This work analyzes the hierarchical Bayesian optimization algorithm (hBOA) on minimum vertex cover for standard classes of random graphs and transformed SAT instances. The performance of hBOA is compared with that of the branch-and-bound problem solver (BB), the simple genetic algorithm (GA) and the parallel simulated annealing (PSA). The results indicate that BB is significantly outperformed by all the other tested methods, which is expected as BB is a complete search algorithm and minimum vertex cover is an NP-complete problem. The best performance is achieved by hBOA; nonetheless, the performance differences between hBOA and other evolutionary algorithms are relatively small, indicating that mutation-based search and recombination-based search lead to similar performance on the tested classes of minimum vertex cover problems.

1. Hybrid Evolutionary Algorithms on Minimum
Vertex Cover for Random Graphs
Martin Pelikan1 , Rajiv Kalapala1 , and Alexander K. Hartmann2
1
Missouri Estimation of Distribution Algorithms Laboratory (MEDAL)
University of Missouri, St. Louis, MO
http://medal.cs.umsl.edu/
{pelikan,rkdnc}@cs.umsl.edu
2
Computational Theoretical Physics
Institut f¨r Physik
u
Universit¨t Oldenburg
a
a.hartmann@uni-oldenburg.de
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

2. Motivation
Background
Minimum vertex cover (MVC) is an important problem
MVC is NP-complete.
Many real-world applications can be formulated as MVC.
Example areas: Bioinformatics, communications.
But not much work on MVC in evolutionary computation.
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.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

3. Outline
1. Minimum vertex cover.
2. Algorithms.
3. Tested problem instances.
4. Experiments.
5. Summary and conclusions.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

4. Minimum Vertex Cover (MVC)
Minimum vertex cover
Given a graph (nodes+edges), a vertex cover is a subset of
nodes that contains at least one node of each edge.
A minimum vertex cover is a vertex cover of minimum size.
Input graph Vertex cover Minimum vertex cover
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

5. Different Flavors of MVC
Types of MVC
Decision problem:
Does a given graph have a vertex cover of given size?
Optimization problem:
What is the minimum vertex cover?
Some properties of MVC
MVC is NP-complete.
Difficult MVC instances have many local optima.
For some classes of graphs, difficulty of MVC well understood.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

6. Compared Algorithms
Compared algorithms
Branch and bound (BB)
Hybrid evolutionary algorithms
Hierarchical BOA (hBOA)
Genetic algorithm (GA)
Parallel simulated annealing (PSA)
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

7. Branch and Bound (BB)
Basic idea
Traverse the entire search
space (try all subsets).
Each level decides on one
node (in or out).
Each leaf encodes a unique
subset of nodes.
Branches that lead to
provably suboptimal
solutions are cut.
Why?
BB is inefficient, but can
verify the global optimum.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

8. Hybrid Evolutionary Algorithms
Representation
Candidate solutions are binary vectors.
Each bit determines presence/absence of one node.
Each string specifies a subset of nodes (allows invalid covers).
Hybridization with simple repair operator
A candidate solution may not represent a valid cover.
Applies single-bit flips to ensure valid covers.
Removes nodes from cover if possible.
Compared algorithms
Hierarchical BOA (hBOA).
Genetic algorithm (GA) with uniform crossover and bit-flip
mutation.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

9. Parallel Simulated Annealing (PSA)
Basic idea
Execute multiple runs of simulated annealing (SA) in parallel.
Each run of SA
Start with the full cover (all nodes included).
Each step adds or removes a node with equal probability.
Removal only allowed if the cover remains valid.
Addition of a node is executed with some probability.
Probability of accepting additions decreases with time
(controlled by temperature).
Why?
PSA and parallel tempering known to perform well on MVC.
Shows the effectiveness of local operators.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

10. Test Problems
Tested problem instances
G (n, m): Random graphs with fixed average node degree.
G (n, p): Random graphs with fixed proportion of edges.
TSAT: Random graphs corresponding to hard SAT instances.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

11. Graphs G (n, m)
Definition
Given c ∈ [0, 1], G (n, m) denotes graphs G = (V , E ) with
|E | = c|V |.
All graphs are sampled equal probability.
How to generate G (n, m) graphs
Start with a graph with no edges.
Add c|V | edges randomly.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

12. Graphs G (n, p)
Definition
Given p ∈ [0, 1], G (n, p) denotes graphs G = (V , E ) with
|V |
|E | = p .
2
All graphs are sampled equal probability.
How to generate G (n, p) graphs
Start with a graph with no edges.
|V |
Add p edges randomly.
2
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

13. Graphs TSAT
Definition
TSAT graphs correspond to SAT instances of model RB (Xu
& Li, 2000) but are generated directly.
How to generate TSAT graphs
Parameters: α = 0.8, r = 2.7808, p = 0.25.
Generate n disjoint cliques of size nα .
Randomly select two cliques and generate pn2α random edges
between these two cliques (no repetition).
Repeat the previous step (with repetitions) rn ln n − 1 times.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

14. Description of Experiments
Problem instances
For each graph type, vary size of the graphs.
Generate 1000 random graphs for each graph type and size.
Parameters of hybrid EAs
Population size determined by bisection method (10 runs).
Probability of crossover = 0.6, probability of bit-flip = 1/n.
Replacement: Restricted tournament replacement (RTR).
Parameters of PSA
Number of parallel runs = n.
Temperature schedule determined empirically to minimize
running time.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

15. Results on G (n, m) with m = 2n
10
10
BB, c=2
9
10 PSA, c=2
Number of evaluations/steps GA, c=2
8
10 hBOA, c=2
7
10
6
10
5
10
4
10
3
10
2
10
50 100 150 200 250
Number of nodes
hBOA outperforms GA.
PSA scales best.
BB is exponential.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

16. Results on G (n, m) with m = 4n
8
10
BB, c=4
PSA, c=4
7
10
Number of evaluations/steps GA, c=4
hBOA, c=4
6
10
5
10
4
10
3
10
2
10
50 100 150 200 250
Number of nodes
hBOA outperforms GA.
PSA scales best.
BB is exponential.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

17. Performance of hBOA on G (n, m) w.r.t. c = m/n
5
10
hBOA, n=250
hBOA, n=200
hBOA, n=150
4
Number of evaluations
10 hBOA, n=100
hBOA, n=50
3
10
2
10
1
10
0.5 1 2 4
c = number of edges / number of nodes
Greater c leads to greater complexity.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

18. Results on G (n, p) with p = 0.5
BB
7
10 PSA
Number of evaluations/steps
GA
6
10 hBOA
5
10
4
10
3
10
2
10
1
10
50 100 150 200 250
Number of nodes
hBOA and GA perform very similarly.
PSA scales best.
BB performs quite well.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

19. Results on TSAT
8
10 BB
PSA
7
Number of evaluations/steps
10 GA
hBOA
6
10
5
10
4
10
3
10
2
10
1
10
25 50 100 200
Number of nodes
All algorithms clearly exponential, but results a bit noisy.
hBOA and GA perform very similary
PSA scales best.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

20. Discussion of Results
Results on G (n, m)
For all algorithms, greater c leads to greater complexity.
...because graphs are lightly connected.
hBOA outperforms GA; PSA scales best; BB is exponential.
Results on G (n, p)
For all algorithms, greater p leads to smaller complexity.
...because graphs are heavily connected.
hBOA and GA similar; PSA scales best; BB is exponential.
Results on TSAT
All algorithms clearly exponential, but results a bit noisy.
hBOA and GA similar; PSA scales best.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

21. Summary and Conclusions
Summary
Described several classes of random graph problems for MVC.
Tested various algorithms on these problem classes.
Conclusions
All incomplete algorithms performed well, outperforming BB.
Both mutation and crossover work very well.
Problems can be used to test other algorithms.
Future research
What makes MVC instances difficult/easy for EDAs/GEAs?
Do other related problems lead to similar conclusions?
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs

22. Acknowledgments
Acknowledgments
NSF; NSF CAREER grant ECS-0547013.
VolkswagenStiftung (Germany) within the program
Nachwuchsgruppen an Universit¨ten.
a
University of Missouri; High Performance Computing
Collaboratory sponsored by Information Technology Services;
Research Award; Research Board.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs