In this chapter we will take a closer look at the principles and
alternative methods of searching for solutions in the presence of
constraints. Let us first recall what we are talking about.
We assume we have the standard pattern of a constraint program:
The model part contains the logical model of our problem. It defines
the variables and the constraints.
Every variable has a domain of values that it can take
(in this context, we only consider domains with a finite number of values).
Once the model is set up, we go into the search phase.
Search is necessary since generally the implementation of the constraints
is not complete, i.e. not strong enough to logically infer directly
the solution to the problem. Also, there may be multiple solutions
which have to be located by search, e.g. in order to find the best one.
In the following, we will use the following terminology:
12.1.1 Overview of Search Methods
Figure 12.1 shows a search space with N (here 16)
possible total assignments, some of which are solutions.
Search methods now differ in the way in which these assignments
We can classify search methods according to different criteria:
Figure 12.1: A search space of size 16
Here is a selection of search methods together with their properties:
Complete vs incomplete exploration
- complete search means that the search space
is investigated in such a way that all solutions are guaranteed to be found.
This is necessary when the optimal solution is needed (one has to prove
that no better solution exists). Incomplete search may be sufficient when
just some solution or a relatively good solution is needed.
- Constructive vs move-based
- this indicates whether the method advances
by incrementally constructing assignments (thereby reasoning about partial
assignments which represent subsets of the search space) or by moving
between total assignments (usually by modifying previously explored
- some methods have a random element while others follow
|Full tree search
The constructive search methods usually organise the search space by
partitioning it systematically. This can be done naturally with a
search tree (Figure 12.2). The nodes in this tree
represent choices which partition the remaining search space into two
or more (usually disjoint) sub-spaces. Using such
a tree structure, the search space can be traversed systematically and
completely (with as little as O(N) memory requirements).
Figure 12.4 shows a sample tree search, namely a depth-first
As opposed to that, figure 12.3 shows an example of an
incomplete move-based search which does not follow a fixed search space
structure. Of course, it will have to take other precautions to avoid
looping and ensure termination.
Figure 12.2: Search space structured using a search tree
Figure 12.3: A move-based search
A few further observations:
Move-based methods are usually incomplete. This is not surprising given
typical sizes of search spaces.
A complete exploration of a huge search space
is only possible if large sub-spaces can be excluded a priori, and this
is only possible with constructive methods which allow one to reason about
whole classes of similar assignments.
Moreover, a complete search method must remember which parts of the
search space have already been visited.
This can only be implemented with
acceptable memory requirements if there is a simple structuring of the
space that allows compact encoding of sub-spaces.
Figure 12.4: A tree search (depth-first)
12.1.2 Optimisation and Search
Many practical problems are in fact optimisation problems, ie. we are
not just interested in some solution or all solutions, but in
the best solution.
Fortunately, there is a general method to find the optimal solution
based on the ability to find all solutions.
The branch-and-bound technique works as follows:
The branch_and_bound library provides generic predicates
which implement this technique:
Find a first solution
- Add a constraint requiring a better solution than the best
one we have so far (e.g. require lower cost)
- Find a solution which satisfies this new constraint.
If one exists, we have a new best solution and we repeat step 2.
If not, the last solution found is the proven optimum.
Since search space sizes grow exponentially with problem size,
it is not possible to explore all assignments except for the
very smallest problems.
The only way out is not to look at the whole search space.
There are only two ways to do this:
This is the simplest predicate in the branch_and_bound library:
A solution of the goal Goal is found that minimizes the value of
Cost. Cost should be a variable that is affected, and
eventually instantiated, by the execution of Goal. Usually, Goal
is the search procedure of a constraint problem and Cost is the variable
representing the cost.
- bb_min(+Goal, -Cost, ++Options)
A more flexible version where the programmer can take more control
over the branch and bound behaviour and choose between different
strategies and parameter settings.
In the following sections we will first investigate the considerable
degrees of freedom that are available for heuristics within the framework of
systematic tree search, which is the traditional search method
in the Constraint Logic Programming world.
Prove that certain areas of the space contain no solutions.
This can be done with the help of constraints. This is often referred
to as pruning.
- Ignore parts of the search space that are unlikely to contain
solutions (i.e. do incomplete search), or at least postpone their exploration.
This is done by using heuristics.
A heuristic is a particular traversal order of the search space
which explores promising areas first.
Subsequently, we will turn our attention to move-based methods
which in ECLiPSe can be implemented using the facilities of the