2 The Ant Colony Optimization Metaheuristic
Ametaheuristic refers to a master strategy that guides and modifies other heuristics to produce
solutions beyond those that are normally generated in a quest for local optimality.
—Tabu Search, Fred Glover and Manuel Laguna, 1998
Combinatorial optimization problems are intriguing because they are often easy to
state but very di‰cult to solve. Many of the problems arising in applications are
NP-hard, that is, it is strongly believed that they cannot be solved to optimality
within polynomially bounded computation time. Hence, to practically solve large
instances one often has to use approximate methods which return near-optimal solu-
tions in a relatively short time. Algorithms of this type are loosely called heuristics.
They often use some problem-specific knowledge to either build or improve solutions.
Recently, many researchers have focused their attention on a new class of algo-
rithms, called metaheuristics. A metaheuristic is a set of algorithmic concepts that
can be used to define heuristic methods applicable to a wide set of di¤erent problems.
The use of metaheuristics has significantly increased the ability of finding very high-
quality solutions to hard, practically relevant combinatorial optimization problems
in a reasonable time.
A particularly successful metaheuristic is inspired by the behavior of real ants.
Starting with Ant System, a number of algorithmic approaches based on the very
same ideas were developed and applied with considerable success to a variety of
combinatorial optimization problems from academic as well as from real-world
applications. In this chapter we introduce ant colony optimization, a metaheuristic
framework which covers the algorithmic approach mentioned above. The ACO
metaheuristic has been proposed as a common framework for the existing applica-
tions and algorithmic variants of a variety of ant algorithms. Algorithms that fit
into the ACO metaheuristic framework will be called in the following ACO
algorithms.
2.1 Combinatorial Optimization
Combinatorial optimization problems involve finding values for discrete variables
such that the optimal solution with respect to a given objective function is found.
Many optimization problems of practical and theoretical importance are of combi-
natorial nature. Examples are the shortest-path problems described in the previous
chapter, as well as many other important real-world problems like finding a mini-
mum cost plan to deliver goods to customers, an optimal assignment of employees
to tasks to be performed, a best routing scheme for data packets in the Internet, an

optimal sequence of jobs which are to be processed in a production line, an alloca-
tion of flight crews to airplanes, and many more.
A combinatorial optimization problem is either a maximization or a minimization
problem which has associated a set of problem instances. The term problem refers to
the general question to be answered, usually having several parameters or variables
with unspecified values. The term instance refers to a problem with specified values
for all the parameters. For example, the traveling salesman problem (TSP), defined
in section 2.3.1, is the general problem of finding a minimum cost Hamiltonian cir-
cuit in a weighted graph, while a particular TSP instance has a specified number of
nodes and specified arc weights.
More formally, an instance of a combinatorial optimization problem P is a triple
ðS; f ;WÞ,where S is the set of candidate solutions, f is the objective function which
assigns an objective function value f ðsÞ to each candidate solution s A S, and W is a
set of constraints. The solutions belonging to the set
~
SJS of candidate solutions
that satisfy the constraints W are called feasible solutions. The goal is to find a glob-
ally optimal feasible solution s

. For minimization problems this consists in finding a
solution s

A
~
S with minimum cost, that is, a solution such that f ðs

Þa f ðsÞ for all
s A
~
S; for maximization problems one searches for a solution with maximum objec-
tive value, that is, a solution with f ðs

Þb f ðsÞ for all s A
~
S.Note that in the follow-
ing we focus on minimization problems and that the obvious adaptations have to be
made if one considers maximization problems.
It should be noted that an instance of a combinatorial optimization problem is
typically not specified explicitly by enumerating all the candidate solutions (i.e., the
set S) and the corresponding cost values, but is rather represented in a more concise
mathematical form (e.g., shortest-path problems are typically defined by a weighted
graph).
2.1.1 Computational Complexity
A straightforward approach to the solution of combinatorial optimization problems
would be exhaustive search, that is, the enumeration of all possible solutions and the
choice of the best one. Unfortunately, in most cases, such a naive approach becomes
rapidly infeasible because the number of possible solutions grows exponentially with
the instance size n,where the instance size can be given, for example, by the num-
ber of binary digits necessary to encode the instance. For some combinatorial opti-
mization problems, deep insight into the problem structure and the exploitation of
problem-specific characteristics allow the definition of algorithms that find an opti-
mal solution much quicker than exhaustive search does. In other cases, even the best
algorithms of this kind cannot do much better than exhaustive search.
26 Chapter 2 The Ant Colony Optimization Metaheuristic