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Data Implementation Evaluation Implementing Fixed-Parameter Algorithms Falk Hffner Institut fr Softwaretechnik und Theoretische Informatik, TU Berlin 20 November 2012 Falk Hffner (TU Berlin) Implementing Fixed-Parameter Algorithms

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SLIDE 1

Data Implementation Evaluation

Implementing Fixed-Parameter Algorithms

Falk Hüffner

Institut für Softwaretechnik und Theoretische Informatik, TU Berlin

20 November 2012

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SLIDE 2

Data Implementation Evaluation

Real-world instances

Recommendation

Use real-world data. Easier to “sell” Analysis might lead to new insights and approaches More fun

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SLIDE 3

Data Implementation Evaluation

Real-world instance sources

Databases

Biological networks Social networks

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SLIDE 4

Data Implementation Evaluation

Real-world instance sources

Databases

Biological networks Social networks

Web crawling

DBLP coauthor graph Song similarity graph (last.fm) Stock correlation graph Wikipedia inter-language links

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SLIDE 5

Data Implementation Evaluation

Real-world instance sources

Databases

Biological networks Social networks

Web crawling

DBLP coauthor graph Song similarity graph (last.fm) Stock correlation graph Wikipedia inter-language links

Cooperation

Statistics visualization Power line network

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SLIDE 6

Data Implementation Evaluation

Parameters

solution size distance from tractable instances (e. g. treewidth) structural parameters (e. g. vertex cover size)

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SLIDE 7

Data Implementation Evaluation

Parameters

solution size distance from tractable instances (e. g. treewidth) structural parameters (e. g. vertex cover size)

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SLIDE 8

Data Implementation Evaluation

Randomly generated instances

Advantages of randomly generated instances

Can have any number and size Can track relation between performance and parameters

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SLIDE 9

Data Implementation Evaluation

Randomly generated instances

Advantages of randomly generated instances

Can have any number and size Can track relation between performance and parameters Types: Application oriented:

Phylogenetic trees Web graphs DNA sequences

general

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SLIDE 10

Data Implementation Evaluation

Example: Colorful Components

COLORFUL COMPONENTS

Instance: An undirected graph G = (V , E ) and a coloring of the vertices χ : V → {1, . . . , c}. Task: Delete a minimum number k of edges such that all connected components are colorful, that is, they do not contain two vertices of the same color. Parameter: k

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SLIDE 11

Data Implementation Evaluation

Example: Colorful Components

COLORFUL COMPONENTS

Instance: An undirected graph G = (V , E ) and a coloring of the vertices χ : V → {1, . . . , c}. Task: Delete a minimum number k of edges such that all connected components are colorful, that is, they do not contain two vertices of the same color. Parameter: k Random model: c: number of colors; n: number of vertices; pv: probability that a component contains a vertex of a certain color; pe: edge probability within component; px: edge probability between components.

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SLIDE 12

Data Implementation Evaluation

A simple solver

Recommendation

Implement a solver that is as simple as possible.

Advantages

first impression on what solutions look like base line for finding bugs Typically simplest: Branching Integer Linear Programming (ILP)

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SLIDE 13

Data Implementation Evaluation

Graph Bipartization

Graph Bipartization: Find a minimum size set of vertices in a graph whose removal results in the graph being bipartite.

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SLIDE 14

Data Implementation Evaluation

Graph Bipartization

Graph Bipartization: Find a minimum size set of vertices in a graph whose removal results in the graph being bipartite.

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SLIDE 15

Data Implementation Evaluation

ILP for Graph Bipartization

c1, . . . , cn : binary variables (cover) s1, . . . , sn : binary variables (color)

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SLIDE 16

Data Implementation Evaluation

ILP for Graph Bipartization

c1, . . . , cn : binary variables (cover) s1, . . . , sn : binary variables (color) minimize

n

  • i =1

ci

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SLIDE 17

Data Implementation Evaluation

ILP for Graph Bipartization

c1, . . . , cn : binary variables (cover) s1, . . . , sn : binary variables (color) minimize

n

  • i =1

ci

  • s. t. ∀{v, w} ∈ E : (sv = sw) ∨ cv ∨ cw

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SLIDE 18

Data Implementation Evaluation

ILP for Graph Bipartization

c1, . . . , cn : binary variables (cover) s1, . . . , sn : binary variables (color) minimize

n

  • i =1

ci

  • s. t. ∀{v, w} ∈ E : (sv = sw) ∨ cv ∨ cw

which can be expressed as an ILP constraint as

  • s. t. ∀{v, w} ∈ E : sv + sw + (cv + cw) 1

∀{v, w} ∈ E : sv + sw − (cv + cw) 1

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SLIDE 19

Data Implementation Evaluation

Implementation language

Recommendation

Use a high-level programming language.

Advantages

more rapid development of typically exponential speedups, but only constant-factor slowdown persistent data structures allow simpler and less error-prone implementation of branching algorithms

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SLIDE 20

Data Implementation Evaluation

Debugging

Recommendation

Verify that your solution is a solution.

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SLIDE 21

Data Implementation Evaluation

Debugging

Recommendation

Verify that your solution is a solution.

Recommendation

Make automated tests that compare the result of the simple solver with that of the optimized solver, on randomly generated test cases.

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SLIDE 22

Data Implementation Evaluation

Debugging

Recommendation

Verify that your solution is a solution.

Recommendation

Make automated tests that compare the result of the simple solver with that of the optimized solver, on randomly generated test cases.

Recommendation

Use a test case minimization tool (e. g. http://delta.tigris.org/).

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SLIDE 23

Data Implementation Evaluation

Debugging

Recommendation

Verify that your solution is a solution.

Recommendation

Make automated tests that compare the result of the simple solver with that of the optimized solver, on randomly generated test cases.

Recommendation

Use a test case minimization tool (e. g. http://delta.tigris.org/).

Recommendation

Use version control.

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SLIDE 24

Data Implementation Evaluation

Data reduction

Recommendation

Implement data reduction rules.

Advantages

can be combined with approximation, heuristics, fixed-parameter, or other exact algorithms

  • ften very effective, even solve the whole instance

normally, the more, the better

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SLIDE 25

Data Implementation Evaluation

Heuristic speedups

Heuristic speedups in branching algorithms: Heuristic branching priorities Lower bounds

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SLIDE 26

Data Implementation Evaluation

Benchmark set

Recommendation

Create a benchmark set using the randomized generator and parameter settings that match those measured in the real-world instances.

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SLIDE 27

Data Implementation Evaluation

Benchmark set

Recommendation

Create a benchmark set using the randomized generator and parameter settings that match those measured in the real-world instances. c ∈ {3, 5, 8}, n ∈ {60, 100, 170}, pv ∈ {0.4, 0.6, 0.9}, pe ∈ {0.4, 0.6, 0.9}, px ∈ {0.01, 0.02, 0.04}.

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SLIDE 28

Data Implementation Evaluation

Comparison of algorithms

Time measurements depend on Machine Compiler options Program name Weather . . .

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SLIDE 29

Data Implementation Evaluation

Comparison of algorithms

Time measurements depend on Machine Compiler options Program name Weather . . . For exponential-time algorithms, averages of running time are useless.

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SLIDE 30

Data Implementation Evaluation

Comparison of algorithms

10

  • 2

10

  • 1

10 10

1

10

2

time (s) 20 40 60 80 100 instances solved (%)

1 2 3 4 5 6

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SLIDE 31

Data Implementation Evaluation

Evaluating one algorithm

40 60 80 100 120 140

n

10

  • 2

10

  • 1

10 10

1

10

2

time (s) 5 10 15 20 25 30 35

c

10

  • 2

10

  • 1

10 10

1

10

2

time (s) 0.0 0.2 0.4 0.6 0.8 1.0

pv

10

  • 2

10

  • 1

10 10

1

10

2

time (s) 0.0 0.2 0.4 0.6 0.8 1.0

pe

10

  • 2

10

  • 1

10 10

1

10

2

time (s) 0.03 0.04 0.05 0.06 0.07 0.08

px

10

  • 2

10

  • 1

10 10

1

10

2

time (s)

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SLIDE 32

Data Implementation Evaluation

Evaluate solution quality

Solutions are optimal, but might not match real-world truth due to model deficiencies.

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SLIDE 33

Data Implementation Evaluation

Evaluate solution quality

2 4 6 8 10 12 14 16 18 20 22 cluster size 20 40 60 80 100 total number

min-cut without DR min-cut with DR column generation

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SLIDE 34

Data Implementation Evaluation

Finally

Recommendation

Make source and data available under a free license.

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