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
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.
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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
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Introduction Data Set
Data Set
We can explore more…
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Introduction Data Set
Graph Visualization: Graph 01
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Introduction Data Set
Graph Visualization: Graph 02 & 03
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Introduction Data Set
Graph Visualization: Graph 04
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Introduction Data Set
Graph Visualization: Graph 05
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Introduction Data Set
Graph Visualization: Graph 06
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Introduction Data Set
Graph Visualization: Graph 07
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Introduction Data Set
Color Sequence Visualization
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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.
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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
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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
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Algorithms Randomized Algorithms
Simulate
Parameters: Policy πsimu, mappingfrom permitted set p tochoice n.
Algorithm: Repeatedly samplingnodes according to πsimu
and performs transitionsTransit until reachingthe final state.
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Algorithms Randomized Algorithms
Repeat
Parameters: A positive integer N > 0,a search component S
Algorithm: It repeats performing Sfor N times.
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Algorithms Randomized Algorithms
LookAhead
Parameters: A search component SAlgorithm: For each n ∈ p, it
performs S onTransit(t, n).
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Algorithms Randomized Algorithms
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).
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Algorithms Randomized Algorithms
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)
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Algorithms Randomized Algorithms
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))))
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Algorithms Greedy Algorithm
Greedy Strategy
argmaxc∈p
Reward({c} ∪ Neighbors(c,G))
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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
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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
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Thank you!
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