Algorithms to Approximately Count and Sample Conforming Colorings of Graphs Sarah Miracle and Dana Randall Georgia Institute of Technology (B,B)(B,B) (R,B)(R,B) (G,R)(G,R) (R,G)(R,G) (R,B)(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)(B,B) (R,B)(R,B) (G,R)(G,R) (R,G)(R,G) (R,B)(R,B) (B,B)(B,B) (R,B)(R,B) (G,R)(G,R) (R,G)(R,G) (R,B)(R,B) (B,B)(B,B) (R,B)(R,B) (G,R)(G,R) (R,G)(R,G) (R,B)(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 F A 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)(B,B) (R,R)(R,R) (G,G)(G,G) (R,R)(R,R) (G,G)(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 F A 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 M C –Provide conditions under which we have a polynomial time algorithm to find a conforming coloring M C mixes rapidly a FPRAS for approximately counting –An example where M C and M L are slow Our Results k ≥ max(Δ,3) –A polynomial time algorithm to find a conforming coloring –The local chain M L connects the state space –M L 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 )). (Δ x (t) is the total variation distance) Definition: Given ε, the mixing time is τ(ε) = max min {t: Δ x (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 Coloring k ≥ 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(Δn 2 ) algorithm for finding one. Example: k = 3 Flower Color-fixed (R,G)(R,G) (B,R)(B,R) (G,B)(G,B)(G,B)(G,B) How to determine if Ω is degenerate?

The Markov chain M L : Starting at σ 0, Repeat: - With prob. ½ do nothing; - Pick v  V and c  {1,…,k}; - Recolor v color c if it results in a valid conforming coloring. The Local Markov Chain M L k ≥ max(Δ,3) Example: k = 2

When does M L connect Ω ? k ≥ max(Δ,3) Thm: If k ≥ max(Δ,3), M L connects the state space Ω.

Use path coupling [Dyer, Greenhill ‘98] to show M L is rapidly mixing. Use M L to design a FPRAS [Jerrum, Valiant, Vazirani ‘86] Rapid Mixing of M L and a FPRAS k ≥ max(Δ,3) Thm: If k ≥ max(Δ,3), then M L is rapidly mixing and there exists a FPRAS for counting the number of conforming k-colorings.

–A new “component” chain M C –Provide conditions under which we have a polynomial time algorithm to find a conforming coloring M C mixes rapidly a FPRAS for approximately counting –An example where M C and M L are slow Our Results k ≥ max(Δ,3) –A polynomial time algorithm to find a conforming coloring –The local chain M L connects the state space –M L mixes rapidly –A FPRAS for approximately counting k = 2 (Δ = max degree including multi-edges and self-loops)

k = 2 What’s Special about k = 2? Generalizes independent sets The local chain M L does not connect the state space Ω

The Component Chain M C k = 2 A path P = v 1, v 2, …, v x is b-alternating if 1.F 1 (v 1,v 2 ) = b 2.For all 1 ≤ i ≤ x-2, F i+1 (v i,v i+1 )≠F i+1 (v i+1,v i+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) (β,α) (α,β) The Component Chain M C

The Markov chain M C : Starting at σ 0, Repeat: - Pick a component C i in C u.a.r.; - With prob. ½ color C i : ρ(C i ), if valid; - With prob. ½ color C i : ρ’(C i ), if valid; - Otherwise, do nothing. k = 2 The Component Chain M C

Positive Results k = 2 1.We give an O(n 3 ) algorithm for finding a 2- coloring 2.The chain M C is rapidly mixing 3.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 M C on Other Graphs k = 2 Thm: There exists a bipartite graph G with Δ = 4 and edge 2-coloring F for which M C and M L take exponential time to converge. (α coloring ) (α,β) (β, α )

The “Bottleneck” G colored α. S 1 colored α. G colored α. S 1 colored α. G colored β. S 2 colored β. G colored β. S 2 colored β. S 1 colored α. S 2 colored β. S 1 colored α. S 2 colored β. k = 2 (α,β) (β, α )

Open Problems 1.When k > 2, is there an analog to M C 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?

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