Hybrid Evolutionary Algorithms on Minimum Vertex Cover for Random Graphs Martin Pelikan 1 , Rajiv Kalapala 1 , and Alexander K. Hartmann 2 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¨ ur Physik Universit¨ at Oldenburg [email protected]Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs
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Hybrid Evolutionary Algorithms on Minimum Vertex Cover for Random Graphs
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.
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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
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.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs
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
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
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs
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.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs
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)
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs
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.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs
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.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs
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.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs
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.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs
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.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs
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.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs
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.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs
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.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs
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.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs
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.
Martin Pelikan, Rajiv Kalapala, Alexander K. Hartmann Hybrid EAs on Minimum Vertex Cover for Random Graphs