# Metaheuristic Algorithms and Tree Decomposition

• Thomas Hammerl
• Nysret Musliu
• Werner Schafhauser
Chapter

## 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

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## Authors and Affiliations

• Thomas Hammerl
• 1
• Nysret Musliu
• 2
• Werner Schafhauser
• 3
1. 1.ViennaAustria
2. 2.Inst. Information SystemsVienna University of TechnologyViennaAustria
3. 3.XIMESViennaAustria