An Efficient Label-setting Algorithm for the Bi-objective Shortest Path
Problem
Antoine Giret
1
, Yannick Kergosien
1
, Gael Sauvanet
2
and Emmanuel Neron
1
1
Universit´e Franc¸ois Rabelais de Tours, CNRS, LI EA 6300, OC ERL CNRS 6305,
64 avenue Jean Portalis, 37200 Tours, France
2
La Compagnie des Mobilit´es, 30 rue Andr´e Theuriet, 37000 Tours, France
Keywords:
Label-setting Algorithm, Pareto front, Bi-Objective Shortest Path Problem, Cycling, Exact Method.
Abstract:
In this paper, we consider a classical Bi-objective Shortest Path problem (BSP) that takes into account both
distance and insecurity criteria. This work is in collaboration with an enterprise who provide a web platform
called G´eov´elo that aims to propose routes for cycling. We propose a new exact method to solve a BSP,
called Label Setting algorithm with Dynamic update of Pareto Front (LSDPF), which aims to find all non-
dominated solutions of the problem. Different exploration strategies havebeen proposed and tested. Numerical
experiments on real data sets and on instances of the literature were conducted. Comparison with recent
benchmarks algorithms solving BSP - the bounded Label Setting algorithm by (Raith, 2010) and the pulse
algorithm by (Duque et al., 2015) - shows that our method outperform these benchmarks algorithms.
1 INTRODUCTION
For economical and ecological reasons, alternative
transportation such as cycling is expanding. Pub-
lic actors have initiated several programs to adapt
the road network for cycling (e.g., greenway, cycling
lane), because users seek to be able to move safely.
For a cyclist, travelling time and distance are not
the only criteria to consider; they may also consider
safety, difficulty and tourist attractions.
This work is in collaboration with La Compag-
nie des Mobilit
´
es who provide a web platform called
G´eov´elo (http://www.geovelo.fr). G´eov´elo aims to
propose routes for cycling. These routes consider two
criteria which are safety (minimising lack of secu-
rity), and distance. Both criteria are very conflicting,
mainly because most road networks are designed for
cars.
The studied problem is a Bi-Objective Shortest
Path Problem: let G = (V,A) be a directed graph
where V is the set of nodes (|V| = n) and A the set of
arcs (|A| = m). {c
1
ij
,c
2
ij
} denotes the costs associated
with each arc (i, j), where c
1
ij
represents its distance
and c
2
ij
its insecurity. The problem consist in finding
a path P from a source node s to a target node t mini-
mizing both criteria {
∑
(i, j)∈P
c
1
ij
,
∑
(i, j)∈P
c
2
ij
}. The re-
sult of the problem is a set of strictly non-dominated
solutions also called Pareto front defined in (Ulungu
and Teghem, 1994).
The Bi-Objective Shortest Path Problem, noted
BSP, is a specific case of the Multi-Objective Shortest
Path Problem, that is one of the most studied prob-
lems among the Multi-Objective Combinatorial Op-
timisation (MOCO) problems. (Serafini, 1987) has
proved that the BSP is N P-Hard. (Tarapata, 2007)
presents a state-of-the-art MOSP problem.
The first methods solving BSP are label-correcting
and label-setting methods proposed by (Hansen,
1980) and (Martins, 1984). Labelling methods can
be seen as a generalization of existing methods solv-
ing the mono-objective problem (Bertsekas, 1998)
and they can be decomposed into two types: label-
correcting and label-setting. The difference between
these types is how the label queue is managed. Label-
setting methods often use a lexicographic order to
ensure that an explored label corresponds to a non-
dominated path (see (C.T. Tung, 1992)). Label-
correcting methods use a simple order to manage la-
bel queue exploration (e.g., FIFO) to simplify the
data structure of the queue (see (Brumbaugh-Smith
and Shier, 1989) and (Skriver and Andersen, 2000)).
There are two different selection types for the label-
correcting methods: a label is managed separately
in label selection, whereas all labels on a selected
node are extended at the same time in node selec-
tion. (Guerriero and Musmanno, 2001) compares dif-
ferent node and label selection strategies for the label-
correcting method. Node selection is generally used
Giret, A., Kergosien, Y., Sauvanet, G. and Neron, E.
An Efficient Label-setting Algorithm for the Bi-objective Shortest Path Problem.
DOI: 10.5220/0005757101970203
In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems (ICORES 2016), pages 197-203
ISBN: 978-989-758-171-7
Copyright
c
 2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
197

with label-correcting methods, whereas label selec-
tion must be used with label-setting methods.
Another type of method to solve BSP is Ranking
method (Climaco and Martins, 1982) that are based
on the kth-shortest path problem ((Carlyle and Wood,
2005)). These methods find the shortest path opti-
mizing one objective, then the second shortest path,
and so on. In addition to labelling and ranking meth-
ods, the two-phase approach is also commonly used
to solve BSP problems. (Ulungu and Teghem, 1995)
introduced this concept. The two-phase method is ba-
sically used to compute separately supported (i.e., lo-
cated on the boundary of the convex hull) and non-
supported solutions. All solutions in the first phase
can be computed by a single objective method with a
weighted sum objective. In the second phase, a bi-
objective method enumerates all non-supported so-
lutions. During this phase, the search space is re-
stricted by the solutions computed during the first
phase. (Raith and Ehrgott, 2009) has shown that this
approach is efficient and proposed a comparison of
the solution strategies for finding all efficient paths.
(Raith, 2010) focuses on label-correcting and label-
setting algorithm (bLSET) and propose an accelera-
tion technique, improving the efficiency of these al-
gorithms. He showed that it is not always neces-
sary to continue the search to the target node to con-
firm that it is dominated. According to (Demeyer and
al., 2013), bLSET algorithm presented better compu-
tational times among labelling approaches proposed
during this period.
Recently, (Duque et al., 2015) proposed a new ex-
act method, called Pulse algorithm, for the BSP and
large-scale road networks. Pulse algorithm is based
on recursive method using pruning strategies that ac-
celerate the graph exploration. The results shown that
the proposed algorithm outperform the bLSET algo-
rithm on very-largescale instances from the DIMACS
dataset.
This work is an extension of (Sauvanet and Neron,
2010) and aims to propose a new exact method to
solve BSP. The proposed method, called Label Set-
ting algorithm with Dynamic update of Pareto Front
(LSDPF), is based on existing methods previously
cited. However, some improvement techniques have
been added. The paper is organized as follows. Sec-
tion 2 presents the LSDPF algorithm with several
exploration strategies. Some numerical results are
given in section 3 in order to test the different ex-
ploration strategies and a parameter of the algorithm.
Then computational experiments compare our pro-
posed method to the bLSET algorithm developed by
(Raith, 2010) and the Pulse algorithm developed by
(Duque et al., 2015). Finally section 4 concludes the
paper and gives some future work directions.
2 LABEL SETTING ALGORITHM
WITH DYNAMIC UPDATE OF
PARETO FRONT (LSDPF)
The LSDPF algorithm is based on a two-phase
method introduced by (Ulungu and Teghem, 1995).
The first phase aims to compute dominated solu-
tions by solving several Mono-Objective Shortest
Path Problems. The second phase aims to get only
non-dominatedsolutionsand is based on classic label-
setting algorithm which has been introduced in (Mar-
tins, 1984). The specificity of the proposed algorithm
is that the final Pareto front can dynamically evolve at
each iteration of the label-setting algorithm, and not
necessary when the target node is reached.
2.1 First Phase
As explained shortly before, the first phase consists in
running a set of Mono-Objective Shortest Path Prob-
lems (using a simple adaptation of (Dijkstra, 1959)),
and is composed by two steps.
The first step provides an upper bound of a given
criterion by solving a Mono-Objective Shortest Path
Problem from s to t, minimizing the other criterion.
These upper bounds allow to reduce the exploration
of the graph for the next step.
The second step consists in solving reverse one-
to-all Mono-ObjectiveShortest Path Problems from t,
where each objective function is defined by a linear
combination of both criteria:
α
∑
(i, j)∈P
c
1
ij
+ (1− α)
∑
(i, j)∈P
c
2
ij
We note A the set of all α values tested. After each
one-to-all resolution, each traversed node is charac-
terized by a pair of values (d
α
i
,s
α
i
), representing a fea-
sible path from i to t with a distance equal to d
α
i
and
an insecurity equal to s
α
i
. This search is stopped at
a node i when both d
α
i
and s
α
i
are greater than upper
bounds found in the first step. Only non dominated
pairs - in Pareto sense - of values (d
α
i
,s
α
i
) are saved.
Two specific cases are important: α = 1 and α =
0. Both allows to determine the lower and the upper
bound for each criterion at each node:
• For α = 1, is obtained at each node i ∈ V the lower
bound of the distance criterion defined by (LB
1
i
=
d
1
i
), which is associated to an upper bound of the
insecurity criterion defined by (UB
2
i
= s
1
i
),
ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems
198

• For α = 0, is obtained at each node i ∈ V the
lower bound of the insecurity criterion defined by
(LB
2
i
= s
0
i
), which is associated to an upper bound
of the distance criterion defined by (UB
1
i
= d
0
i
).
Finally, the set of pairs (d
α
s
,s
α
s
) is used to initialize the
Pareto front. Additional one-to-one Mono-Objective
Shortest Path from s to t can be done (i.e. using more
values of α), in order to extend the initial Pareto Front
(cf figure 1).
Figure 1: Initialisation of the Pareto front.
2.2 Second Phase
The second phase is based on a label-setting algo-
rithm with additional improvements related to the Bi-
Objective Shortest Path problem. These improve-
ments allow to reduce the number of labels created
and thus finding the final Pareto front faster. The pro-
posed method (LSDPF) is presented in the algorithm
1. Let us define:
• s and t respectively the source and target nodes.
• i and j the indexes of nodes and u, v, w and x the
indexes of labels.
• l
u
a label, defined by a tuple (i,vd, vs) where:
– i is the node associated to the label.
– vd and vs represent respectively the values of
distance and insecurity of a feasible path from
s to i.
• LN
i
the set of labels at node i.
• Q contains unexplored labels.
• l
v
≺
p
l
u
means that the label l
v
is strictly domi-
nated by l
u
in the Pareto sense.
The final Pareto front contained all non-dominatedso-
lutions is obtained at node t: it corresponds to the set
of labels LN
t
.
The main steps of the LSDPF algorithm are:
• Line 1: the FIRSTPHASE function represents the
first phase of the proposed method explained in
Algorithm 1:Label Setting algorithm with Dynamic update
of Pareto Front (LSDPF).
1: FIRSTPHASE()
2: LN
i
←
/
0, ∀i = {1,...,n}
3: LN
s
← {(s,0, 0)}
4: LN
t
← {∪
∀α
(d
α
s
,s
α
s
)}
5: Q ← {(s,0,0)}
6: while Q 6=
/
0 do
7: (i,vd, vs) ←NEXTLABEL(Q)
8: Q ← Q\(i,vd,vs)
9: for all (i, j) ∈ A do
10: ( j, vd
′
,vs
′
) ← ( j,vd + c
1
ij
,vs+ c
2
ij
)
11: l
u
← ( j, vd
′
,vs
′
)
12: if ∄l
v
∈ LN
j
/l
v
≺
p
l
u
then
13: if ∄l
v
∈ LN
t
/l
v
≺
p
( j, vd
′
+ LB
1
j
,vs
′
+
LB
2
j
) then
14: if j 6= t then
15: Q ← Q∪ l
u
16: end if
17: LN
j
← LN
j
∪ l
u
18: UPDATELABELS(Q, LN
j
, l
u
)
19: DPF(LN
t
, ( j,vd
′
,vs
′
))
20: end if
21: end if
22: end for
23: end while
24: return LN
t
Algorithm 2: UPDATELABELS.
1: function UPDATE(Q, LN
j
, l
u
)
2: for all l
v
∈ LN
j
/l
u
≺
p
l
v
do
3: LN
j
← LN
j
\ l
v
4: if l
v
∈ Q then
5: Q ← Q\ l
v
6: end if
7: end for
8: end function
the section 2.1. It allows to determine all pairs
(d
α
i
,s
α
i
). The Pareto front is initialized in line 4.
• Line 7: the NEXTLABEL function defines the or-
der to explore the label, which affects the evolu-
tion of the Pareto front and thus the convergence
of the algorithm. Three strategies explained later
have been tested.
• Line 12: the current label is explored if and only
if it is not dominated by another label existing on
the same node.
• Line 13: the current label l
u
is exploredif and only
if it is promising, i.e. if ( j,vd
′
+ LB
1
j
,vs
′
+ LB
2
j
)
is not dominated by a label existing on the target
node. ( j, vd
′
+ LB
1
j
,vs
′
+ LB
2
j
) represents a not
An Efficient Label-setting Algorithm for the Bi-objective Shortest Path Problem
199

Algorithm 3: DPF.
1: function DPF(LN
t
, ( j,vd
′
,vs
′
))
2: for all α ∈ A do
3: l
v
← (t, vd
′
+ d
α
j
,vs
′
+ s
α
j
)
4: if ∄l
w
∈ LN
t
/l
w
≺
p
l
v
then
5: LN
t
← LN
t
∪ l
v
6: for all l
x
∈ LN
t
/l
v
≺
p
l
x
do
7: LN
t
← LN
t
\ l
x
8: end for
9: end if
10: end for
11: end function
necessary feasible path from s to t, containing two
subpaths. The first subpath is given by the label l
u
(which is a feasible path from s to j), and the sec-
ond subpath corresponds to the lower bounds of
both criteria to reach t from j. In other words, the
label l
u
cannot give better solutions than those in
the current Pareto front. During the search, the
Pareto front will iteratively contain better solu-
tions involving that more and more labels will not
be explored. This explains why the initialization
of the Pareto front is important as well as its up-
dates (line 19).
• Line 18: the UPDATELABELS function is pre-
sented in the algorithm 2. The goal of this func-
tion is to update Q and LN
j
, according to the cur-
rent explored label l
u
. All dominated labels in Q
and LN
j
are deleted since they are not interesting
ones.
• Line 19: the DPF function is presented in the al-
gorithm 3. It allows to dynamically update the
Pareto front. From the current explored label l
u
,
several feasible paths from s to t can be computed
thanks to each solution of Mono-Objective Short-
est Path Problems determined in the first phase (cf
figure 2). This kind of path is decomposed into
two subpaths: a subpath from s to j given by the
label l
u
, and a second one from j to t given by the
solution of the linear combination determined by
α during the Mono-Objective Shortest Path Prob-
lem. This step aims to improve the Pareto front
throughout the search, and thus reducing the num-
ber of explored labels (line 13 of algorithm 1).
2.3 Exploration Strategies
The exploration strategy determines in which order
the labels will be explored (goal of the NEXTLABEL
function in the algorithm). From a strategy to another
one, the number of explored labels can be different
Figure 2: Dynamic update of the Pareto front.
as well as the computation time of the LSDFP algo-
rithm. The selected strategy is thus important, three
strategies are presented and have been tested.
2.3.1 Strategy 1
The first strategy is simple. It consists in selecting the
label that contains the minimum vd
′
value among all
labels presents in Q. If several labels have the same
minimum value vd
′
, the label with the smallest value
of vs
′
is selected among these labels. The graphic evo-
lution of the set of explored nodes/labels can be repre-
sented by a circle around the source node, which be-
come increasingly larger. Only distance criterion as
first criterion has been tested (and not the insecurity
one). Indeed, the range of distance values is bigger
than the range of insecurity values in cycling context.
Considering the distance criterion first, the search will
be guided to more interesting labels.
2.3.2 Strategy 2
The second strategy aims to explore the most promis-
ing label first. From the set of labels presents in Q, the
next label to treat is the label with the smallest value
(vd
′
+ LB
1
j
). In case of equalities, the label with the
smallest value (vs
′
+ LB
2
j
) is selected. As previously,
only distance criterion is tested as first criterion for
the the same reason. The representation graph of ex-
plored labels is close to an ellipsoid around the source
and target nodes.
2.3.3 Strategy 3
The last strategy uses a linear combination of vd
′
+
LB
1
j
and vs
′
+ LB
2
j
. Unlike other strategies, this one
will take into account both criteria to guide the explo-
ration. Thus the next label presents in Q to explore is
the label minimizing the function:
β∗ (vd
′
+ LB
1
j
) + (1− β) ∗ (vs
′
+ LB
2
j
)
ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems
200

3 COMPUTATIONAL
EXPERIMENTS
The LSDPF algorithm has been implemented
in C++ language using the library Boost
http://www.boost.org. Computational experi-
ments have been performed on an Intel Core i7 vPro
processor quad-core with 8GB of RAM and using
Linux system. First, numerical results are given for
the three strategies and several values of A on real
data sets taking into account the study case (Cycling).
Next, the efficiency of the LSDPF algorithm with
the best exploration strategy is compared to efficient
benchmarks algorithms on instances of the literature.
3.1 Comparison of Exploration
Strategies
To compare the different exploration strategies pre-
sented in section 2.3, we used real roads networks that
come from the cooperative OpenStreetMap project
(http://www.openstreetmap.org). Each node of the
network represents an intersection and each arc be-
tween two intersections is composed by two val-
ues: the travel distance and the insecurity of the
road. Three graphs have been extracted: Paris
(France), Berlin (Germany) and San Francisco Bay
Area (USA). The Paris graph used represents the
city of Paris and its 30 adjacent towns, and contains
29,086 nodes and 64,538 arcs. The second graph con-
sidered in this study corresponds to the city of Berlin.
This one is interesting because Germany is a coun-
try in which the OpenStreetMap community is active
and thus safety informations are accurately reported.
It contains 59,673 nodes and 145,840 arcs. The San
Francisco graph is interesting because the road net-
work structure of US cities is rather different and big-
ger than European cities. It contains 174,975 nodes
and 435,959 arcs. For each graph, we randomly gen-
erated 50 instances (i.e. pair of source and target
nodes) for Paris and Berlin graphs, and 25 instances
for the San Francisco one.
The table 1 presents the average execution time
(in milliseconds) of LSDFP algorithm for each strat-
egy and each graph. The parameter β is equal to 0.5.
The first column indicates the graph name, columns
from 2 to 4 shows the average execution time of each
strategy, and last column shows the average number
of non-dominated solutions on the Pareto front.
The same interpretation can be done for all the
graphs: the second strategy seems to be better than
the first one as we might expect but also than third
one, which can be surprising. The second strategy
improves by approximately 20% the execution time
Table 1: Average execution time (ms) of LSDPF for each
strategy.
Graph Str. 1 Str. 2 Str. 3 |S|
Paris 194,8 158,6 403,6 117
Berlin 206,2 150,8 246 127
SF 87040,8 71815,2 715942,4 1280
of the first strategy and between 40% and 90% exe-
cution times of the third strategy. Results of the third
strategy are not as good as expected. The linear com-
bination of both criteria doesn’t seems guide the ex-
ploration towards good solutions quickly.
The table 2 presents the average number of ex-
plored labels during the LSDPF algorithm, for each
strategy and each graph. The first column indicates
the graph name, columns from 2 to 4 shows average
number of explored labels. An explored label means
that it has been added to the Q set (line 15 of LSDPF
algorithm).
Table 2: Average number of explored labels for each strat-
egy.
Graph Str. 1 Str. 2 Str. 3
Paris 110754 72118 66464
Berlin 124773 73400 68796
SF 14679416 11071874 10450375
The strategy exploring the biggest number of la-
bels is the strategy one. The strategy 2 and 3 have
approximatively the same number of explored labels.
The result of the comparison between the strategies
1 and 2 is that when the number of explored label
decreases, the execution time decrease. However the
same interpretation cannot be done for the difference
found between the strategies 1 and 3 and between the
strategies 2 and 3. Even if the strategy 3 explores less
labels, its average of execution times are bigger than
others, meaning the linear combination of both cri-
teria cannot be used to quickly guide the exploration
towards good solutions.
3.2 Comparison of Several A Values
Only two strategies are used for the numerical exper-
iments of several A values. Three cases have been
tested:
• A = {0;1} (| A | = 2)
• A = {0;0.5;1} (| A | = 3)
• A = {0;0.25;0.5;0.75;1} (| A | = 5)
Table 5 presents the average execution time (in ms)
of the first phase for each case. Table 3 shows the
average execution time (in ms) of the second phase
for each case and each strategy (strategy 1 noted S1
An Efficient Label-setting Algorithm for the Bi-objective Shortest Path Problem
201

strategy 2 noted S2). The total average execution time
(in ms) of LSDPF algorithm is presented in table 4.
Finally, the number of explored labels (in thousands)
of LSDPF algorithm is given in table 6.
Table 3: Average execution time of second phase (ms) for
several values of A several strategies.
Graph | A | = 2 | A | = 3 | A | = 5
S1 S2 S1 S2 S1 S2
Paris 165 129 154 163 142 156
Berlin 146 91 139 113 110 114
SF 86844 71666 64430 75884 67118 82026
Table 4: Average execution time of LSDPF (ms) for several
values of A and several strategies.
Graph | A | = 2 | A | = 3 | A | = 5
S1 S2 S1 S2 S1 S2
Paris 195 159 193 201 207 221
Berlin 206 151 223 197 257 261
SF 87041 71862 64641 76094 67488 82396
From previous tables 5 to 6, it seems not interest-
ing to use more than 2 α values whatever the strategy
1 or 2. The fact that the execution time increase when
the number of α values increases in a given strategy
can be explained by the additional consumption of
CPU time needed by DPF function. Moreover, the
degradation of execution time of the strategy2 is more
significant in comparison with the strategy 1.
3.3 Efficiency of LSDPF
In order to compare our results with benchmarks al-
gorithms solving the BSP - the bounded Label Set-
ting algorithm (blSET) (Raith, 2010) and the pulse
algorithm (Duque et al., 2015) - computational ex-
periments have been performed on instances from
the 9
th
DIMACS challenge. Three graphs have been
used. The first graph corresponds to New York City.
This graph contains 264,346 nodes and 733,846 arcs.
The second corresponds to another representation of
the San Francisco Bay Area and contains 321,270
nodes and 800,172 arcs. The third corresponds to
the state of Florida and contains 1,070,376 nodes and
2,712,798 arcs. Criteria considered for these graphs
correspond to the physical distance and transit time
between nodes. To fairly compare algorithms, the
same pairs of source and target nodes considered in
(Duque et al., 2015) have been used for each instance.
As (Duque et al., 2015), the 30 instances of each
graph has been clustered into the same three equal
sized groups, denoted S (small), M (medium) and L
(large), based on the number of non-dominated solu-
tions found on the Pareto front.
The table 7 compares the average execution time
Table 5: Average execution time of first phase (ms) for sev-
eral values of A.
Graph | A | = 2 | A | = 3 | A | = 5
Paris 30 39 65
Berlin 60 84 147
SF 196 211 370
Table 6: Average number of explored labels (in thousands)
for several values of A several strategies.
Graph | A | = 2 | A | = 3 | A | = 5
S1 S2 S1 S2 S1 S2
Paris 111 72 89 71 78 70
Berlin 125 73 100 72 79 71
SF 14679 11072 12355 11023 11289 10965
of the bLSET algorithm, the Pulse algorithm, and LS-
DPF algorithm. All times are in milliseconds. The
first column indicates the cluster name, composed
of the instance name and the letter from the group
to which it belongs. Column 2 shows the average
number of non-dominated solutions on the the Pareto
front. Column 3, 4 and 5 presents the average exe-
cution time of the bLSET algorithm, Pulse algorithm
and LSDPF algorithm. Values followed by ∗ means
that at least one of the instances could not be solved in
less than one hour and so the search has been stopped.
Table 7 shows that execution times of bLSET are
average bigger than other algorithms. Regarding the
Pulse and LSDPF algorithms, the Pulse algorithm is
particularly efficient on instances with a small num-
ber of solutions unlike other methods. For these in-
stances, LSDPF takes up to 7 times more to find the
entire Pareto front but execution times remain reason-
able since they are on average lower than 3 seconds.
Nevertheless, we can see that LSDPF has the same av-
erage order execution times to solve instances what-
ever the number of solutions which is not the case
of the two other algorithms where it appear to have
an exponential time progression. Moreover they may
not find the entire Pareto front in the case of larger in-
stances (see table 8). To conclude, LSDPF is clearly
the faster method to find the whole Pareto front for
medium and larger instances.
Finally, the criterion used for the shortest paths
search on these instances are distance and time, that
are not completely independents criteria. The Pareto
front thus contains fewer non-dominated solutions
that it is possible to have using more conflicting crite-
ria, such as distance and insecurity for example.
4 CONCLUSION
In this paper, we consider a classical Bi-objective
ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems
202

Table 7: Averaging times (ms) for the bLSET, the Pulse al-
gorithm and the LSDPF on 9
th
DIMACS challenge graphs.
Cluster |S| bLSET Pulse LSDPF
NY-S 34.10 62390 320 2169
NY-M 147.40 301160 52320 3692
NY-L 422.70 881260 1367660* 945
BAY-S 8.80 6780 160 509
BAY-M 49.90 55240 5700 508
BAY-L 171.80 317430 105550 360
FLA-S 14.70 330120 350 2343
FLA-M 94.10 566150* 347910 1189
FLA-L 552.30 2627430* 888590* 2776
Table 8: Number of instances unsolved in less than an hour
for bLSET, Pulse algorithm and the LSDPF.
Cluster bLSET Pulse LSDPF
NY-L 0 3 0
FLA-M 1 0 0
FLA-L 6 1 0
Shortest Path Problem (BSP) which considers both
distance and insecurity inspired from a cycling con-
text. We proposed an exact algorithm based on label-
setting methods and called LSDPF, in order to find all
non-dominated solutions of the problem. One of the
specificities of LSDPF algorithm is to dynamically
update the whole Pareto front duringthe search allow-
ing to reduce the number of explored labels. Different
strategies have been proposed and tested. Numerical
experiments on real data sets and on instances of the
literature were conducted. Comparison with recent
benchmarks algorithms to solve BSP (the bounded
Label Setting algorithm (blSET) (Raith, 2010) and the
pulse algorithm (Duque et al., 2015)) shows that our
method outperform the benchmarks algorithms.
The first perspective of this work is to general-
ize our methods to several conflictual criteria on same
type of instances. Another promising research direc-
tion is to use some graph contractions in the case of
multi-criteria.
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