Local Probing Applied to Scheduling
Olli Kamarainen and Hani El Sakkout
IC-Parc
Imperial College of Science, Technology and Medicine, London SW7 2AZ, UK
{ok1,hhe}@icparc.ic.ac.uk
Abstract. This paper describes local probing, an algorithm hybridiza-
tion form that combines backtrack search enhanced with local consistency techniques (BT+CS) with local search (LS) via probe backtracking. Generally BT+CS can be eective at nding solutions for (or proving the infeasibility of) tightly constrained problems with complex and
overlapping constraints, but lacks good optimization characteristics. By
contrast, LS can be superior at optimizing problems that are loosely constrained, or that have constraints which are satisable by simple neighbourhood procedures, but it also has several weaknesses of its own. It is
weaker on problems with a complex constraint satisfaction element, and
cannot prove problem infeasibility, causing prolonged execution times
and ambiguous search outcomes for even trivially infeasible problems.
We show these divergent characteristics on a general resource constrained
scheduling problem class, extended with a widely applicable objective
function. We then detail a local probing hybrid that marries the strengths
of constraint satisfaction techniques, including good satisfaction characteristics and proofs of problem infeasibility, with the superior optimization characteristics of LS. This local probing hybrid achieves satcompleteness, without incorporating all the constraints into the LS neighbourhood function. Finally, we discuss the principal questions that must
be answered in creating local probing hybrids for other problems.
1 Introduction
1.1 Paper Overview
The paper is structured as follows. This section motivates the paper, and introduces the hybrization form and the application domain. Section 2 details local
probing's application to scheduling. Experimental results and comparisons to
other algorithms are described in Sect. 3. In Sect. 4, the key questions that must
be answered in creating a new local probing hybrid are listed and discussed.
Lastly, Sect. 5 gives the conclusions of this study.
1.2 Motivation
Backtrack search supported by local consistency techniques (BT+CS) is often
good at solving tight problems with complex constraints. However, it is generally
less suitable for optimization.
In this paper, local search (LS) is hybridized with BT+CS. In LS, typically
one or more complete but partially consistent or sub-optimal assignments are
improved at each search step. This gives rise to LS's strengths vis-a-vis conventional constraint satisfaction methods. Firstly, its search moves are more
informed than those of backtrack search, since the quality of a complete assignment is more easily measurable, by comparison with the quality of BT+CS's
partial assignments.1 This is particularly valid when assessing the quality of the
assignment w.r.t. a global optimization function. Secondly, the absence of systematicity allows assignments to be modied in any order, and so early search
moves do not necessarily skew search to focus only on particular sub-spaces.
This avoids a principal pitfall of BT+CS, where early search moves can skew
the search by restricting it to particular sub-spaces, until they are exhaustively
investigated.2
However, a traditional LS algorithm suers from a number of drawbacks.
Firstly, its lack of search memory can mean that it constantly returns to the
same search sub-spaces, causing it to get trapped in local optima.3 Secondly,
it lacks a natural way in which the eective local consistency procedures of
constraint satisfaction can be easily integrated, since these work best in combination with hard search decisions (such as variable assignments) that lead to further local-consistency deductions. This is a disadvantage when solving complex
constraints, i.e. ones that cannot be adequately captured by simple LS neighbourhood procedures, nor adequately dealt with by LS violation-minimization
optimization criteria. Thirdly, when the problem is infeasible, it cannot prove
infeasibility, and continues to search unsuccessfully for a solution, eventually
timing-out or giving up without informing the user whether the problem is infeasible or not, even when infeasibility is trivial to determine. This characteristic
is unacceptable when, for instance, the algorithm must be applied to a large
batch of problems in a reasonable amount of time, or if proofs of infeasibility are
valuable to the user.
The above considerations lead many researchers to conclude that the dierent
characteristics of conventional LS and BT+CS algorithms are complementary. It
is reasonable to assume that the hybridization of these methods can be fruitful
for certain problems.
1.3 Hybridization Form: Probe Backtracking
In the literature, various approaches to hybridizing BT+CS and LS, have been
investigated (e.g. [2, 4, 17, 2325, 27, 28]).4 Here the focus is the hybridization of
the two by means of probe backtrack search [7, 9]. Probe backtracking (PBT)
is a general hybridization form that has been shown to yield ecient hybrid
algorithms for problem classes similar to the one investigated here. Previously
Informedness was discussed in [21].
Moreover, if we consider as a measure of search move informedness the length of the
partial assignment undergoing extension, early BT+CS search decisions are even less
informed than decisions deeper in the search tree, greatly increasing the chance that
search is skewed to unproductive areas.
3
Many variations of basic LS strategies have tried to tackle this defect (e.g. tabu
search [11] and guided local search [26]).
4
See [10] for a general survey.
1
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Fig. 1. An illustration of probe backtrack search
though probe backtracking was used to hybridize BT+CS with linear and mixed
integer solvers, rather than with LS [1, 9].5
In probe backtracking, a constraint satisfaction and optimization problem
is solved by a high-level BT+CS algorithm that hybridizes another algorithm.
All the problem constraints are classied into two sets: easy and hard. The
easy constraints are those constraints that are easily solved by the hybridized
algorithm, known as the prober. At any search node, the backtrack search applies the prober to a sub-problem consisting of the easy constraints and an
objective function. The prober will generate a complete assignment that satises all the easy constraints, and is good (and, for some probers, super-optimal)
with respect to the objective function.6 If the assignment also happens to satisfy
all the hard constraints, a solution has been generated, and the search may
terminate, or continue with a tighter cost bound if better solutions are sought
as part of the branch and bound process. If the assignment has been found to
violate one or more hard constraints, then a new easy constraint is identied
and posted to the prober. This easy constraint is chosen to force the prober to
generate a dierent assignment that reduces the violation of one of the violated
hard constraints. The posting of this constraint moves the backtrack search to
a new node, and the process repeats. On backtracking, the negation of this easy
constraint is posted (the negation must also be an easy constraint). Figure 1
illustrates the behaviour of probe backtracking when the prober returns an assignment that violates one or more hard constraints. A pseudo-code description
can be found in [9].
1.4 Application to Scheduling
Dierent versions of probe backtrack search have been applied to commercial
dynamic scheduling problems [9], to earliness/tardiness scheduling problems [3],
and to a generic scheduling problem extended with a piecewise linear objective
function [1]. The latter is selected here as a testbed for the technique that is
developed, because it generalizes the rst two.
A similar evolution from integrating LP-based methods towards other solvers has
been discussed in [15].
6
A super-optimal assignment is a complete, partially consistent assignment that is no
worse w.r.t. the objective function than any optimal solution.
5
The selected problem is a scheduling problem that is based on the kernel
resource feasibility problem (KRFP) [6, 9]. The KRFP generalizes most scheduling
benchmarks, including job shop scheduling, ship loading, bridge building, and
resource constrained project scheduling.
In the KRFP, we have several types of resources and a given quantity of each.
We have also a xed number of non-preemptive activities. Each activity requires
a quantity of a specic resource type during its execution. An arbitrary set of
metric temporal constraints relate the start and end times of activities. The aim
is to schedule activities so that all the temporal constraints are satised, and
the demand on any resource does not exceed its resource quantity at any time
on the scheduling horizon.
While the KRFP itself represents only a constraint satisfaction problem, an
objective function is often needed. Many real world optimization criteria such
as revenue and cost optimization in transportation can be approximated by
a piecewise linear function. In the problem we investigate, a piecewise linear
function maps the start time (or end) of each of the activities of the KRFP to
an activity cost. The objective function is then a minimization of the sum of the
values of these functions. The kernel resource feasibility problem with a piecewise
linear objective function is abbreviated by PL-KRFP.
Like many large scale combinatorial optimization problems (LSCO's), the PLKRFP can be clearly divided into dierent sub-problems. The probe backtrack
approach presented in [9] keeps temporal constraints within the sub-problem addressed by the prober, while relaxing the resource constraints and handling them
at the backtrack search level. If the solution to the sub-problem does not satisfy
the relaxed resource constraints, additional temporal precedence constraints are
added to the sub-problem in order to relieve violations. This approach was also
used in several algorithms described in [1]. In all these algorithms, the probers
are based on linear programming (LP) or mixed-integer programming (MIP).
1.5 Overview of Approach
In local probing, an LS algorithm is used as the prober instead. Here, simulated
annealing (SA, [19]) is the LS algorithm that is hybridized to illustrate the
potential of this form.7
By comparison with the earlier probe backtracking hybrids based on LP and
MIP, which rely on the prober solving a sub-problem that is linear, or highly
linear, local probing hybrids are widely applicable, because the prober can solve
sub-problems with arbitrary objective functions and arbitrary easy constraints.
However, one disadvantage of local probing hybrids is that, because an LS
prober is normally incomplete, the overall algorithm loses completeness. We
distinguish though two levels of completeness for constraint satisfaction and
optimization problems:
7
Note that most other LS algorithms including hill climbing, tabu search ([11]), guided
local search ([26]), and genetic algorithms ([12, 14]) could have been hybridized in
an analogous fashion. SA is selected for this preliminary study because of its simplicity. Arguably, the results of SA are representative since a more sophisticated LS
algorithm would also lead to a better hybrid.
sat-completeness, where the algorithm returns a solution, or proves no solution exists, in a nite number of steps; and
opt-completeness, where the algorithm returns an optimal solution, or proves
no solution exists, in a nite number of steps.
We show how sat-complete local probing hybrids can be constructed.
The local probing algorithm that is presented here solves the PL-KRFP, and
uses an LS prober that utilizes a generic neighbourhood operator. The operator
is capable of satisfying an arbitrary set of easy temporal constraints. It needs
to do so because neighbourhood is dynamic, i.e. the set of temporal constraints
that must be satised by LS changes from one backtrack search node to the next.
This neighbourhood operator has been found to be much more eective than
other tested alternatives [18]. It uses a linear solver to nd temporally feasible
neighbours in the proximity of the current node by solving an LP problem with
a linear minimal perturbation objective function.
Our local probing hybrid is compared with (1) a dierent, LP-based, probe
backtracking hybrid previously specialized for the PL-KRFP, and (2) a pure
SA algorithm tuned for the same problem. The results show that applying local
probing can lead to better overall algorithm performance by comparison with
the other hybrid algorithm. They also show that local probing improves the SA
algorithm by addressing its weaknesses: it forces SA to escape from infeasible
local optima by dynamically reshaping its neighbourhood function; it supports
it by applying local consistency methods where they are more eective; and
it restores algorithm sat-completeness. Finally, the paper discusses the broader
potential of local probing.
2 Local Probing for Scheduling
2.1 Overview
The constraints of the PL-KRFP consist of three classes: resource constraints,
temporal constraints and constraints linking the objective function to the temporal decision variables. The neighbourhood operator of the LS algorithm can
be designed such that it satises certain constraint classes, thereby guaranteeing
that the probes never violate these classes. It is possible to take any subset of
the problem constraints into the neighbourhood operator, and thus, ensure probe
feasibility with respect to that subset. The question is which kind of sub-problem
is easy for the neighbourhood operator to satisfy and makes the overall algorithm ecient at the same time. This is important since the class of constraints
satised by the neighbourhood operator includes the ones that will be posted by
probe backtracking as it drives the search towards globally feasible solutions.
In this paper, the temporal constraints are dealt with by the neighbourhood
operator while the LS objective function aims to minimize the overall cost, and
penalizes the violation of the hard resource constraints. The local probing
procedure is summarized in Fig. 2. If the algorithm returns a solution, a cost
bound constraint is posted and the algorithm is repeated. The cost bound is not
dealt with in the neighbourhood operator; rather, before returning a probe, LS
prober checks that the cost bound is satised (and fails if it is not).
1. begin local_probing
2.
if not(solve temporal sub-problem by LS prober)
3.
then return FALSE
4.
else if not(nd violated resource constraints)
5.
then return TRUE
6.
else begin
7.
select violated resource constraint HC;
8.
select temporal constraint EC that reduces violation of HC;
9.
post EC;
10.
if local_probing
11.
then return TRUE;
12.
remove EC;
13.
post not(EC);
14.
if local_probing
15.
then return TRUE;
16.
remove not(EC);
17.
return FALSE;
18.
end;
19. end local_probing
Fig. 2. Local probing algorithm.
In probe backtracking, dierent types of constraint propagation and search
ordering heuristics can be applied. When making the decision on which constraint to post to the sub-problem in order to repair the violations, we select
rst the resource constraint that is violated the most, then we select a temporal constraint that would reduce violation (by reducing resource overlap), but
which would cause minimal change to the proposed schedule. On backtracking,
the negation of this constraint is posted instead. Posting the temporal constraint
triggers local consistency procedures which reduce the domains of the variables
further. The details are described in [9]. The implementation relies on the hybridization facilities of the ECLiPSe constraint logic programming platform, e.g.
attributed variables, tentative values and waking priorities [16].
2.2 Simulated Annealing as an LS Prober
As an LS strategy, we use simulated annealing (SA) [19], which is easy to implement and may avoid local minima. It starts from an initial assignment8, and,
at each search step, the neighbourhood operator suggests for the sub-problem a
candidate solution that satises all the temporal sub-problem constraints.
Search continues until a termination condition is satised. In SA, a worse
neighbour may be accepted. The acceptance is randomized and depends on the
quality of a neighbour and a temperature parameter, which is decreased during
the search depending on the selected cooling strategy. The probability of moving
to worse neighbours decreases with the temperature parameter.
Hard resource constraints are treated as soft constraints by adding a penalty
component to the objective function, namely, the resource utilization area that
exceeds the resource limit over the time horizon, multiplied by a constant. The
results presented suggest that this approach works well together with the repairbased search capability of probe backtracking.
2.3 Minimal Perturbation Neighbourhood Operator
We require an ecient neigbourhood operator, which nds feasible neighbour
solutions. It should be generic and capable of satisfying dierent classes of con8
The initial assignment in the very beginning is the initial schedule. Subsequently,
the previously generated probe is used.
straints, especially temporal constraints in our scope. In order to utilize the
eciency benets of LP within the neighbourhood operator of local probing,
an operator based on minimal perturbation search [7, 9] is introduced. First, it
assigns a value to only one randomly selected variable9. This value selection is
backtrackable and all the feasible values in the domain of the selected variable
can be explored. Once the variable has been instantiated, the remaining variables
are solved as a minimal perturbation problem by LP. In a minimal perturbation
problem the aim is to nd the temporally feasible solution which minimizes the
changes to the previous assignment. The objective function and the minimal
perturbation constraints are modelled as follows:
min
XN x
i=1
d i;
s.t.
i? i
i? i
(1)
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(2)
=1
(3)
The absolute change between variable and its initial value is a non-linear
expression. It is represented by a new variable x = j ? j, and the total change
over all the non-instantiated variables can be minimized by adding linear
constraints (2) and (3). The objective function (1) of the minimal perturbation
problem is linear and the solution can be produced by a Simplex-based LP solver
extremely quickly.
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x
d i
d i
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2.4 Sat-completeness
Probe backtrack search will guarantee to satisfy any constraints that are monitored for violation and repaired within its backtrack procedure, provided that
a number of conditions are met. Specically: (1) if a constraint is violated, it
must be detected and scheduled for repair by probe backtracking within a nite
number of steps; (2) the probe backtrack procedure must identify all easy constraints whose satisability will determine whether the hard constraint can be
satised; (3) it must systematically post these easy constraints (and on backtracking, their negations, to fully partition the search space) to the prober until
either the constraint is satised, or it is proved impossible to satisfy; and (4) the
prober must guarantee to satisfy the posted easy constraints if this is possible.
We do not refer to the rst three conditions in the context of the KRFP
resource constraints and their relationship with temporal precedence constraints,
since they have been fully described in the context of previous complete probe
backtracking hybrids [9].
Condition (4) is satised because the neighbourhood operator will always nd
a temporally feasible assignment if one exists. The neighbourhood operator will
select a variable and systematically attempt each of its values. If a temporally
feasible solution exists with the variable set to a particular value, the LP solver
9
The domain of the selected variable is divided into two sets, by randomly selecting a
value x and placing values better than x in the better set S1 and worse (or equal)
values in the worse set S2 . S1 is explored rst by using random value selection,
before exploring S2 .
will nd one. Thus the neighbourhood operator ensures that the LS prober
guarantees to satisfy the set of easy constraints if they are satisable.
Thus local probing is sat-complete: it nds a solution if one exists, and proves
infeasibility if no solutions exist. However, the algorithm is not opt-complete since
the assignments it returns are not super-optimal.
3 Empirical Study
3.1 Algorithms used for comparison
We compared a tuned local probing hybrid (denoted Probe(Loc)) with two other
algorithms: an LP-based probe backtracking hybrid from [1] (Probe(LP)), and a
pure SA algorithm (SA).
Probe(LP). The best performing LP-based probe backtracking algorithm from
[1] was chosen (we refer you to this paper for a full algorithm description).
The algorithm is similar to our local probing instance in that it is a probe
backtracking hybrid specialized for solving the PL-KRFP, but diers in that its
prober is LP-based and opt-complete. The prober of Probe(LP) applies an LP
solver to a linear relaxation of the piecewise linear objective function. Possible
discrepancies between the linearly relaxed objective and the allowed values of
the true piecewise linear function are resolved by a small search procedure which
branches over the temporal variables' domains, until this discrepancy has been
eliminated. This allows the prober to return assignments that satisfy all the
easy temporal constraints, and are super-optimal. Note that Probe(LP) does not
include any resource constraint violation penalties within the prober's objective
function, resulting in a prober that has a greedy characteristic: it nds the best
quality prober solution from the cost point of view, independent of how many
hard resource constraints it violates.10
SA. SA was selected to represent pure LS because SA is used as our local
probing prober, allowing fair comparisons of performance: the SA implementation utilizes precisely the same neighbourhood operator and objective function
that are used in the prober.
A manual tuning process was applied to rene SA's parameters until it produced good quality results: the parameters applied in SA were those that produced the best results on a representative sample of PL-KRFP instances, w.r.t.
both the number of solved problems, and the quality of the generated solutions.
SA starts with the initial temperature of 100000, and after every 10000 neighbour candidate evaluations the temperature is lowered by multiplying it with
constant 0.85 (geometric reduction).11 This search continues until it times-out.
There are no easy ways to penalize resource violation in an LP objective function, due
to the limited constraint and objective function classes it can deal with. Therefore,
in the hybrids of [1], resource constraint satisfaction was handled completely at the
probe backtracking level.
11
A neighbour candidate is accepted if it is better than the current assignment or
p < exp(Ccurr ?Cnew )=T where: p is a random number between 0 and 1; Cnew is the
cost of the neighbour candidate; Ccurr is the cost of the current assignment; and T
is the temperature.
10
Probe(Loc). The SA prober of Probe(Loc) was tuned in a similar fashion, but
the best parameters in this case were dierent because resource satisfaction is
supported by the backtrack search component. The best combination are used
in Probe(Loc), and they comprise: a cooling factor of 0.9; a cooling step every
10 neighbour candidate evaluations; and an SA procedure that terminates after
100 cooling steps.
3.2 Test Problems
The tests were carried out on 1200 PL-KRFP benchmark problems. Each instance contains an initial schedule, comprising scheduled activities which all use
the same type of resource. The initial schedule needs to be changed because
(a) it is made infeasible by a reducing the number of available resources, and
(b) it needs to be optimized with respect to a piecewise linear objective function. The problem instances contain randomly-generated temporal constraints
between arbitrary activity start and end points. The temporal constraints are
always satised by the initial schedule, but must continue to be enforced as
the schedule changes. A temporal constraint density parameter controls their
density, and is proportional to the probability of a temporal constraint existing
between any pair of start/end time variables.12 Each activity is associated with
a piecewise linear function containing at least ve local minima. The function
represents the varying cost of the activity as its start time takes dierent values.
The objective is to minimize the sum of the activity costs.
The tests were run over 300 dierent schedule and temporal constraint density combinations. The number of activities varied over {50, 100}, and the temporal constraint densities over {0.3, 0.6, 0.9}. Each combination was the basis
for four problem instances: dierent time window constraints were used for the
activities, allowing them to change by no more than {10, 20, 50, 100}
time units from their initial position. The number of available resources was
40 per cent lower than the number used by the input schedule. The scheduling
horizon was limited to 200 time units.13
The test were run on Pentium II 450 MHz PCs, and the algorithms timedout after 1800 seconds of CPU time. The three algorithms were implemented on
ECLiPSe 5.3, and the LP sub-problems were solved using the default settings of
CPLEX 6.5.
3.3 Results
Any algorithm run on a problem instance results in one of the following outcomes:
search completion with an infeasibility proof; search completion with a solution,
or a time-out with a solution; or, nally, a time-out without a solution.
Table 1 presents, for each algorithm, the number of instances (out of 1200)
for which it (a) obtained an infeasibility proof; (b) found a solution (whether or
not the search completed); or (c) timed-out without a solution.
The results for proofs of infeasibility show the trivial result that SA is not
able to prove that a solution cannot be found, while the other two algorithms
12
The probability is equal to density=(2N ? 1), where N is the number of activities.
13
See http://www.icparc.ic.ac.uk/~ok1/pl-krfp_benchmarks.html for the benchmarks.
Table 1. Results over all instances.
Method
SA
Probe(LP)
Probe(Loc)
Proved infeasible Solution found Timeout w/o solution
0 (0.0%) 417 (34.8%)
783 (65.3%)
299 (24.9%) 313 (26.1%)
588 (49.0%)
285 (23.8%) 429 (35.8%)
486 (40.5%)
do. Interestingly though, they also seem to indicate that Probe(LP) is slightly
better at proving infeasibility than Probe(Loc) within the selected time limit.
This must be due to the fact that the LP-based prober of Probe(LP) is faster
at generating probes than the SA prober of Probe(Loc). Nevertheless, the two
algorithms remain comparable overall, because the former's prober returns assignments that generally violate more hard resource constraints (its objective
function is unable to penalize resource constraint violation).
By the solutions found measure, the results show that Probe(Loc) slightly
outperforms SA and is clearly better than Probe(LP). For SA and Probe(Loc), the
resource-violation penalty term seems to work well on solvable problems, giving
them a clear advantage over Probe(LP), whose prober is unable to recognize
resource violations and does not incrementally improve a partially consistent
solution. The greater exibity of LS over linear programming when modelling
disjunctive constraints and objective criteria is the source of this advantage.
The third measure, timeout w/o solution, seems to be one of the best overall
indicators of algorithm performance, since it quanties the problem instances
where the algorithm was unable to generate a meaningful answer. By this measure Probe(Loc) is the best of the three algorithms (40.5% by comparison with
49.0% for Probe(LP), and 65.3% for SA).
A dierent measure of performance is solution quality. Probe(Loc) or Probe(LP)
proved that no solution exists for 356 instances out of the 1200, and in 399 cases,
all three algorithms timed-out without a solution. However, at least one algorithm found a solution for the remaining 445 cases, and of these, 300 were solved
by all three. These 300 instances, even though they represent only 25 per cent
of the whole problem set, provide us with a vehicle to compare fairly the cost
minimization abilities of the three algorithms.
Table 2 shows the average cost improvement relative to the cost of the initial
(input) schedule, expressed as a percentage change.14 Even though the dierences between algorithms are not great, by this measure Probe(LP) clearly outperforms both Probe(Loc) and SA. Of course, the 300 instances that were solved
by each algorithm represent the easiest part of the problem set, where the
hard resource constraints are less relevant, and Probe(LP)'s greedy, super-optimal
probes are most eective. This advantage though must be weighed against the
fact that Probe(LP) solves by far the fewest problems (out of the 445 problems
solved by at least one algorithm, it fails to solve 132, versus 28 for SA, and 16
for Probe(Loc)).
14
Recall that the initial input schedule is infeasible due to the imposed 40% resource
utilization reduction, and so some returned solutions are associated with a rise in
cost.
Table 2. Average cost improvement relative to initial schedule cost, for problem instances solved by all three algorithms.
Method
Improvement (%)
-21.31
-24.57
-22.81
Table 3. Average CPU time in secs. needed to get a rst solution over all instances
(CPU-all), and over the instances solved by all three algorithms (CPU-easy).
SA
Probe(LP)
Probe(Loc)
Method
SA
Probe(LP)
Probe(Loc)
CPU-all CPU-easy
1228
70
924
43
839
35
Table 3 presents the average CPU times needed to get a rst solution or
prove infeasibility. Where an algorithm timed-out, the time-out period was used
instead.15 For each algorithm, the rst column covers all 1200 instances. The
second column covers just the easy 300 instances that were solved by all three
algorithms. It seems that Probe(Loc) not only nds solutions more often, but is
faster at doing so than Probe(LP) and SA.
Finally, Fig. 3 shows how the algorithm performance measures vary as the
problem tightness varies.16 Problem tightness is inversely proportional to timewindow size, and directly proportional to temporal constraint density. It is clear
from these gures that Probe(Loc) shadows the performance of SA on feasible,
loose problems and that of Probe(LP) on tight and infeasible problems, leading
to better overall performance.
4 Discussion
4.1 Discussion on Experimental Results
Constraint satisfaction characteristic The results demonstrate that local
probing can improve SA's performance by addressing its three principal weaknesses: it forces SA to escape from infeasible local optima by dynamically reshaping its neighbourhood function; it supports it by applying local consistency methods where they are more eective; and it restores algorithm sat-completeness.
Addressing the rst two weaknesses leads to improved performance on medium
tightness problems (e.g. in the tests, instances where density=0.6). Addressing
the last enables proofs of infeasibility, and eliminates SA's needless timeouts on
infeasible problems (e.g. where density=0.9).
The results also show the benets of LS's exibility by comparison with LPbased probers the ability to easily penalize violations of hard constraints
in the objective function means that local probing is able to capitalize on SA's
ability to solve relatively loosely-constrained problems more often (e.g. where
15
16
We preferred to use the time-out period rather than some arbitrary penalty.
Please note that the directions of the axes have been changed for dierent measures
to improve the observation angle.
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Fig. 3. Infeasibility proofs, solutions found, and time-outs without a solution over
temporal constraint density and time-window size.
density=0.3). This results in a large increase in the number of solutions found
within the timeout, and a shorter time to the rst solution. Nevertheless the
LP-based prober's speed enabled it to prove infeasibility slightly more often.
Optimization characteristic Although LP-based probers are not as widely
applicable as LS probers, when they are applicable their ability to quickly generate super-optimal assigments can enable a high-quality optimization characteristic. The results support this statement, showing that the LP-based prober
returned better quality solutions for those problems that it was able to solve. For
such loosely-constrained problems, the time spent in each call of the prober of
Probe(Loc) is too short to enable it to nd the best quality solutions. However,
it cannot be increased dramatically though without impacting the performance
on more tightly-constrained problems. This trade-o indicates that a dynamic
prober termination condition, that varies the time spent in the LS prober according to the number of violated hard constraints, might prove very eective.
This is also true when comparing Probe(Loc) to SA. While the results show
that the local probing hybrid is on average slightly better at optimization, for
problems where constraint satisfaction is less of an issue, it makes sense to fully
utilize the computation time for optimization, by allowing the prober to take
more time.
4.2 Building other local probing algorithms
Given the performance improvements obtained on the PL-KRFP, it is likely that
other problems could also be solved better by applying a local probing hybrid.
However, in building a local probing hybrid, the algorithm creator faces many
design dimensions. As for other probe backtracking approaches these include
the backtrack search heuristics, the local consistency techniques, and the
branch and bound procedure that is applied at the backtrack search level
of probe backtracking.
The creator of local probing hybrids faces many other choices though. Such
choices mostly relate to the LS strategy, and include the basic algorithm type,
e.g. hill climbing, tabu search [11], or genetic algorithm [12, 14]; its initialization procedure, e.g. heuristic or randomized; and the applied neighbourhood
operator, e.g. rst improvement, steepest descent.17 The operator might also
be expanded systematically in order to nd feasible neighbours, e.g. variable
neighbourhood search in [22], large neighbourhood search in [25] and the neighbourhood operators in [18]. Other algorithmic dimensions include learning: e.g.
the tabu lists of tabu search, or the penalty terms of guided local search (GLS,
[26]); the iteration structure, e.g. SA's temperature schedules, and the selection processes of genetic algorithms; and lastly the termination condition:
perhaps a maximum run-time; a best solution quality (optimal, gap-to-lowerbound, improvement-too-slow); or a maximum number of neighbour candidate
evaluations.
However, among the most critical choices in a local probing hybrid are the
constraint classes that must be satised by the neighbourhood operator, and
those classes modelled by the LS prober's objective function. These choices distribute the responsibility for solving the problem constraints, sharing them between the LS prober, and the higher level BT+CS algorithm.
Recall that in probe backtracking, the problem constraints are divided into
easy and hard constraint sets, as explained in the introduction. Here LS is the
prober that solves easy sub-problem. Thus it must be capable of: (1) satisfying
the easy constraints, while seeking to (2) optimize the objective function, and
possibly (3) minimize the violation of hard constraints.
Achieving (1) depends on nding neighbourhood operators that can satisfy
the easy constraints, since these are normally the only constraints that an LS
algorithm guarantees to satisfy.
Achieving (2) is possible by including the problem's objective criteria in the
LS objective function. To achieve (3), it is necessary to extend the objective
function to penalize hard constraint violation. This reduces the risk of hard
constraint violation, but clearly the probe backtracking procedure must still
monitor and repair such constraints to remain sat-complete, as there is no guarantee that they will all be satised by the LS prober.
4.3 Related Work
In this paper, the resource constraints of the scheduling problem are relaxed,
and a temporal sub-problem which is easy to satisfy is solved separately. The
approach was used in [8], and elsewhere (e.g. [5]). Here, the neighbourhood
17
N.B. the genetic algorithm cross-over and mutation operators can be viewed as
specialized neighbourhood operators.
operator of LS satises the temporal constraints, and the cost function and the
resource constraints are modelled in an extended objective function. In [18], the
LS probers did not penalize hard constraint violation, but some were able to
satisfy the cost bound constraint.
Local probing belongs to the class of LS/BT+CS hybridizations where LS
is performed in search nodes of the global search tree. In [24], a constructive
approach continues until a dead-end is reached, then an LS algorithm modies
the current partial solution passing it back to the construction routine. In [4], a
vehicle routing problem with side constraints is solved by performing LS after
each insertion of a customer during the construction process (Incremental Local
Optimization). In [28], a two-phase algorithm applies LS to generate a partial
solution and uses BT to extend it to a complete one, repeating this process until
a solution is found. In one approach presented in [27], an LS procedure, utilizing
special constraints based on invariants of Localizer [20], is run at each node in
the constructive search tree to select the next variable to be instantiated.
The minimal perturbation neighbourhood operator used to satisfy the temporal constraints in the LS prober of Probe(Loc) is related to a hybridization
class where BT+CS is used in neighbourhood exploration. For instance, in [25],
limited discrepancy search ([13]) is applied to incrementally explore the neighbourhood. In the genetic algorithm approach presented in [2], instead of values,
sub-domains of original variables are used as meta-variables, which are evaluated by solving the corresponding sub-CSPs by BT+CS. In [23], obtaining a
neighbour is a constrained optimization problem that is solved by a branch and
bound algorithm. In [17], a complete solution is constructed by tabu search using
BT+CS to choose neighbours from sets of alternative partial assignments.
5 Conclusion
This paper demonstrated that local probing hybrids can marry the strengths of
constraint satisfaction techniques, including good satisfaction characteristics and
proofs of problem infeasibility, with the superior optimization characteristics of
local search. The paper detailed a local probing algorithm that solved a highly
generic scheduling problem class, and showed it to be more eective than an
LP-based hybrid, and a traditional pure local search algorithm, both of which
were specialized for this class.
Acknowledgements. The authors would like to thank Farid Ajili and Neil
Yorke-Smith for their help and valuable feedback on this work.
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