Prize Collecting Travelling Salesman Problem
Fast Heuristic Separations
Kamyar Khodamoradi and Ramesh Krishnamurti
School of Computing Science, Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada
Keywords:
Prize Collecting TSP, Linear Programming, Generalized Subtour Elimination Constraints, Primitive Comb
Inequalities, Integrality Gap, Shrinking Heuristic.
Abstract:
The Prize Collecting Travelling Salesman Problem (PCTSP) is an important generalization of the famous
Travelling Salesman Problem. It also arises as a sub problem in many variants of the Vehicle Routing Problem.
In this paper, we provide efficient methods to solve the linear programming relaxation of the PCTSP. We
provide efficient heuristics to obtain the Generalized Subtour Elimination Constraints (GSECs) for the PCTSP,
and compare its performance with an optimal separation procedure. Furthermore, we show that a heuristic to
separate the primitive comb inequalities for the TSP can be applied to separate the primitive comb inequalities
introduced for the PCTSP. We evaluate the effectiveness of these inequalities in reducing the integrality gap for
the PCTSP.
1 INTRODUCTION
The Prize Collecting Travelling Salesman Problem
(PCTSP) is a variant of the Travelling Salesman Prob-
lem (TSP), and is important in its own right (Wolsey,
1998). It also arises as a sub problem that needs to be
solved in order to extract a column in column genera-
tion formulations of various vehicle routing problems.
PCTSP is an NP-hard problem, and has received a lot
of attention in the literature. The PCTSP comprises a
complete graph with a depot node. In addition to costs
(distances) between nodes, each node has a ‘prize’
associated with it. The objective is to derive a tour
which includes the depot, and maximizes the sum of
the prizes associated with the nodes in the tour, less
the cost of the tour. It can be considered as a general-
ization of the TSP, since the TSP is the PCTSP with a
prize of 0 associated with each node. Several variants
of the PCTSP have been studied in the literature. The
version of the PCTSP we study in this paper was first
introduced by Balas (Balas, 1989) in a more general
setting to model the scheduling of the daily activities
of a steel rolling mill.
In the version introduced by Balas, a tour profits
(to the extent of a prize associated with the node) from
each node it visits, while it is penalized (to the extent
of a penalty associated with the node) for every node
it does not visit. The profit/penalty for each node cor-
responds to the prize associated with the node. The
objective is to obtain a tour such that the total prize
collected exceeds a prescribed amount, while minimiz-
ing the sum of the travel cost and penalties. The travel
cost for a tour is the sum of the distances between
consecutive nodes in the tour. Balas (Balas, 1989)
derives the cuts for the PCTSP corresponding to the
subtour elimination constraints for the TSP. The cuts
we use in this paper, called the Generalized Subtour
Elimination Constraints (GSECs) was first proposed
by Goemans (Goemans, 1994) in the context of the
Steiner tree problem. See also Wolsey (Wolsey, 1998)
for a clear treatment of our version of the PCTSP. In
a continuation of his work on PCTSP, Balas (Balas,
1995) derives, among other cuts, the cuts correspond-
ing to the primitive comb inequalities for the TSP. The
separation heuristic we use for these cuts is a heuristic
given by Padberg and Hong (Padberg and Hong, 1980)
for detecting blossoms for the TSP. We show that these
heuristics can also be used as separation procedures for
the primitive comb inequalities for the PCTSP. We use
these heuristics to obtain LP solutions with improved
integrality gap.
The first known solution procedure to solve the
PCTSP exactly was a branch-and-bound procedure
(Fischetti and Toth, 1988). Fischetti et al. (Fischetti
et al., 1997) also study a branch-and-cut algorithm for
the symmetric case of Generalized Traveling Salesman
problem, in which the nodes are split into clusters, and
any feasible solution to the problem should cover at
380
Khodamoradi, K. and Krishnamurti, R.
Prize Collecting Travelling Salesman Problem - Fast Heuristic Separations.
DOI: 10.5220/0005758103800387
In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems (ICORES 2016), pages 380-387
ISBN: 978-989-758-171-7
Copyright
c
 2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

least one node from each cluster. The symmetric prop-
erty refers to the fact that the cost of going from a node
u to another node v is the same is the cost of v to u .
In more recent work, Chaves and Lorena (Chaves and
Lorena, 2008) propose a hybrid metaheuristic for the
problem and compare its performance with CPLEX.
Bérubé, Gendreau, Potvin (Bérubé et al., 2009) pro-
pose a branch-and-cut algorithm to solve a variant of
the PCTSP. In this variant, the objective is to obtain a
tour with minimum travel cost, subject to the constraint
that the total prize collected exceeds a predetermined
amount. They incorporate many valid inequalities for
their variant of the PCTSP, and evaluate the perfor-
mance of a branch-and-cut algorithm by introducing
these inequalities.
Our version of the PCTSP arises directly as a sub
problem in a variant of the vehicle routing problem,
called the Skill Vehicle Routing Problem (Cappanera
et al., 2011). In this paper, we focus on the LP solution
procedure for this version of the PCTSP. Furthermore,
we use fast separation heuristics for both the GSECs
and the primitive comb inequalities. For the GSECs,
we adapt a fast heuristic, and compare its performance
with an exact procedure, both in the quality of its solu-
tion, as well as in its running time. For the primitive
comb inequalities, we show that a well-known separa-
tion heuristic for the TSP (Padberg and Hong, 1980)
also works as a separation procedure for the PCTSP.
We also compare the GSEC cuts with the primitive
comb inequalities in their quality of solutions.
2 NOTATION AND PROBLEM
FORMULATION
We let G = (S,E) denote the complete graph repre-
senting the instance of the problem, with node r ∈ S
denoting the depot which every tour must visit. Cost
c
e
associated with each edge e ∈ E is the Euclidean
distance between the two endpoints of edge e . In the
PCTSP, the salesman may choose to visit city j ∈ S . If
he does so, then he obtains a prize β
j
but incurs a travel
cost c
e
if he traverses edge e = (i, j) . The salesman
must start and end his tour at node r , and maximize
the total prize he collects, less the cost of the tour.
We provide below the ILP formulation for the
PCTSP (Wolsey, 1998). In the formulation below, we
let decision variables y
j
be 1 if the salesman visits city
j (and 0 otherwise), and x
e
be 1 if he traverses edge
e (and 0 otherwise). We also let set S
0
= S ∪ {r} and
E
0
= E\{δ(r)} , where δ(r) is the set of edges incident
on the depot node r.
2.1 Problem Formulation
max
∑
j∈S
β
j
y
j
−
∑
e∈E
c
e
x
e
(2.1)
subject to
∑
e∈δ(i)
x
e
= 2y
i
∀i ∈ S
(2.2)
∑
e∈E(V)
x
e
6
∑
i∈V \{k}
y
i
∀k ∈ V,V ⊆ S
0
(2.3)
y
r
= 1 (2.4)
x
e
∈ {0, 1},y
j
∈ {0, 1} ∀e ∈ E, ∀ j ∈ S
(2.5)
The integer variable y
l
is set to 1 ( 0 ) if node
l ∈ S is included (not included) in the tour. Simi-
larly, the integer variable x
e
is set to 1 ( 0 ) if edge
e ∈ E is included (not included) in the tour. Note that
∑
j∈S
β
j
y
j
=
∑
j∈T
β
j
and
∑
e∈E
c
e
x
e
=
∑
e∈T
c
e
= C(T ) .
Constraint 2.2 ensures that if node l is included in the
tour, then two edges of the tour must be incident on it.
Constraint 2.3 is the generalized sub tour elimination
constraint (GSEC). Constraints of this form are used
to prevent any sub tour that does not include root node
r . These are generalizations of the sub tour elimina-
tion constraints for the standard travelling salesman
problem.
Clearly, there are an exponential number of con-
straints included in the GSECs and we cannot include
them all to solve the sub problem. To get around
this problem, we include these constraints as they are
needed (whenever there is a sub tour that does not
include node r ). It is easy to detect such sub tours
when the decision variables take 0/1 values. However,
because we solve the LP relaxation of the sub prob-
lem, fractional values may be assigned to the decision
variables y
j
and x
e
. We outline below the method that
can be used to detect for such sub tours (which do not
include node r ) when fractional values are assigned to
the decision variables.
Let the LP solution to the sub problem be given
by (x∗,y∗) . Then the generalized sub tour elimination
constraint is violated for a subset W ⊆ S
0
of nodes if
the inequality
∑
e∈E
0
(W )
x
∗
e
>
∑
l∈W \{k}
y
∗
l
is satisfied.
Such a subset W can be extracted by solving the fol-
lowing integer program. We call the problem below
the separation problem for GSECs. The formulations
for the separation problem for GSECs for the PCTSP
are given in Wolsey 1998 (Wolsey, 1998).
Prize Collecting Travelling Salesman Problem - Fast Heuristic Separations
381

3 THE SEPARATION PROBLEM
FOR GENERALIZED SUBTOUR
ELIMINATION CONSTRAINTS
A formulation for the separation problem for GSECs
is given below (Wolsey, 1998).
ζ
k
= max
∑
e∈E
0
x
∗
e
w
e
−
∑
j∈W\{k}
y
∗
j
z
j
(3.1)
subject to
w
e
6 z
i
,w
e
6 z
j
∀e = (i, j) ∈ E
0
(3.2)
w
e
> z
i
+ z
j
− 1 ∀e = (i, j) ∈ E
0
(3.3)
w
e
∈ {0, 1}, z
j
∈ {0, 1} ∀e ∈ E
0
,∀ j ∈ W
(3.4)
z
k
= 1 (3.5)
Constraint 3.3 above can be dropped because it
is implied by Constraint 3.2 if x
∗
e
> 0 for e ∈ E
0
(Wolsey, 1998). It turns out that the constraint matrix
for the above separation problem (after Constraint 3.3
is dropped) is totally unimodular. Thus, in polyno-
mial time, we solve the LP relaxation of the separation
problem to obtain constraint(s) to introduce into the
sub problem.
The optimal solution to the LP relaxation of the
modified separation problem is integral. This optimal
solution corresponds to the subset W ⊆ S
0
and node k ∈
W such that the inequality
∑
e∈E
0
(W )
x
∗
e
6
∑
l∈W \{k}
y
∗
l
,
corresponding to Constraint 2.3 of the sub problem
is violated. This inequality is introduced into the sub
problem and the LP relaxation of the PCTSP is again
solved. This is repeated until there is no subset W ⊆ S
0
and node k ∈ W such that the inequality
∑
e∈E
0
(W )
x
∗
e
6
∑
l∈W \{k}
y
∗
l
is violated.
In practice, solving an LP for each k ∈ W can be
very time consuming. Therefore, a shrinking heuristic
is used to speed up the separation algorithm. The
shrinking heuristic is described below.
3.1 A Shrinking Heuristic Algorithm
for the Separation of GSECs
The shrinking heuristic we describe below helps find
many violated GSECs quickly. We generalize the
heuristic developed by Crowder and Padberg (Crow-
der and Padberg, 1980) and Land (Land, 1979) for the
TSP. Recall that S
0
= S \{r} and E
0
= E\{δ(r)} . In
this approach, the GSEC inequalities are transformed
into the equivalent cut-set inequalities. In the follow-
ing, E(V
1
, V
2
) is used to denote the set of edges which
have one endpoint in V
1
and the other in V
2
.
∑
e∈E(V,S
0
\V )
x
e
> 2y
k
∀k ∈ V,V ⊆ S
0
(3.6)
Note that the cut-set inequalities can be derived from
Constraints 2.2 and 2.3. These inequalities allow us
to transform the problem to a network flow problem
in which we seek a minimum cut in a rather special
sense. We are looking for violated cut-set constraints.
Equivalently, we wish to find a subset V ⊆ S
0
of ver-
tices for which,
∑
e=(i, j), i∈V, j∈S
0
\V
x
e
< 2y
k
for some
k ∈ V . Alternatively, we look for a cut (V, S
0
\V ) that
minimizes
∑
e∈E(V,S
0
\V )
x
e
− 2max
k∈V
{y
k
}. (3.7)
There exists a k for which Inequality 3.6 is violated
if and only if the value of a cut that minimizes 3.7 is
negative. To find such a cut, we repeatedly reduce the
size of the graph by shrinking subsets of vertices into a
single vertex, provided that certain conditions are met.
Consider two subsets, V
1
,V
2
⊆ S
0
. If V
1
and V
2
are
to be merged, the value contributed to the cut value
by V
1
should not be worsened after the merge. This
means that we should have:
∑
e∈E(V
1
,S
0
\V
1
)
x
e
− 2max
k∈V
1
{y
k
} >
∑
e∈E(V
1
∪V
2
,S
0
\(V
1
∪V
2
))
x
e
− 2 max
k∈V
1
∪V
2
{y
k
}, (3.8)
By symmtery, a similar inequality can be written for
V
2
. Without going into too much detail, we note that
we can merge (or shrink) V
1
and V
2
if
∑
e∈E(V
1
,V
2
)
x
e
>
∑
i∈V
2
y
i
+ max
k∈V
1
{y
k
} − max
k∈V
1
∪V
2
{y
k
} −
∑
e∈E(V
2
)
x
e
.
(3.9)
and
∑
e∈E(V
1
,V
2
)
x
e
>
∑
i∈V
1
y
i
+ max
k∈V
2
{y
k
} − max
k∈V
1
∪V
2
{y
k
} −
∑
e∈E(V
1
)
x
e
.
(3.10)
The inequalities (3.9) and (3.10) is the basis of our
shrinking heuristic. We start with the original graph
G = (S
0
,E
0
), and consider pairs of vertices for shrink-
ing. As we proceed with the shrinking algorithm, a
vertex v
i
may become a super node and represent a set
of vertices in the original graph G . Thus, we associate
a value m
i
to every vertex of the graph v
i
which denotes
the maximum value of y
k
’s in the corresponding com-
ponent of v
i
in the original graph. Whenever we merge
two vertices v
i
and v
j
connected via edge e into a new
vertex v
i
0
, we set the new values y
i
0
= y
i
+ y
j
− x
e
, and
ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems
382

m
i
0
= max{m
i
,m
j
} and also adjust the edge weights
for the newly created node. The following pseudo-
code best describes our shrinking algorithm:
Algorithm 1: The Shrinking Heuristic.
Data: Graph G = (S
0
V
0
) , Edge weights w : E
0
→ R
>0
,
and vertex values y : S
0
→ R
>0
Result: The cut V that maximizes
∑
e∈E(V,S
0
\V )
x
e
−
max
k∈V)
{y
k
}
Initialize each node to be a separate component
for every edge e = (i, j) in E
0
do
if (x
e
> y
i
− max{0, m
i
− m
j
}) and
(x
e
> y
j
− max{0, m
j
− m
i
}) then
Merge nodes i and j into a new node `
for every node u in S
0
do
if u is adjacent to i or j then
Let e
0
denote the edge (`, u)
x
e
0
← x
(u,i)
+ x
(u, j)
end
end
y
`
← y
i
+ y
j
− x
e
m
`
← max{m
i
, m
j
}
end
end
4 PRIMITIVE COMB
INEQUALITIES FOR THE TSP
To improve the quality of fractional solution to the
PCTSP, valid inequalities can be introduced. Primitive
comb inequalities are a class of valid inequalities. In
this section, we study these valid inequalities for our
version of the PCTSP. Later in Section 6, we investi-
gate how effective the primitive comb inequalities are
in improving the quality of fractional solutions to the
PCTSP.
4.1 Formulation
Comb inequalities are valid inequalities that have been
introduced successfully to obtain LP solutions with
reduced integrality gaps for the TSP. In addition to this,
exact separation algorithms for the comb inequalities
have been proposed for the TSP. Because the LPs are
solved many times in a branch and bound algorithm to
solve the TSP optimally, efficient heuristics have also
been proposed for the separation problem which find
violated comb inequalities.
In this section, we look at inequalities for the sim-
plest form of such structures known as primitive combs.
A general comb consists of a set of nodes H known
as the handle, and a set of t teeth, each tooth being a
set of nodes that spans out of the handle. A primitive
comb restricts each tooth to be a single edge. Primitive
comb inequalities have been shown to improve the
quality of the fractional solution to the PCTSP.
Let us first look at the primitive comb inequalities
for the Travelling Salesman polytope. In what follows,
x(A, B) =
∑
e=(i, j):i∈A, j∈B
x
e
, where x
e
is 1 if the edge
e is incorporated in the tour, and x
e
is 0 otherwise.
x(H, H) +
t
∑
j=1
x(T
j
, T
j
) 6
|
H
|
+
t − 1
2
, (4.1)
where H , the handle, and T
j
for j = 1, 2, . . . , t , the
teeth, are sets of nodes satisfying the following condi-
tions (Gutin and Punnen, 2002):


H ∩ T
j


=


T
j
\H


= 1 j = 1, . .., t, (4.2)
with t > 3 odd
T
i
∩ T
j
=
/
0 i, j = 1, ..., t (4.3)
Padberg and Hong (Padberg and Hong, 1980),
among others, provide a heuristic separation algorithm
for primitive comb inequalities for the TSP. We refer
to this as the odd-component heuristic.
4.2 Odd-component Heuristic for TSP
In this heuristic, given an optimal LP solution x
∗
to a
TSP instance over a complete graph (V, E) , the graph
G
∗
1/2
is constructed. This graph has the vertex set V
and edge set {e ∈ E : 0 < x
∗
e
< 1} . Let the resulting
graph comprise of q components, whose vertex sets
are V
1
, V
2
, ... , V
q
. For each component i,1 6 i 6 q , the
heuristic determines the subset of edges T
i
in the set
δ(V
i
) (the set of edges which cross the set V
i
) whose
LP values are 1 . T
i
is thus given by T
i
= {e ∈ E : x
∗
e
=
1 ∧ e ∈ δ(V
i
)} . If the cardinality of this set is odd,
then the following simple procedure determines a set
that either violates the subtour inequality for TSP, or
a primitive comb inequality. If two edges in set T
i
share a vertex v ∈ V \V
i
, then vertex v is included in
the set V
i
and the procedure is repeated until the edges
in T
i
are pairwise disjoint. Such a set V
i
violates the
subtour inequality if |T
i
| = 1 , and the primitive comb
inequality if |T
i
| > 3 . Note that if two teeth T
i
and T
j
intersect outside if H , adding the common vertex to H
would get rid of exactly two teeth, therefore the parity
of the teeth of H is preserved. Therefore if a handle has
odd number of teeth (which indicated either a violated
GSEC or a violated primitive comb inequality) before
the removal of a pair of teeth, it will still have odd
many teeth afterwards.
Balas (Balas, 1995) shows an equivalent version
of the comb inequalities are facet defining for the
Prize Collecting Travelling Salesman Problem - Fast Heuristic Separations
383

Prize Collecting Travelling Salesman polytope (Balas,
1995). The following section provides the details of
these comb inequalities for our version of the PCTSP.
5 PRIMITIVE COMB
INEQUALITIES FOR PCTSP
5.1 Formulation
Balas shows in (Balas, 1995) that the following in-
equality defines a facet of the Prize Collecting TS
polytope. Therefore inequality (4.1) translates to in-
equality (5.1) in the case of the Prize Collecting TSP.
x(H, H) +
t
∑
j=1
x(T
j
, T
j
) + z(H) 6
|
H
|
+
t − 1
2
,
(5.1)
Here, z(H) =
∑
v∈H
z(v) , and z(v) is 1 if the vertex v is
left out of the optimal tour (we need to a pay a penalty
for it), and z(v) is 0 otherwise. When compared to the
formulation of the sub problem (Objective Function 2.1
and Constraints 2.2–2.5), one can write y(v) = 1 −z(v)
for all vertices v.
We show below that a connected component heuris-
tic which has been used by Hong (Hong, 1972) and
Land (Land, 1979) can be applied to separate the prim-
itive comb inequalities introduced by Balas for the
PCTSP.
5.2 Odd-component Heuristic for
PCTSP
The odd-component heuristic for Prize Collecting TSP
(Padberg and Hong, 1980) works with an equivalent
formulation of the primitive comb inequalities for the
Travelling Salesman polytope:
x(δ(H)) +
t
∑
j=1
x(δ(T
j
)) > 3t + 1. (5.2)
To be able to use the same heuristic, we first trans-
late Balas’s inequality into a similar form:
x(δ(H)) +
t
∑
j=1
x(δ(T
j
)) > 3t + 1 − 2
t
∑
j=1
z(T
j
).
Lemma 1. The following inequality defines a facet of
the Prize Collecting TS polytope.
x(δ(H)) +
t
∑
j=1
x(δ(T
j
)) > 3t + 1 − 2
t
∑
j=1
z(T
j
).
Proof. By (Balas, 1995), we know that inequality (5.1)
is facet defining. Also, from Constraint 2.2, the frac-
tional degree of each node v is 2 · y(v) = 2 − 2 · z(v) .
Therefore, we can write
2 x(H, H) + x(δ(H)) =
∑
v∈H
deg(v) = 2
|
H
|
− 2 z(H).
Thus,
x(H, H) =
|
H
|
− z(H)−
1
2
x(δ(H)).
Similarly, we have that
2 x(T
j
, T
j
) + x(δ(T
j
)) =
∑
v∈T
j
deg(v) = 4 − 2 z(T
j
),
for j = 1, . . . , t, which yields
x(T
j
, T
j
) = 2 − z(T
j
) −
1
2
x(δ(T
j
)).
Therefore,
x(H, H) +
t
∑
j=1
x(T
j
, T
j
) + z(H) =
|
H
|
−
1
2
x(δ(H)) + 2t −
t
∑
j=1
z(T
j
) −
1
2
t
∑
j=1
x(δ(T
j
))
(5.3)
From equation (5.3) and inequality (5.1),
1
2
x(δ(H)) +
1
2
t
∑
j=1
x(δ(T
j
)) > 2t −
t
∑
j=1
z(T
j
) −
t − 1
2
.
Thus,
x(δ(H)) +
t
∑
j=1
x(δ(T
j
)) > 3t + 1 − 2
t
∑
j=1
z(T
j
).
We show below why the odd-component heuris-
tic used by Padberg and Hong to separate primitive
comb inequalities for the TSP can be used to separate
primitive comb inequalities for the PCTSP without any
modifications.
Assume that after running the odd-component
heuristic on G
∗
1/2
, a component V
i
has an odd num-
ber of teeth. We remove all non-disjoint teeth and add
their common vertex to V
i
. Let the handle H be V
i
. We
assume H has t > 3 teeth as we are more interested
in finding violated primitive comb inequalities rather
than violated GSECs. By the structure of G
∗
1/2
, all
the outgoing edges of H have weight exactly equal
to 1. Thus, x(δ(H)) = t . Consider a tooth of H , say
T
k
= (u
k
, v
k
) (See Figure 1). Because of the degree
constraints, and the fact that the edge (u
k
, v
k
) is tak-
ing a capacity q from each of the two endpoints, we
ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems
384

can write x(δ(T
k
)) = 2(1 − y
u
k
)− 1 + 2(1 −y
v
k
−1) =
2t − 2(y
u
k
+ y
v
k
. Summing over all t edges and adding
x(δ(H)) we get:
x(δ(H)) +
t
∑
j=1
x(δ(T
j
)) = 3t − 2
t
∑
j=1
y(T
j
)
This is exactly 1 short of satisfying the corresponding
primitive comb inequality, therefore any component
of G
∗
1/2
with odd number of teeth would correspond to
a violated inequality that we can introduce as a cut.
u
1
u
2
u
t
v
1
v
2
v
t
Figure 1: A component of G
∗
1/2
.
In the next section, we empirically analyze the per-
formance of this heuristic alongside with the shrinking
heuristic for GSECs.
6 COMPUTATIONAL RESULTS
In this section, we report the efficacy of the two
adapted heuristics and analyze the behaviour of the
corresponding separations. The heuristics have been
implemented in C++ and the LP was solved using
CPLEX 12.5. Then the code was run on an Intel 2.8
GHz processor with a maximum time limit of 14,400
seconds (4 hours).
In this section, we first describe the instance that
we have used. Then we discuss the experiments we
designed to test the behaviour and performance of the
two heuristics, namely the shrinking heuristic for sepa-
ration of the GSECs and the odd-component heuristic
for separation of the primitive comb inequalities. We
next explain how we compared these results and an-
alyze their relative gap with respect to the optimal
integral solution whenever such a solution is available.
We summarize the results in Tables 1 and 2.
6.1 Instances
A total of 42 instances out of the TSPLIB library
(Reinelt, 1991) have been transformed for our experi-
ments. Out of the 43 instances, 10 were VRP instances
and 32 symmetric TSP instances. In the VRP instances,
each node is associated with a demand value. We use
this demand as the prize of covering that node. For
the TSP instances, which do not come with demands,
we merely generate node prizes independently and
uniformly at random in the range of 1 to 200.
6.2 Designing the Experiments
We analyze the adapted heuristics based on three differ-
ence aspects: i. the solution quality, ii. the number of
cuts they detect, iii. the running time. The major factor
in the success of a heuristic is indeed if it can produce
near optimal solutions in a timely manner. Neverthe-
less, the number of cuts a heuristic can generate can
give us insight about how well that heuristic is adapted
for a certain type of polytope. If near optimality can
be achieved with fewer cuts, we have some empiri-
cal indications that the heuristic is well-suited for the
polytope of the given problem. For this reason, we
also report the number of cuts for both the heuristics,
as well as for an optimal procedure that uses LP for
separation of GSECs. In what follows, we explain the
method used for making the comparisons.
Shrinking Heuristic. To evaluate the efficacy of the
shrinking heuristic, we compare them against a pro-
cedure that uses linear programming for separation of
the GSECs, that is, a formulation that solves a cut-set
equivalent of (3.1)–(3.5) for each node to find the vio-
lated GSECs. Three parameters mentioned above are
reported for the heuristic GSEC separation, as well as
the LP separation.
Odd-component Heuristic. To evaluate this heuristic,
we first solve instances with the GSEC heuristic only,
and then use odd-component heuristic to find violated
primitive comb constraints. This way, we can measure
how the solution quality may improve and how many
new cuts are introduced, and then weigh it against the
extra time we need to spend for running the second
heuristic. The improvements are also reported in the
right-most column of Table 2. the third column of
Table 2 shows the number of extra cuts detected by
using the second heuristic on top of the first. We also
solve the ILP version of the problem using the built-in
branch-and-bound method of CPLEX using regular
Subtour Elimination Constraints (SECs) and compare
them against the optimal LP solutions obtained by
using both heuristics.
Note that to separate SEC inequalities, all we need
to do is to find an integral subtour in a solution, which
can be done in polynomial time using a connected
component algorithm. This method can be slow for
larger instances, therefore in many cases we have not
been able to solve the ILP optimally in the time limit
of 4 hours. For such instances, the gap percentages
has been left blank. Also, in Table 1, “t.l.” stands for
too long, and is used for running times that are greater
than 14,400 seconds.
Prize Collecting Travelling Salesman Problem - Fast Heuristic Separations
385

Table 1: Computational results for the heuristic GSEC sepa-
ration.
LP GSEC Sep. Heuristic GSEC Sep.
Instance Obj. Cuts Time Obj. Cuts Time
eil13 3228.61 0 <1 3228.61 0 <1
ulysses22 -1964.93 16 1 -1964.93 13 <1
bayg29 -44.35 33 4 -44.35 14 <1
eil33 -2924.76 44 7 -2924.76 18 <1
att48 1358.75 0 7 1358.75 0 <1
hk48 551.65 1 7 551.65 1 <1
eil51 -4454.6 2 32 -4454.6 2 <1
st70 -6433.23 94 423 -6433.23 26 1
eilB76 -7442.17 28 964 -7442.31 11 3
eilC76 -6898.96 27 354 -6900.36 7 1
eilD76 -6774.68 33 434 -6774.74 15 2
pr76 5568.42 0 50 5568.42 0 <1
gr96 -9531.81 83 2740 -9531.81 26 2
rat99 9260.24 80 4802 -9260.87 15 14
rd100 -10119.1 189 2501 -10119.1 35 2
eil101 -8988.51 73 1108 -8988.51 17 5
eilA101 -9160.86 60 3153 -9160.86 17 1
eilB101 -9710.61 61 1107 -9710.61 18 2
lin105 N/A t.l. -87.93 354 4371
pr107 339.84 0 238 339.84 0 <1
gr120 -10451.1 85 3367 -10451.1 31 80
gr137 -13070.5 112 4306 -13070.5 26 19
pr144 532.69 11 4677 532.69 6 <1
ch150 N/A t.l. -8891.09 75 68
pr152 984.54 13 10780 984.54 13 2
rat195 N/A t.l. -18301.7 39 972
d198 N/A t.l. -19183.9 31 3
korB200 N/A t.l. -138.68 111 7
gr202 N/A t.l. -19859.1 48 70
ts225 1296.11 0 8466 1296.11 0 14
gr229 N/A t.l. -20520.4 67 8
gil262 N/A t.l. -24784.5 96 27
a280 N/A t.l. -24933.6 60 67
lin318 N/A t.l. -1883.21 365 1906
fl417 N/A t.l. -41406.8 48 6
gr431 N/A t.l. -41751 123 823
pr439 N/A t.l. 322.41 20 70
pcb442 N/A t.l. -43489.5 142 338
d493 N/A t.l. -48934.3 43 25
p654 N/A t.l. -65478.8 71 1068
d657 N/A t.l. -64671.9 125 1162
gr666 N/A t.l. -65273.1 169 54
It is natural to assume the heuristics would be most
beneficial for large enough instances. As Tables 1 and
2 show, there is not much room for improvement for
small instances as usually even the linear programming
formulation can come up with an integral solutions
without the aid of GSEC or primitive comb inequality
constraints.
As soon as the size of the instance goes beyond
50 points, the time spent on the LP separation shows
a huge increase while the cost of heuristic separation
is still relatively low. This is not so surprising as the
LP separation has to solve an entire linear program-
ming problem for each node of the graph. For small
instances, the overhead caused by a few extra prob-
lems is negligible, but for larger instances it can be
prohibitive. Also, for the instances for which an in-
tegral solution is available, the gap between the two
Table 2: Computational results for the heuristic Primitive
Comb Inequalities separation.
Heuristic GSEC + Primitive Comb
Instance Obj. Extra Cuts Time % Gap Improv.
eil13 3228.61 0 <1 0 0
ulysses22 -1964.93 5 <1 0 0
bayg29 -44.35 0 1 0 0
eil33 -2924.76 0 1 0 0
att48 1358.75 0 1 0 0
hk48 551.65 0 1 2.01 0
eil51 -4451.36 10 1 0.03 3.24
st70 -6431.62 6 2 0 1.61
eilB76 -7441.27 4 4 0 1.04
eilC76 -6900.17 6 1 0.03 0.19
eilD76 -6773.72 7 7 0 1.02
pr76 5568.42 0 1 0 0
gr96 -9529.7 36 5 0.02 2.11
rat99 -9260.32 4 21 0.04 0.55
rd100 -10119.1 20 3 0 0
eil101 -8988.02 4 6 0.03 0.49
eilA101 -9160.59 2 2 0.01 0.27
eilB101 -9710.34 2 7 0 0.27
lin105 -87.93 0 5565 0
pr107 339.84 0 <1 0 0
gr120 -10451.1 2 185 0 0
gr137 -13068.3 10 31 0.01 2.2
pr144 532.69 0 <1 0
ch150 -8874.46 8 85 0.53 16.63
pr152 984.54 0 2 0
rat195 -18296.1 119 6567 0.09 5.6
d198 -19183.9 5 6 0 0
korB200 -138.68 0 8 0
gr202 -19858.2 38 281 0 0.9
ts225 1296.11 0 20 0 0
gr229 -20516.7 16 13 0.02 3.7
gil262 -24777 32 49 0.04 7.5
a280 -24921.1 49 122 0.02 12.5
lin318 -1842.91 94 4420 40.3
fl417 -41406.8 40 8 0
gr431 -41748.6 38 2041 0.01 2.4
pr439 322.41 0 108 0
pcb442 -43489.5 20 963 0
d493 -48934.3 39 38 0
p654 -65478.8 35 1317 0
d657 -64671.9 21 1482 0
gr666 -65266 80 113 0.03 7.1
heuristic solutions combined and the optimal solution
is very small. In most cases the gap is very close to
0 and only for one instance it goes as high as %2 . In
larger instances, both heuristics perform consistently
well while the LP separation of GSECs and the ILP
formulation do not return with a solution in the period
of 4 hours.
A comparison between using GSEC heuristic sepa-
ration alone and using both heuristics together reveals
that although there are cases for which the solution
quality can benefit significantly from running both
heuristics instead of one, the GSEC separation seems
to be promising on its own in many cases. The rela-
tively long extra time spent on the separation of prim-
itive combs results in the introduction of a few extra
cuts indeed. However, the improvement in the solution
quality is not very significant. This is perhaps to be
ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems
386

expected since we run the primitive comb separation
after all the GSEC cuts are introduced. As a result, the
LP has already closed the gap between the fractional
and the optimal integral objective value, and therefore,
improvements at this point come in very small por-
tions. We conjecture that by using different sequences
of running the heuristics, one can expect to speed up
the running time and get the best of the two heuristics.
7 CONCLUSIONS
The Prize Collecting Travelling Salesman Problem
(PCTSP) is an important generalization of the famous
Travelling Salesman Problem. It also arises as a sub
problem in many variants of the Vehicle Routing Prob-
lem. In this paper, we provide efficient methods to
solve the linear programming relaxation of the PCTSP.
We provide efficient heuristics to obtain the Gener-
alized Subtour Elimination Constraints (GSECs) for
the PCTSP, and compare its performance with linear
programming, which can be used to solve the sepa-
ration problem for GSECs for the PCTSP optimally.
In this paper, we also show that the connected com-
ponent heuristic of Padberg and Hong can be applied
to separate the primitive comb inequalities introduced
by Balas for the PCTSP. We evaluate the effectiveness
of introducing these inequalities for reducing the inte-
grality gap for the PCTSP. Through empirical analysis,
we have been able to verify the importance of two
separation heuristics in finding near optimal solutions
to the Prize Collecting TSP in a timely manner. As
the instance sizes grow, the two heuristics show great
promise by finding a significant number of violated
inequalities.
In this paper, we have looked at two basic heuris-
tic. As a possible future work, one can continue this
line of work by adapting/designing other heuristics for
broader classes of cuts. One extension can be incorpo-
rating more general comb inequalities. Furthermore,
the order of introducing the cuts can have a great im-
pact on the running time. One can study various ways
of mixing different separation procedures to find the
one that is most suitable for specific types of instances.
REFERENCES
Balas, E. (1989). The prize collecting traveling salesman
problem. Networks, 19(4):621–636.
Balas, E. (1995). The prize collecting traveling salesman
problem ii: Polyhedral results. Networks, 25(4):199–
216.
Bérubé, J.-F., Gendreau, M., and Potvin, J.-Y. (2009).
A branch-and-cut algorithm for the undirected prize
collecting traveling salesman problem. Networks,
54(1):56–67.
Cappanera, P., Gouveia, L., and Scutellà, M. G. (2011). The
skill vehicle routing problem. In Proceedings of the
5th International Conference on Network Optimization,
2011, number LNCS 6701, pages 354–364. Springer-
Verlag.
Chaves, A. A. and Lorena, L. A. N. (2008). Hybrid meta-
heuristic for the prize collecting travelling salesman
problem. In Proceedings of the EvoCOP 2008, number
LNCS 4972, pages 123–134. Springer-Verlag.
Crowder, H. and Padberg, M. W. (1980). Solving large-scale
symmetric travelling salesman problems to optimality.
Management Science, 26:495–509.
Fischetti, M., Salazar, J. J., and Toth, P. (1997). A
branch-and-cut algorithm for the symmetric gener-
alized travelling salesman problem. Operations Re-
search, 45(3):378–393.
Fischetti, M. and Toth, P. (1988). An additive approach for
the optimal solution of the prize-collecting travelling
salesman problem. In Golden, B. L. and Assad, A. A.,
editors, Vehicle routing: Methods and studies, pages
319–343. Elsevier Science Publishers.
Goemans, M. X. (1994). The steiner polytope and related
polyhedra. Mathematical Programming, 63:157–182.
Gutin, G. and Punnen, A. P., editors (2002). The Traveling
Salesman Problem and its Variations. Combinatorial
optimization. Kluwer Academic, Dordrecht, London.
Hong, S. (1972). A Linear Programming Approach for
the Traveling Salesman Problem. PhD thesis, John
Hopkins University, Baltimore, Maryland, USA.
Land, A. (1979). The solution of some 100-city travelling
salesman problems. Technical report, London School
of Economics, London, UK.
Padberg, M. W. and Hong, S. (1980). On the symmetric
traveling salesman problem: a computational study.
Mathematical Programming Study, 12:778–107.
Reinelt, G. (1991). Tsplib: A traveling salesman problem
library. ORSA J. Comput., pages 376–384.
Wolsey, L. A. (1998). Integer programming. Wiley inter-
science series in discrete mathematics and optimization.
Wiley.
Prize Collecting Travelling Salesman Problem - Fast Heuristic Separations
387