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A Practical Analysis of Kernelization Techniques for the Maximum Cut Problem NII Shonan Meeting | March 5, 2019 Damir Ferizovic, Sebastian Lamm , Matthias Mnich, Christian Schulz, Darren Strash DEPARTMENT OF INFORMATICS: INSTITUTE OF THEORETICAL

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DEPARTMENT OF INFORMATICS: INSTITUTE OF THEORETICAL INFORMATICS

A Practical Analysis of Kernelization Techniques for the Maximum Cut Problem

NII Shonan Meeting | March 5, 2019 Damir Ferizovic, Sebastian Lamm, Matthias Mnich, Christian Schulz, Darren Strash

KIT – The Research University in the Helmholtz Association

www.kit.edu

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Max-Cut: Definition and Example

Given G = (V, E), find S ⊆ V such that |E(S, V \ S)| is maximum Notation: mc(G) := max

S⊆V |E(S, V \ S)|

v3 v1 v2 v4 v5 v6 v7

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 2/20

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Max-Cut: Definition and Example

Given G = (V, E), find S ⊆ V such that |E(S, V \ S)| is maximum Notation: mc(G) := max

S⊆V |E(S, V \ S)|

v3 v1 v2 v4 v5 v6 v7

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 2/20

S = {v2, v3, v4, v6}

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Max-Cut: Definition and Example

Given G = (V, E), find S ⊆ V such that |E(S, V \ S)| is maximum Notation: mc(G) := max

S⊆V |E(S, V \ S)|

v3 v1 v2 v4 v5 v6 v7

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 2/20

S = {v2, v3, v4, v6}

→ |E(S, V \ S)| = 7

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Max-Cut: Definition and Example

Given G = (V, E), find S ⊆ V such that |E(S, V \ S)| is maximum Notation: mc(G) := max

S⊆V |E(S, V \ S)|

v3 v1 v2 v4 v5 v6 v7

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 2/20

S = {v2, v3, v4, v6}

→ |E(S, V \ S)| = 7 → maximum?

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Max-Cut: Definition and Example

Given G = (V, E), find S ⊆ V such that |E(S, V \ S)| is maximum Notation: mc(G) := max

S⊆V |E(S, V \ S)|

v3 v1 v2 v4 v5 v6 v7

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 2/20

S = {v3, v4, v6, v7}

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Max-Cut: Definition and Example

Given G = (V, E), find S ⊆ V such that |E(S, V \ S)| is maximum Notation: mc(G) := max

S⊆V |E(S, V \ S)|

v3 v1 v2 v4 v5 v6 v7

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 2/20

S = {v3, v4, v6, v7}

→ |E(S, V \ S)| = 8

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Max-Cut: Definition and Example

Given G = (V, E), find S ⊆ V such that |E(S, V \ S)| is maximum Notation: mc(G) := max

S⊆V |E(S, V \ S)|

v3 v1 v2 v4 v5 v6 v7

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 2/20

S = {v3, v4, v6, v7}

→ |E(S, V \ S)| = 8 → maximum?

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Max-Cut: Importance of Studying it

Member of Karp’s 21 NP-complete problems Used in...

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 3/20

Circuit design Statistical physics Social networks

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Kernelization: Definition and Example

Kernelization: Compress graph while preserving optimality

v3 v1 v2 v4 v5 v6 v7 G0 = G :

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 4/20

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Kernelization: Definition and Example

Kernelization: Compress graph while preserving optimality

v3 v1 v2 v4 v5 v6 v7 G0 = G :

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 4/20

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Kernelization: Definition and Example

Kernelization: Compress graph while preserving optimality

v3 v1 v2 v6 v7 G1 :

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 4/20

mc(G0) = mc(G1) + 2

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Kernelization: Definition and Example

Kernelization: Compress graph while preserving optimality

v3 v1 v2 v6 v7 G1 :

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 4/20

mc(G0) = mc(G1) + 2 mc(G1) = 6

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Kernelization: Definition and Example

Kernelization: Compress graph while preserving optimality

v3 v1 v2 v4 v5 v6 v7 G0 = G :

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 4/20

mc(G0) = 6 + 2 = 8

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Max-Cut: Current Research on Kernelization

Previous work mostly of theoretical nature

Analyze problem k + mc-lowerbound(G) Different reformulations (Etscheid and Mnich 2018, Madathil, Saurabh, and Zehavi 2018, Prieto 2005)

Research on practicality missing

Present for other problems

INDEPENDENT SET, VERTEX COVER (Hespe, Schulz, and Strash 2018,

Akiba and Iwata 2016)

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 5/20

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Max-Cut: Reduction Rule 8 in Etscheid and Mnich 2018

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 6/20

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Max-Cut: Reduction Rule 8 in Etscheid and Mnich 2018

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 6/20

K4 K3 K5 K1 S

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Max-Cut: Reduction Rule 8 in Etscheid and Mnich 2018

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 6/20

K4 K3 K5 K1 S

1

NG(x) ∩ S = NG(X) ∩ S

2

|X| > |K|+|NG(X)∩S|

2

≥ 1

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Max-Cut: Reduction Rule 8 in Etscheid and Mnich 2018

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 6/20

K4 K3 K5 K1 S

1

NG(x) ∩ S = NG(X) ∩ S

2

|X| > |K|+|NG(X)∩S|

2

≥ 1 ✗

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Max-Cut: Reduction Rule 8 in Etscheid and Mnich 2018

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 6/20

K4 K3 K5 K1 S

1

NG(x) ∩ S = NG(X) ∩ S

2

|X| > |K|+|NG(X)∩S|

2

≥ 1

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Max-Cut: Reduction Rule 8 in Etscheid and Mnich 2018

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 6/20

K4 K3 K5 K1 S

1

NG(x) ∩ S = NG(X) ∩ S

2

|X| > |K|+|NG(X)∩S|

2

≥ 1

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Performance of Work from Etscheid and Mnich 2018

Kernelization mostly driven by rule 8 Weak-points in practice

Reliance on clique-forest Parameter k large in practice

Kernel size O(k) too large O(k · |E(G)|) time too slow

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 7/20

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Overview of Our Contributions

Implemented and evaluated work of Etscheid and Mnich 2018 Generalized existing reduction rules

Rules not dependent on a subgraph anymore

Developed new reduction rules

Simplistic but significant improvement in practice Identified inclusions

Efficient implementation

Timestamping system

Benchmark over a variety of instances

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 8/20

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Overview of Our Contributions

Five new unweighted reduction rules

Rule to compress induced 3-paths Two rules reducing cliques (R8, S2) Two antagonizing rules – merge and divide of cliques

Briefly investigated: Weighted path compression

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 9/20

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External/Internal Vertices v1 v2 v3 v4 v6 v7 v5

S

⇐ ⇒ ∈ Cext(G[S]) ⇐ ⇒ ∈ Cint(G[S])

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 10/20

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Rule Generalization: R8 – “Sharing Adjacencies”

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 11/20

K4 K3 K5 K1 S

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Rule Generalization: R8 – “Sharing Adjacencies”

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 11/20

S

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Rule Generalization: R8 – “Sharing Adjacencies”

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 11/20 1

NG(X) ∪ X = NG(x) ∪ {x}

2

|X| > max{|NG(X)|, 1}

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Rule Generalization: R8 – “Sharing Adjacencies”

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 11/20

v1 v2 v3 v4 v5

1

NG(X) ∪ X = NG(x) ∪ {x}

2

|X| > max{|NG(X)|, 1}

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Rule Generalization: R8 – “Sharing Adjacencies”

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 11/20

v1 v2 v3 v4 v5

1

NG(X) ∪ X = NG(x) ∪ {x} NG(X) ∪ X = NG(x) ∪ {x} NG(X) ∪ X = NG(x) ∪ {x}

2

|X| > max{|NG(X)|, 1}

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New Reduction Rule: S2 – “Semi-Isolated Cliques”

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 12/20

Clique G[S] with |Cext(G[S])| ≤

  • |S|

2

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New Reduction Rule: S2 – “Semi-Isolated Cliques”

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 12/20

Clique G[S] with |Cext(G[S])| ≤

  • |S|

2

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New Reduction Rule: S2 – “Semi-Isolated Cliques”

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 12/20

Clique G[S] with |Cext(G[S])| ≤

  • |S|

2

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New Reduction Rule: S2 – “Semi-Isolated Cliques”

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 12/20

Clique G[S] with |Cext(G[S])| ≤

  • |S|

2

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New Reduction Rule: S2 – “Semi-Isolated Cliques”

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 12/20

Clique G[S] with |Cext(G[S])| ≤

  • |S|

2

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Techniques Used for Performance

Avoid time-intensive checks

Vertex v internal in clique: ∀w ∈ NG(v) : Deg(v) ≤ Deg(w)

Avoid checking the same reduction rules

Timestamp of most recent change in neighborhood for each vertex Keep timestamp T for each rule: All vertices with timestamp ≤ T already processed Update vertex on change

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 13/20

T T + 3 v2 T − 1 v1 T − 4 v3

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Experiments on KaGen Graphs

Random graphs by KaGen, 150 per each graph type. |V| = 2048 Total runtime: 16 sec. (68 min. by Etscheid and Mnich 2018!)

Kernelization efficiency for KaGen graphs; metric: e(G) = 1 − |V(Gker)|

|V(G)|

2 4 6 8 0% 20% 40% 60% 80% 100% Graph Density: |E(G)|

|V(G)|

e(G)

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GNM RGG2D RGG3D RHG Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 14/20

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Experiments on KaGen Graphs

Improvement over Etscheid and Mnich 2018. |V| = 2048 Discrepancy between theory and practice

Absolute difference in efficiency: eabsDiff = e(Gnew) − e(Gold)

2 4 6 8 0% 20% 40% 60% 80% 100% Graph Density: |E(G)|

|V(G)|

eabsDiff = e(Gnew) − e(Gold)

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GNM RGG2D RGG3D RHG Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 15/20

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Experiments on KaGen Graphs

Improvement on our results with weighted path compression

Absolute difference in efficiency: eabsDiff = e(GnewWeighted) − e(Gnew)

2 4 6 8 0% 10% 20% 30% 40% 50% 60% Graph Density: |E(G)|

|V(G)|

eabsDiff = e(Gnew) − e(Gold)

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GNM RGG2D RGG3D RHG Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 16/20

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Experiments – BiqMac Solver

Name

|V(G)|

degavg e(G) TBM(G) TBM(Gker) ego-facebook 2888 1.03 1.00

  • 0.01

[∞] road-euroroad 1174 1.21 0.79

  • rt-twitter-copen

761 1.35 0.85

  • 1.77

[∞] bio-diseasome 516 2.30 0.93

  • 0.07

[∞] ca-netscience 379 2.41 0.77

  • 0.67

[∞] g000302 317 1.50 0.21 1.88 0.74 [2.53] g001918 777 1.59 0.12 31.11 17.45 [1.78] g000981 110 1.71 0.28 531.47 21.53 [24.68] imgseg 105019 3548 1.22 0.93 f 13748.62 [∞] imgseg 35058 1274 1.42 0.37

  • imgseg 374020

5735 1.52 0.82 f

  • Times in seconds. 10 hour time limit with 5 iterations.

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 17/20

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Experiments – Localsolver

Name

|V(G)|

degavg e(G) TLS(G) TLS(Gker) ego-facebook 2888 1.03 1.00 20.09 0.09 [228.91] road-euroroad 1174 1.21 0.79

  • rt-twitter-copen

761 1.35 0.85

  • 834.71

[∞] bio-diseasome 516 2.30 0.93

  • 4.91

[∞] ca-netscience 379 2.41 0.77

  • 956.03

[∞] g000302 317 1.50 0.21 0.58 0.49 [1.17] g001918 777 1.59 0.12 1.47 1.41 [1.04] g000981 110 1.71 0.28 10.73 4.73 [2.27] imgseg 105019 3548 1.22 0.93 234.01 22.68 [10.32] imgseg 35058 1274 1.42 0.37 34.93 24.71 [1.41] imgseg 374020 5735 1.52 0.82 1739.11 72.23 [24.08] Times in seconds. 10 hour time limit with 5 iterations.

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 18/20

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Experiments – Localsolver

Solution size over time by Localsolver: initial vs. kernelized

100 200 300 400 500 600 80% 85% 90% 95% 100% Time in Seconds Solution Size (% of max found) ca−coauthors−dblp web−Stanford web−it−2004 Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 19/20

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Conclusion

Summary Previous work: Good in theory, bad in practice Set of new (unweighted) reduction rules Sparse graphs highly reducible Significant benefits for existing solvers Future Work Add parallelism? New (weighted) reduction rules? Hybrid approach: Use solver for reductions?

Motivation Analysis of Existing Work Our Contribution Results Conclusion Sebastian Lamm – Kernelization and Max-Cut March 5, 2019 20/20