hybrid evolutionary algorithms on minimum vertex cover for random graphs

Hybrid Evolutionary Algorithms on MinimumVertex Cover for Random Graphs

Martin Pelikan1, Rajiv Kalapala1, and Alexander K. Hartmann2

1Missouri Estimation of Distribution Algorithms Laboratory (MEDAL)University of Missouri, St. Louis, MO

http://medal.cs.umsl.edu/pelikan,[email protected]

2Computational Theoretical PhysicsInstitut fur Physik

Universitat [email protected]

Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

http://medal.cs.umsl.edu/

pelikan,[email protected]

[email protected]

Motivation

Background

I Minimum vertex cover (MVC) is an important problemI MVC is NP-complete.I Many real-world applications can be formulated as MVC.I Example areas: Bioinformatics, communications.

I But not much work on MVC in evolutionary computation.

I Few interesting test instances available online.

Purpose

1. Generate a broad range of random MVC problem instances.

2. Determine optimum of all instances using a complete method.

3. Test various evolutionary algorithms on these MVC instances.


Outline

1. Minimum vertex cover.

2. Algorithms.

3. Tested problem instances.

4. Experiments.

5. Summary and conclusions.


Minimum Vertex Cover (MVC)

Minimum vertex cover

I Given a graph (nodes+edges), a vertex cover is a subset ofnodes that contains at least one node of each edge.

I A minimum vertex cover is a vertex cover of minimum size.

Input graph Vertex cover Minimum vertex cover


Different Flavors of MVC

Types of MVC

I Decision problem:Does a given graph have a vertex cover of given size?

I Optimization problem:What is the minimum vertex cover?

Some properties of MVC

I MVC is NP-complete.

I Difficult MVC instances have many local optima.

I For some classes of graphs, difficulty of MVC well understood.


Compared Algorithms

Compared algorithms

I Branch and bound (BB)I Hybrid evolutionary algorithms

I Hierarchical BOA (hBOA)I Genetic algorithm (GA)

I Parallel simulated annealing (PSA)


Branch and Bound (BB)

Basic idea

I Traverse the entire searchspace (try all subsets).

I Each level decides on onenode (in or out).

I Each leaf encodes a uniquesubset of nodes.

I Branches that lead toprovably suboptimalsolutions are cut.

Why?

I BB is inefficient, but canverify the global optimum.


Hybrid Evolutionary Algorithms

Representation

I Candidate solutions are binary vectors.I Each bit determines presence/absence of one node.I Each string specifies a subset of nodes (allows invalid covers).

Hybridization with simple repair operator

I A candidate solution may not represent a valid cover.I Applies single-bit flips to ensure valid covers.I Removes nodes from cover if possible.

Compared algorithms

I Hierarchical BOA (hBOA).I Genetic algorithm (GA) with uniform crossover and bit-flip

mutation.


Parallel Simulated Annealing (PSA)

Basic idea

I Execute multiple runs of simulated annealing (SA) in parallel.I Each run of SA

I Start with the full cover (all nodes included).I Each step adds or removes a node with equal probability.I Removal only allowed if the cover remains valid.I Addition of a node is executed with some probability.I Probability of accepting additions decreases with time

(controlled by temperature).

Why?

I PSA and parallel tempering known to perform well on MVC.

I Shows the effectiveness of local operators.


Test Problems

Tested problem instances

I G (n,m): Random graphs with fixed average node degree.

I G (n, p): Random graphs with fixed proportion of edges.

I TSAT: Random graphs corresponding to hard SAT instances.


Graphs G (n, m)

Definition

I Given c ∈ [0, 1], G (n,m) denotes graphs G = (V ,E ) with

|E | = c |V |.

I All graphs are sampled equal probability.

How to generate G (n, m) graphs

I Start with a graph with no edges.

I Add c |V | edges randomly.


Graphs G (n, p)

Definition

I Given p ∈ [0, 1], G (n, p) denotes graphs G = (V ,E ) with

|E | = p

(|V |2

).

I All graphs are sampled equal probability.

How to generate G (n, p) graphs

I Start with a graph with no edges.

I Add p(|V |

2

)edges randomly.


Graphs TSAT

Definition

I TSAT graphs correspond to SAT instances of model RB (Xu& Li, 2000) but are generated directly.

How to generate TSAT graphs

I Parameters: α = 0.8, r = 2.7808, p = 0.25.

I Generate n disjoint cliques of size nα.

I Randomly select two cliques and generate pn2α random edgesbetween these two cliques (no repetition).

I Repeat the previous step (with repetitions) rn ln n − 1 times.


Description of Experiments

Problem instances

I For each graph type, vary size of the graphs.

I Generate 1000 random graphs for each graph type and size.

Parameters of hybrid EAs

I Population size determined by bisection method (10 runs).

I Probability of crossover = 0.6, probability of bit-flip = 1/n.

I Replacement: Restricted tournament replacement (RTR).

Parameters of PSA

I Number of parallel runs = n.

I Temperature schedule determined empirically to minimizerunning time.


Results on G (n, m) with m = 2n

50 100 150 200 250

102

103

104

105

106

107

108

109

1010

Number of nodes

Num

ber

of e

valu

atio

ns/s

teps

BB, c=2PSA, c=2GA, c=2hBOA, c=2

I hBOA outperforms GA.

I PSA scales best.

I BB is exponential.


Results on G (n, m) with m = 4n

50 100 150 200 25010

2

103

104

105

106

107

108

Number of nodes

Num

ber

of e

valu

atio

ns/s

teps

BB, c=4PSA, c=4GA, c=4hBOA, c=4

I hBOA outperforms GA.

I PSA scales best.

I BB is exponential.


Performance of hBOA on G (n, m) w.r.t. c = m/n

0.5 1 2 410

1

102

103

104

105

c = number of edges / number of nodes

Num

ber

of e

valu

atio

ns

hBOA, n=250hBOA, n=200hBOA, n=150hBOA, n=100hBOA, n=50

I Greater c leads to greater complexity.


Results on G (n, p) with p = 0.5

50 100 150 200 25010

1

102

103

104

105

106

107

Number of nodes

Num

ber

of e

valu

atio

ns/s

teps

BBPSAGAhBOA

I hBOA and GA perform very similarly.

I PSA scales best.

I BB performs quite well.


Results on TSAT

25 50 100 20010

1

102

103

104

105

106

107

108

Number of nodes

Num

ber

of e

valu

atio

ns/s

teps

BBPSAGAhBOA

I All algorithms clearly exponential, but results a bit noisy.

I hBOA and GA perform very similary

I PSA scales best.


Discussion of Results

Results on G (n, m)

I For all algorithms, greater c leads to greater complexity.I ...because graphs are lightly connected.

I hBOA outperforms GA; PSA scales best; BB is exponential.

Results on G (n, p)

I For all algorithms, greater p leads to smaller complexity.I ...because graphs are heavily connected.

I hBOA and GA similar; PSA scales best; BB is exponential.

Results on TSAT

I All algorithms clearly exponential, but results a bit noisy.

I hBOA and GA similar; PSA scales best.


Summary and Conclusions

Summary

I Described several classes of random graph problems for MVC.

I Tested various algorithms on these problem classes.

Conclusions

I All incomplete algorithms performed well, outperforming BB.

I Both mutation and crossover work very well.

I Problems can be used to test other algorithms.

Future research

I What makes MVC instances difficult/easy for EDAs/GEAs?

I Do other related problems lead to similar conclusions?


Acknowledgments

Acknowledgments

I NSF; NSF CAREER grant ECS-0547013.

I VolkswagenStiftung (Germany) within the programNachwuchsgruppen an Universitaten.

I University of Missouri; High Performance ComputingCollaboratory sponsored by Information Technology Services;Research Award; Research Board.


hybrid evolutionary algorithms on minimum vertex cover for random graphs

Technology

vertex cover of minimum

graphs tsatdenitiontsat

graphs g n

minimumvertex cover

random graphs martin

vertex cover of given

classes of graphs

hartmann hybrid eas