Springer Handbook of Computational Intelligence pp 1255-1270 | Cite as

# Metaheuristic Algorithms and Tree Decomposition

## Abstract

This chapter deals with the application of evolutionary approaches and other metaheuristic techniques for generating tree decompositions. Tree decomposition is a concept introduced by *Robertson* and *Seymour* [1] and it is used to characterize the difficulty of constraint satisfaction and NP-hard problems that can be represented as a graph. Although, in general, no polynomial algorithms have been found for such problems, particular instances can be solved in polynomial time if the treewidth of their corresponding graph is bounded by a constant. The process of solving problems based on tree decomposition comprises two phases. First, a decomposition with small width is generated. Basically in this phase the problem is divided into several subproblems, each included in one of the nodes of the tree decomposition. The second phase includes solving a problem (based on the generated tree decomposition) with a particular algorithm such as dynamic programming. The main idea is that by decomposing a problem into subproblems of limited size, the whole problem can be solved more efficiently. The time for solving the problem based on its tree decomposition usually depends on the width of the tree decomposition. Thus, it is of high interest to generate tree decompositions having small widths.

Finding the treewidth of a graph is an NP-hard problem [2]. In order to solve this problem, different algorithms have been proposed in the literature. Exact methods such as branch and bound techniques can be used only for small graphs. Therefore, metaheuristic algorithms based on genetic algorithms [3], simulated annealing [4], tabu search [5], iterated local search [6], and ant colony optimization (ACO ) [7, 8] have been proposed in the literature to generate good upper bounds for larger graphs. Such techniques have been applied very successfully and they are able to find the best existing upper bounds for many benchmark problems in the literature.

In this chapter, we will first introduce the concept of tree decomposition, and then give a survey on metaheuristic techniques used to generate tree decompositions. Three approaches based on genetic algorithms, iterated local search, and ACO that were proposed in the literature will be described in detail. Finally, we will also mention briefly two recent approaches that exploit tree decompositions within metaheuristic search.

- ACO
ant colony optimization

- AP
alternating-position crossover

- CSP
constraint satisfaction problem

- CX
cycle crossover

- DM
displacement mutation operator

- EM
exchange mutation operator

- GA
genetic algorithm

- GCP
graph coloring problem

- IHA
iterative heuristic algorithm

- ISM
insertion mutation operator

- IVM
inversion mutation operator

- LS
local search

- MCS
maximum cardinality search

- OX1
order crossover

- OX2
order-based crossover

- PMX
partially-mapped crossover

- POS
position-based crossover

- SAT
satisfiability

- SIM
simple-inversion mutation operator

- SM
scramble mutation operator

- VNS
variable neighborhood search

### References

- [64.1]N. Robertson, P.D. Seymour: Graph minors II: Algorithmic aspects of tree-width, J. Algorithms
**7**, 309–322 (1986)MathSciNetCrossRefMATHGoogle Scholar - [64.2]S. Arnborg, D.G. Corneil, A. Proskurowski: Complexity of finding embeddings in a k-tree, SIAM J. Algebr. Discrete Methods
**8**, 277–284 (1987)MathSciNetCrossRefMATHGoogle Scholar - [64.3]J.H. Holland:
*Adaptation in Natural and Artificial Systems*(Univ. of Michigan Press, Ann Arbor 1975)Google Scholar - [64.4]S. Kirkpatrick, C.D. Gelaff, M.P. Vecchi: Optimization by simmulated annealing, Science
**220**(4598), 671–680 (1983)MathSciNetCrossRefMATHGoogle Scholar - [64.5]F. Glover: Future paths for integer programming and links to artificial intelligence, Comput. Oper. Res.
**5**, 533–549 (1986)MathSciNetCrossRefMATHGoogle Scholar - [64.6]H. Lourenço, O. Martin, T. Stützle: Iterated local search. In:
*Handbook of Metaheuristics*, Vol. 57, ed. by F. Glover, G.A. Kochenberger (Springer, New York 2003) pp. 320–353Google Scholar - [64.7]M. Dorigo: Optimization, Learning and Natural Algorithms, Ph.D. Thesis (Dipartimento di Elettronica, Politecnico di Milano, Italy 1992), in ItalianGoogle Scholar
- [64.8]M. Dorigo, V. Maniezzo, A. Colorni: The ant system: Optimization by a colony of cooperating agents, IEEE Trans. Syst. Man Cybern. B
**26**(1), 29–41 (1996)CrossRefGoogle Scholar - [64.9]S. Lauritzen, D. Spiegelhalter: Local computations with probabilities on graphical structures and their application to expert systems, J. R. Stat. Soc. Ser. B
**50**, 157–224 (1988)MathSciNetMATHGoogle Scholar - [64.10]A.M. Koster, S.P. van Hoesel, A.W. Kolen: Optimal solutions for frequency assignment problems via tree decomposition, Lect. Notes Comput. Sci.
**1665**, 338–350 (1999)MathSciNetCrossRefMATHGoogle Scholar - [64.11]J. Alber, F. Dorn, R. Niedermeier: Experimental evaluation of a tree decomposition based algorithm for vertex cover on planar graphs, Discrete Appl. Math.
**145**, 210–219 (2004)MathSciNetMATHGoogle Scholar - [64.12]A. Koster, S. van Hoesel, A. Kolen: Solving partial constraint satisfaction problems with tree-decomposition, Networks
**40**(3), 170–180 (2002)MathSciNetCrossRefMATHGoogle Scholar - [64.13]J. Xu, F. Jiao, B. Berger: A tree-decomposition approach to protein structure prediction, Proc. IEEE Comput. Syst. Bioinform. Conf. (2005) pp. 247–256Google Scholar
- [64.14]M. Morak, N. Musliu, R. Pichler, S. Rümmele, S. Woltran: Evaluating tree-decomposition based algorithms for answer set programming, Proc. Learn. Intell. Optim. Conf. (LION 6) (2012)Google Scholar
- [64.15]A. Koster, H. Bodlaender, S. van Hoesel:
*Treewidth: Computational Experiments, Electronic Notes in Discrete Mathematics*, Vol. 8 (Elsevier Science, Amsterdam 2001)Google Scholar - [64.16]F. Clautiaux, A. Moukrim, S. Négre, J. Carlier: Heuristic and meta-heurisistic methods for computing graph treewidth, RAIRO Oper. Res.
**38**, 13–26 (2004)MathSciNetCrossRefMATHGoogle Scholar - [64.17]D.R. Fulkerson, O. Gross: Incidence matrices and interval graphs, Pac. J. Math.
**15**, 835–855 (1965)MathSciNetCrossRefMATHGoogle Scholar - [64.18]F. Gavril: Algorithms for minimum coloring, maximum clique, minimum coloring cliques and maximum independent set of a chordal graph, SIAM J. Comput.
**1**, 180–187 (1972)MathSciNetCrossRefMATHGoogle Scholar - [64.19]P. Larranaga, C. Kuijpers, M. Poza, R. Murga: Decomposing Bayesian networks: Triangulation of the moral graph with genetic algorithms, Stat. Comput.
**7**(1), 19–34 (1997)CrossRefGoogle Scholar - [64.20]N. Musliu, W. Schafhauser: Genetic algorithms for generalized hypertree decompositions, Eur. J. Ind. Eng.
**1**(3), 317–340 (2007)CrossRefGoogle Scholar - [64.21]T. Hammerl, N. Musliu: Ant colony optimization for tree decompositions. In:
*EvoCOP*, ed. by P. Cowling, P. Merz (Springer, Berlin, Heidelberg 2010) pp. 95–106Google Scholar - [64.22]U. Kjaerulff: Optimal decomposition of probabilistic networks by simulated annealing, Stat. Comput.
**2**(1), 2–17 (1992)CrossRefGoogle Scholar - [64.23]N. Musliu: An iterative heuristic algorithm for tree decomposition. In:
*Studies in Computational Intelligence, Recent Advances in Evolutionary Computation for Combinatorial Optimization*, Vol. 153, ed. by C. Cotta, J.I. van Hemert (Springer, Berlin, Heidelberg 2008) pp. 133–150CrossRefGoogle Scholar - [64.24]D.S. Johnson, M.A. Trick:
*The Second Dimacs Implementation Challenge: NP-Hard Problems: Maximum Clique, Graph Coloring, and Satisfiability*, Series in Discrete Mathematics and Theoretical Computer Science (American Mathematical Society, Boston 1993)Google Scholar - [64.25]T. Hammerl: Ant Colony Optimization for Tree and Hypertree Decompositions, M.S. Thesis (Vienna University of Technology, Vienna 2009)Google Scholar
- [64.26]M. Dorigo, T. Stützle:
*Ant Colony Optimization*, A Bradford Book (MIT Press, Cambridge 2004)MATHGoogle Scholar - [64.27]B. Bullnheimer, R.F. Hartl, C. Strauss: A new rank based version of the ant system: A computational study, Cent. Eur. J. Oper. Res. Econ.
**7**(1), 25–38 (1999)MathSciNetMATHGoogle Scholar - [64.28]T. Stützle, H. Hoos: Max-min ant system and local search for the traveling salesman problem, IEEE Int. Conf. Evol. Comput. (1997) pp. 309–314Google Scholar
- [64.29]T. Stützle, H. Hoos: Max-min ant system, Future Gener. Comput. Syst.
**16**(9), 889–914 (2000)CrossRefMATHGoogle Scholar - [64.30]M. Dorigo, L.M. Gambardella: Ant colony system: A cooperative learning approach to the traveling salesman problem, IEEE Trans. Evol. Comput.
**1**(1), 53–66 (1997)CrossRefGoogle Scholar - [64.31]N. Musliu: Generation of tree decompositions by iterated local search. In:
*EvoCOP*, ed. by C. Cotta, J. van Hemert (Springer, Berlin, Heidelberg 2007) pp. 130–141Google Scholar - [64.32]K. Shoikhet, D. Geiger: A practical algorithm for finding optimal triangulations, Proc. Natl. Conf. Artif. Intell. (AAAI'97) (1997) pp. 185–190Google Scholar
- [64.33]V. Gogate, R. Dechter: A complete anytime algorithm for treewidth, Proc. 20th Annu. Conf. Uncertain. Artif. Intell. UAI-04 (2004) pp. 201–208Google Scholar
- [64.34]E. Bachoore, H. Bodlaender: A branch and bound algorithm for exact, upper, and lower bounds on treewidth, Lect. Notes Comput. Sci.
**4041**, 255–266 (2006)CrossRefMATHGoogle Scholar - [64.35]R. Tarjan, M. Yannakakis: Simple linear-time algorithm to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs, SIAM J. Comput.
**13**, 566–579 (1984)MathSciNetCrossRefMATHGoogle Scholar - [64.36]K. Kask, A. Gelfand, L. Otten, R. Dechter: Pushing the power of stochastic greedy ordering schemes for inference in graphical models, Proc. Natl. Conf. Artif. Intell. (AAAI) (2011) pp. 54–60Google Scholar
- [64.37]H.L. Bodlaender, A.M.C.A. Koster: Treewidth computations I. Upper bounds, Inf. Comput.
**208**(3), 259–275 (2010)MathSciNetCrossRefMATHGoogle Scholar - [64.38]A. Khanafer, F. Clautiaux, E.-G. Talbi: Tree-decomposition based heuristics for the two-dimensional bin packing problem with conflicts, Comput. Oper. Res.
**39**(1), 54–63 (2012)MathSciNetCrossRefMATHGoogle Scholar - [64.39]M. Fontaine, S. Loudni, P. Boizumault: Guiding VNS with tree decomposition, 23rd IEEE Int. Conf. Tools Artif. Intell. (ICTAI) (IEEE, Boca Raton 2011) pp. 505–512Google Scholar