Algorithms for Graph Coloring Problem

Algorithms for Graph Coloring Problem

Wang Shengyi

National University of Singapore

November 6, 2014

Wang Shengyi (NUS) Algorithms for Graph Coloring Problem November 6, 2014 1 / 24

Introduction Problem Description

Problem Description

• For a graph G = (V, E) and a color sequence c = (c0, c1, . . . , cn), firstlychoose a node v ∈ V and populate with c0

• Populate the rest of G in order of the rest of color sequence c such that onlynew nodes connected to a previously populated node may be populated.

• Each populated G can be called a configuration, which can be seen as afunction fmapping node to color. We can calculate a reward value for sucha configuration f by the following formula:

H =∑

All filled (i,j)∈E

1− δ(f(i), f(j)) where δ(x, y) =

{0 x ̸= y1 x = y

• For any G and c, develop an algorithm to generate a configuration that givesthe maximum reward.


Introduction Data Set

Data Set

Set Number of Vertices Number of Edges Length of Color Sequences01 10 29 1002 153 5533 2003 153 5533 13004 590 658 40005 2969 3372 400006 483 1358 40007 11748 34716 9000



Data Set

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Graph Visualization: Graph 01



Graph Visualization: Graph 02 & 03















Color Sequence Visualization


Algorithms Representation

States

• Each state can be represented as a triple t = (γ, p, σ)• γ is the subgraph which has not been populated.• p is the set of all permitted choices.• σ is a sequence of nodes which have been populated chronologically.


Algorithms Representation

State Transition Function: Transit

• Input: An old state (γ, p, σ) and n ∈ p• Output: An new state (γ′, p′, σ′)

• γ′ = γ removing n and related edges• p′ = p ∪ neighbors of n in γ − {n}• σ′ = σ :: n


Algorithms Randomized Algorithms

Search Component Generators: A Pentomino Style

• A search component S ∈ S is a stochastic algorithm.• Input: t = (γ, p, σ), Output: One or multiple final states t1, t2, . . . , tm• Before running, S would check the budget. For each final state, S comparesthe reward with the best result so far and updates the budget.

• A search component generatorΨ : Θ → S• Given a set of parameters θ ∈ Θ,Ψ(θ) is a search component• Simulate, Repeat, LookAhead, Step and Select



Simulate

Parameters: Policy πsimu, mappingfrom permitted set p tochoice n.

Algorithm: Repeatedly samplingnodes according to πsimu

and performs transitionsTransit until reachingthe final state.



Repeat

Parameters: A positive integer N > 0,a search component S

Algorithm: It repeats performing Sfor N times.



LookAhead

Parameters: A search component SAlgorithm: For each n ∈ p, it

performs S onTransit(t, n).



Step

Parameters: A search component SAlgorithm: For each remaining steps

until the final state, itperforms S first. Then itextracts the local bestchoice nl and performstransitionTransit(tstep, nl).



Select

Parameters: A selection policy πsel, asearch component S

Algorithm: It looks like Step. But ineach step, it chooses nodeaccording to πsel.

πsel: πUCB-1C

s(t, c): Sum of rewards,from Swith Transit(t, c)

n(t, c): Number of times c wasselected in state t

n(t): Sum of n(t, c)

argmaxc∈p

s(t, c)n(t, c)

+ C

√ln n(t)n(t, c)



Compisition

• Is = Step(Repeat(N, Simulate(πrandom)))

• Nmc(0) = Simulate(πrandom)Nmc(l) = Step(LookAhead(Nmc(l− 1)))

• Uct(C) = Step(Repeat(N, Select(πUCB-1C , Simulate(πrandom))))


Algorithms Greedy Algorithm

Greedy Strategy

argmaxc∈p

Reward({c} ∪ Neighbors(c,G))


Result

Results of Iterative Sampling (Is)

N set01 set02 set03 set04 set05 set06 set071 11 50 1981 129 1039 439 972110 16 58 2091 159 1107 473 9777100 19 74 2127 167 1126 483 98781000 19 79 2147 167 1158 489 99384000 19 84 2181 175 1158 506 10006


Result

Result of Different Algorithms for Budget 1000

Algorithms set01 set02 set03 set04 set05 set06 set07Is 19 79 2147 167 1158 489 9938

Nmc(2) 19 79 2171 171 1152 498 9953Nmc(3) 18 75 2158 173 1145 503 9912Uct(0.3) 19 98 2165 171 1141 504 9933Uct(0.5) 19 98 2165 171 1141 500 9933Greedy 19 156 2719 219 1473 737 16118


Thank you!


Technology

Algorithms for Graph Coloring Problem