Graph Theoretic Concepts

Applications

Conforming colorings generalize the following: – Independent sets– Vertex colorings– List colorings– H-colorings– Adapted colorings

Examples in Graph Theory

Conforming colorings generalize the following: – Independent sets– Vertex colorings– List colorings– H-colorings– Adapted colorings

Examples in Graph Theory

Adapted (or Adaptable) Colorings

Given:• a (multi) graph G = (V,E)• an edge coloring FA coloring C of the vertices of G is adapted to F if there is no edge e = (u,v) with F(e) = C(v) = C(u).

(B,B)

(R,R)

(G,G)

(R,R

)

(G,G

)

Adapted (or Adaptable) Colorings

• Adaptable Chromatic Number introduced [Hell & Zhu, 2008]• Adapted List colorings of Planar Graphs [Esperet et al., 2009]• Adaptable chromatic number of graph products [Hell et al., 2009] • Polynomial algorithm for finding adapted 3-coloring given edge 3-

coloring of complete graph [Cygan et al., SODA ‘11]

Given:• a (multi) graph G = (V,E)• an edge coloring FA coloring C of the vertices of G is adapted to F if there is no edge e = (u,v) with F(e) = C(v) = C(u).

Conforming colorings generalize Independent Sets• Set k = 2• Color each edge (B,B)• Each conforming coloring is an independent set

Independent Sets

• Extensive work using Monte Carlo approaches to count and sample for special cases.

• Design a Markov chain for sampling configurations that is rapidly mixing

(i.e. independent sets, colorings)

• These chains can be slow. Non-local ones can be more effective but harder to analyze.

(i.e. Wang-Swendsen-Kotecký )

Approximately Counting and Sampling

– A new “component” chain MC

– Provide conditions under which we have• a polynomial time algorithm to find a conforming coloring

• MC mixes rapidly

• a FPRAS for approximately counting

– An example where MC and ML are slow

Our Resultsk ≥ max(Δ,3) – A polynomial time algorithm to find a

conforming coloring

– The local chain ML connects the state space

– ML mixes rapidly

– A FPRAS for approximately counting

k = 2

(Δ = max degree including multi-edges and self-loops)

Definitions

• A Markov chain is rapidly mixing if τ(ε) is poly(n, log(ε-1)).

(Δx(t) is the total variation distance)

Definition: Given ε, the mixing time is

τ(ε) = max min {t: Δx(t’) < ε, for all t’ ≥ t}.x

• A Markov chain is slowly mixing if τ(ε) is at least exp(n).

Definition: A Fully Polynomial Randomized Approximation Scheme (FPRAS) is a randomized algorithm that given a graph G with n vertices, edge coloring F and error parameter 0< ε ≤ 1 produces a number N such that

P[(1-ε)N ≤ A(G,F) ≤ (1+ε)N] ≥ ¾

Finding a Conforming Coloringk ≥ max(Δ,3)

Thm: When k ≥ max(Δ,3) and Ω is not degenerate, there exists a conforming k-coloring and we give an O(Δn2) algorithm for finding one.

Example: k = 3Flower Color-fixed

(R,G)

(B,R

) (G,B

)

(G,B

)

How to determine if Ω is degenerate?

The Markov chain ML: Starting at σ0, Repeat:

- With prob. ½ do nothing;

- Pick v Î V and c Î {1,…,k};

- Recolor v color c if it results in a valid conforming coloring.

The Local Markov Chain ML

k ≥ max(Δ,3)

Example: k = 2

When does ML connect Ω ?k ≥ max(Δ,3)

Thm: If k ≥ max(Δ,3), ML connects the state space Ω.

• Use path coupling [Dyer, Greenhill ‘98] to show ML is rapidly mixing.

• Use ML to design a FPRAS [Jerrum, Valiant,

Vazirani ‘86]

Rapid Mixing of ML and a FPRASk ≥ max(Δ,3)

Thm: If k ≥ max(Δ,3), then ML is rapidly mixing and there exists a FPRAS for counting the number of conforming k-colorings.

– A new “component” chain MC

– Provide conditions under which we have• a polynomial time algorithm to find a conforming coloring

• MC mixes rapidly

• a FPRAS for approximately counting

– An example where MC and ML are slow

Our Resultsk ≥ max(Δ,3) – A polynomial time algorithm to find a

conforming coloring

– The local chain ML connects the state space

– ML mixes rapidly

– A FPRAS for approximately counting

k = 2

(Δ = max degree including multi-edges and self-loops)

k = 2What’s Special about k = 2?

• Generalizes independent sets• The local chain ML does not connect the state

space Ω

The Component Chain MC

k = 2

• A path P = v1, v2, …, vx is b-alternating if 1. F1(v1,v2) = b2. For all 1 ≤ i ≤ x-2, Fi+1(vi,vi+1)≠Fi+1(vi+1,vi+2)

• Vertices u and v are color-implied if there is a 1-alternating and a 2-alternating path from u to v or u=v

• Color-implied is an equivalence relation and defines a partition of the vertices into color-implied components

Defining Color-Implied Components

k = 2

• Each color implied component C has at most 2 colorings α(C) and β(C).

• Components become vertices• Edges reflect the edges and coloring constraints from F

The Color-Implied Component Graph

(α coloring )

(α,β)

(α,β)

(α,β)

(α,β)

(β,α

)

(β,α)(β

,α)

(α,β)

The Component Chain MC

The Markov chain MC: Starting at σ0, Repeat:

- Pick a component Ci in C u.a.r.;

- With prob. ½ color Ci: ρ(Ci), if valid;

- With prob. ½ color Ci: ρ’(Ci), if valid;

- Otherwise, do nothing.

k = 2

The Component Chain MC

Positive Resultsk = 2

1. We give an O(n3) algorithm for finding a 2-coloring

2. The chain MC is rapidly mixing3. We give a FPRAS for approximately counting

If every vertex v in the color-implied component graph satisfies one of the following

• d(v) ≤ 2 (d(v) = the degree of v)

• d(v) ≤ 4 and v is monochromatic

Rapid Mixing Proof: Uses path coupling and comparison with a similar chain which selects an edge in the component graph and recolors both vertices.

Slow Mixing of MC on Other Graphsk = 2

Thm: There exists a bipartite graph G with Δ = 4 and edge 2-coloring F for which MC and ML take exponential time to converge.

(α coloring ) (α,β) (α,β)

(α,β)(α,β) (β,α)

(β,α)(β,α)

(β,α)

S1

S2

G

The “Bottleneck”

G colored α.S1 colored α.

G colored β.S2 colored β.

S1 colored α.S2 colored β.

k = 2

(α,β)

(α,β)

(α,β

)

(α,β) (β,α)

(β,α)(β,α)

(β,α)

S1S2 G

Open Problems

1. When k > 2, is there an analog to MC

that is faster?2. Other approaches to sampling when

k ≥ 2?3. Can conforming/adapted colorings

give insights into phase transitions for independent sets and colorings?

Thank you!