solving graph problems with boolean methodsbryant/presentations/boolean-methods.pdf · al ith b d i...
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SolvingSolvingSolvingSolvingGraph Problems with Graph Problems with pp
Boolean MethodsBoolean Methods
Randal E. BryantCarnegie Mellon UniversityCarnegie Mellon University
http://www.cs.cmu.edu/~bryant
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The Graph Coloring ProblemThe Graph Coloring ProblemColor the regions of a planar graphColor the regions of a planar graph
Use only 4 colors Use only 4 colors No two adjacent regions can have the same color
Example: Color the states of a U.S. map Example: Color the states of a U.S. map
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The MacGregor GraphThe MacGregor GraphScientific American, Scientific American,
April 1975April 1975April 1975April 1975 Said to be proof that
some planar graphssome planar graphs could not be colored with just 4 colors
An April-fool’s joke, but still difficult to solve by handhand
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Boolean SAT SolversBoolean SAT SolversWhat They DoWhat They Do
Express problem as a set of constraints Search for solution that satisfies all constraints
Encoding Graph Coloring with SATEncoding Graph Coloring with SAT Encode each region with two 0/1-valued variables:
00 Bl 00 Blue 01 Green 10 Red 11 Yellow
For each adjacent region, require at least one of the corresponding variables to have oppositethe corresponding variables to have opposite values
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Encoding Coloring ConstraintsEncoding Coloring Constraints
ai, bi aj, bj
Encode region i with variables ai, bi
For adjacent regions i and j, want:ai ≠ aj bi ≠ bj
Clausal form (and of or’s):
(c ai aj) (c ai aj) (c bi bj) (c bi bj)
c = 0 ai ≠ aj
c = 1 bi ≠ bj
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The ZChaff SAT SolverThe ZChaff SAT Solver From Princeton University
Al ith b D i P t L & L l d Algorithm by Davis Putnam Logemann & Loveland With many refinements
Based on backtracking searchBased on backtracking search Try assigning values to variables When hit contradiction
Create new constraint encoding conflict Backtrack by undoing some of the most recent assignments Resume search with new variable assignments
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Visualizing the Search ProcessVisualizing the Search Process
Black: Neither variable assigned valueSi l l B th i bl i d i i Single color: Both variables assigned, giving unique color.
Blended colors: One variable assigned, the otherBlended colors: One variable assigned, the other unassigned, indicating two possible colors
YouTube: http://www.youtube.com/watch?v=0gt503wK7AI
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The Final ResultThe Final Result
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Another SolutionAnother Solution
Minimum use of greeen Minimum use of greeen(7 times)
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Try It YourselfTry It Yourself
Color the rest of the map using 3 colors
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Minimum Colorings of US MapMinimum Colorings of US Map
Only need to use green twice
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Odd CyclesOdd Cycles
C thi b l d ithCan this be colored with just 3 colors?
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Odd Cycles in US MapOdd Cycles in US Map
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Overlapping Odd CyclesOverlapping Odd Cycles
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Breaking Odd CyclesBreaking Odd Cycles
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Viewing Maps as GraphsViewing Maps as Graphs
48 nodes105 edges
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Coloring a GraphColoring a Graph
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The Macgregor GraphThe Macgregor Graph
110 nodes324 edges
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The Four Color TheoremThe Four Color Theorem Can color any planar graph with just 4 colors.
HistoryHistory Conjectured in 1852 1890: it was shown that 5 colors would suffice 1976: Appel & Haken claimed they had proof
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Proof of Four Color TheoremProof of Four Color Theorem
Proof MethodProof MethodAppel & Haaken showed there were 1,936
graphs that covered all possibitiesWrote computer program to check all of themWrote computer program to check all of them
ReactionReactionMany mathematicians didn’t like this kind of
proofP h b itt d h k d Program has been rewritten and rechecked multiple times, and so the proof is generally accepted.p
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Coloring Other Graph TypesColoring Other Graph TypesSphere: same as planeSphere: same as plane
Plane sphere Reduce exterior edges to points Reduce exterior edges to points
Sphere planeSphere plane Cut hole and stretch out flat
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Coloring A TorusColoring A TorusTorusTorus
7 colors necessary 7 regions, each with g ,
6 neighbors
Also sufficient
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Torus: An Infinite Wallpaper PatternTorus: An Infinite Wallpaper PatternPatternPattern
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Sudoku as a Graph Coloring ProblemSudoku as a Graph Coloring ProblemProblemProblem
3 9 6 7 4 1 2 5 8 3 9 6 7 4 1 2 5 88 5 7 2 3 9 1 4 61 2 4 5 8 6 7 9 35 7 1 6 2 3 9 8 4
9 88 5 7 2 3 9 1 4 61 2 4 5 8 6 7 9 35 7 1 6 2 3 9 8 45 7 1 6 2 3 9 8 4
6 8 3 4 9 7 5 1 29 4 2 1 5 8 3 6 7
5 7 1 6 2 3 9 8 46 8 3 4 9 7 5 1 29 4 2 1 5 8 3 6 7
2 3 9 8 1 4 6 7 54 6 5 9 7 2 8 3 17 1 8 3 6 5 4 2 9
2 3 9 8 1 4 6 7 54 6 5 9 7 2 8 3 17 1 8 3 6 5 4 2 9
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Adding ColorsAdding Colors
1 3 9 6 7 4 1 2 5 823
8 5 7 2 3 9 1 4 61 2 4 5 8 6 7 9 3
45
5 7 1 6 2 3 9 8 46 8 3 4 9 7 5 1 29 4 2 1 5 8 3 6 76
7
9 4 2 1 5 8 3 6 72 3 9 8 1 4 6 7 54 6 5 9 7 2 8 3 18
9
4 6 5 9 7 2 8 3 17 1 8 3 6 5 4 2 9
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Taking Away NumbersTaking Away Numbers3 9 6 7 4 1 2 5 88 5 7 2 3 9 1 4 68 5 7 2 3 9 1 4 61 2 4 5 8 6 7 9 35 7 1 6 2 3 9 8 45 7 1 6 2 3 9 8 46 8 3 4 9 7 5 1 29 4 2 1 5 8 3 6 72 3 9 8 1 4 6 7 54 6 5 9 7 2 8 3 17 1 8 3 6 5 4 2 9
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Graph Structure of SudokuGraph Structure of Sudoku
1 3 4 21 3 4 2
4 2 1 34 2 1 3
2 4 3 12 4 3 1
3 1 2 43 1 2 44 X 4 Sudoku Column constraints
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Graph Structure of SudokuGraph Structure of Sudoku
1 3 4 21 3 4 2
4 2 1 34 2 1 3
2 4 3 12 4 3 1
3 1 2 43 1 2 44 X 4 Sudoku Row constraints
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Graph Structure of SudokuGraph Structure of Sudoku
1 3 4 21 3 4 2
4 2 1 34 2 1 3
2 4 3 12 4 3 1
3 1 2 43 1 2 44 X 4 Sudoku Block constraints
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Graph Structure of SudokuGraph Structure of Sudoku
1 3 4 21 3 4 2
4 2 1 34 2 1 3
2 4 3 12 4 3 1
3 1 2 43 1 2 44 X 4 Sudoku
16 nodesAll constraints
56 edges16 nodes 56 edges
9 x 9 Sudoku: 81 nodes, 810 edges
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Visualizing Solution ProcessVisualizing Solution Process
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Visualizing Solution ProcessVisualizing Solution Process
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Solving A Sudoku PuzzleSolving A Sudoku Puzzle
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Touring the USTouring the US
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Touring the USTouring the US
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Weighted US GraphWeighted US Graph
Shortest driving distances between capitol cities Staying within source and destination states Staying within source and destination states Computed by Don Knuth using Mapquest
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A Capitol TourA Capitol TourA Hamiltonian Path
15 136 il (24 359 k ) t t l15,136 miles (24,359 km) total68,656,026 possible tours
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Limiting Node DegreeLimiting Node Degree
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A Spanning TreeA Spanning Tree
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The Shortest Capitol TourThe Shortest Capitol Tour
11,698 miles (18,826 km) total
A Traveling Salesman Path
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The Longest Capitol TourThe Longest Capitol Tour
18,040 miles (29,033 km) total
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Two Interesting Capitol ToursTwo Interesting Capitol Tours
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Two Interesting Capitol ToursTwo Interesting Capitol Tours
3 time changes 19 time changes
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Touring MacGregorTouring MacGregorA Hamiltonian Cycle
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Lessons LearnedLessons LearnedGraph ColoringGraph Coloring
Maps are a kind of graph Sudoku is a graph coloring problem
Hamiltonian PathsHamiltonian Paths Find a path in graph that goes through every node
once Considered a difficult problem
Boolean MethodsBoolean Methods Can encode wide variety of graph problems Can find solution using SAT solver In worst case, has exponential performance
But gets solution for many interesting problems