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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 ...

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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- 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

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