H.P. WILLIAMS
LONDON SCHOOL OF
ECONOMICS
MODELS FOR SOLVING
THE
TRAVELLING SALESMAN PROBLEM
h.p.williams@lse.ac.uk
STANDARD FORMULATION OF THE
(ASYMMETRIC) TRAVELLING
SALESMAN PROBLEM
Conventional Formulation:
(cities 1,2, …, n) (Dantzig, Fulkerson,
Johnson) (1954). xij is a link in tour
c x
Minimise:
ij
i,j
subject to:
ij
x
 1 all j
x
 1 all i
ij
i
ij
j

i , jS
xij  | S | - 1 all S  {2..., n}
e.g.
2
6
3
x32  x26  x63
 x23  x62  x36  2
0(2n) Constraints
= (2n-1 + n –2)
0(n2) Variables
= n(n – 1)
EQUIVALENT FORMULATION
Replace subtour elimination
constraints with
 x  1 all S  1,2, ,..., n
iS
jS
ij
____
S
S
Add second set of constraints for all i in S and
subtract from subtour elimination constraints for S
OPTIMAL SOLUTON TO A 10 CITY
TRAVELLING SALESMAN PROBLEM
8
6
3
1
2
1
0
5
7
4
Cost = 881
9
FRACTIONAL SOLUTION FROM
CONVENTIONAL (EXPONENTIAL)
FORMULATION)
Cost = 878 (Optimal Cost = 881)
Sequential Formulation (Miller, Tucker, Zemlin (1960))
ui
=
Sequence Number in which city i visited
Defined for i = 2,3, …, n
Subtour elimination constraints replaced by
S:
ui - uj +nxij  n – 1 i,j = 2,3, …, n
Avoids subtours
but allows total tours (containing city 1)
2
u2 – u6+ nx26  n-1
6
u6 – u3+ nx63  n-1
u3 – u2+ nx32  n-1
3
3n
0(n2)
0(n2)
Constraints
Variables
=
=
 3(n – 1)
(n2 – n + 2)
(n – 1) (n + 1)
Weak but can add 'Logic Cuts'
e.g.
uk  1  xij  x jk  x1 j
FRACTIONAL SOLUTION FROM
SEQUENTIAL FORMULATION
Subtour Constraints Violated : e.g.
Logic Cuts Violated: e.g.
Cost = 773 3/5
x  x  1
27
72
u  1  x  x  x
9
27
79
17
(Optimal Cost = 881)
Flow Formulations
Single Commodity (Gavish & Graves (1978))
Introduce extra variables (‘Flow’ in an arc)
Replace subtour elimination constraints by
F1:
y  (n  1) x all i, j
ij
ij
 y  n 1
j
1j
 y   y  1 all j  1
i
ij
k
jk
Can improve (F1’) by amended constraints:
y  ( n  2) x
ij
ij
all i, j  1
Network Flow formulation in yij variables over
complete graph
4
1
1
2
1
n-1
3
1
Graph must be connected. Hence no subtours possible.
2
0( n )
2
0( n )
Constraints
 n(n  2)
Variables
 2n(n  1)
FRACTIONAL SOLUTION FROM SINGLE
COMMODITY FLOW FORMULATION
2
Cost = 794 9
(Optimal solution = 881)
FRACTIONAL SOLUTION FROM MODIFIED SINGLE
COMMODITY FLOW FORMULATION
Cost = 794
43
48
(Optimal solution = 881)
(192=3x64)
Two Commodity Flow (Finke, Claus Gunn (1983))
y is flow of commodity 1 in arc i  j
ij
z is flow of commodity 2 in arc i  j
ij
 y  y
ij
j
F2:
z z
ij
j
z z
j
j
ij
j
y z
ij
j
 1 i  1
 n 1 i  1
ji

1 i 1
  (n  1) i  1
ji
n  1 all i
=
ji
(n  1) xij all i, j
=
ij
1
Commodity 1
1
3
2
1
1
n-1
Commodity 2
n-1
Commodity 2
Commodity 1
0( n ) Constraints  n(n  4)
2
0( n ) Variables  3n(n  1)
2
1
Multi-Commodity (Wong (1980) Claus (1984))
“Dissaggregate” variables
y
is flow in arc destined for k
k
ij
y  x all i, j, k
k
ij
F3
ij
 y 1
 y 1
k
i
k
ik
i
1i
y 0
k
i
i1
 y  0 all k
k
j
kj
 y   y all j, k , j  1, j  k.
k
i
ij
k
i
ji
3
Constraints  n  2n  6n  3
3
Variables
0( n )
0( n )
3
2
 n n  1
2
LP Relaxation of equal strength to Conventional
Formulation.
But of polynomial size.
Tight Formulation of Min Cost Spanning Tree
+ (Tight) Assignment Problem
FRACTIONAL SOLUTION FROM MULTI
COMMODITY FLOW FORMULATION (=
FRACTIONAL SOLUTION FROM CONVENTIONAL
(EXPONENTIAL) FORMULATION)
Cost = 878 (Optimal Cost = 881)
Stage Dependent Formulations
First (Fox, Gavish, Graves (1980))
= 1 if arc i  j traversed at stage t
=
T1:
0 otherwise
y  n
t
ij
i , j ,t
  ty    ty  1 i  2, 3...n
n
n
j 1
t 2
t
ij
n
n
j 1
t 1
t
ji
(Stage at which i left 1 more than stage at which entered)
y  0, t  n
t
i1
y  0, t  1
t
1j
y  0, i  1
1
ij
0n  Constraints  n
0n  Variables  n n  1
3
2
Also convenient to introduce x variables with constraints
ij
x y
ij
t
t
ij
FRACTIONAL SOLUTION FROM 1ST
(AGGREGATED) TIME-STAGED FORMULATION
Cost = 364.5 (Optimal solution = 881)
NB ‘Lengths’ of Arcs can be > 1
Second (Fox, Gavish, Graves (1980))
T2:
Disaggregate to give

yijt  1
all j

yijt  1
all i

yijt  1
i  j ,t
j  i ,t
i , j ,i  j
n
n
  ty
j 1
t 2
t
ij
all t
n
-
n
  ty
j1
t 1
t
ji
 1 i  2,3, ... n
Initial conditions no longer necessary
0(n) Constraints = 4 n – 1
0(n3) Variables
= n2 (n – 1)
FRACTIONAL SOLUTION FROM 2nd
TIME-STAGED FORMULATION
Cost
= 799
164
357
(optimal solution = 881)
(714 = 2 x 3 x 7 x 17)
Third
T3:
(Vajda/Hadley (1960))
t
y ij
interpreted as before

yijt  1
all j

yijt  1
all i

yijt  1
all t
i j
j i
t

yijt   y tjk1

y11 j = 1

yin1 = 1
i j
j 1
i 1
k j
= 0 all j, t
0 (n2) Constraints
= (2n2+ 3)
0 (n3) Variables
= n2(n-1)
FRACTIONAL SOLUTION FROM 2nd TIME
2nd TIME-STAGED FORMULATION
Cost =
804
1
2
Optimal solution = 881
OBSERVATION
Multicommodity Flow Formulation
y y 0
t
ij
i
y
t
ij
t
k
is flow i  j
jk
destined for node t
Time Staged Formulation
y y 0
t 1
t
i
k
ij
j k
y  1 iff go i  j at stage t
t
ij
Are these formulations related?
Can extra variables  y  , introduced
syntactically, be given different semantic
interpretations?
t
ij
COMPARING FORMULATIONS
Minimise:
cx
Subject to: Ax  By  b
x, y  0, x integer
W  {w | wB  0, w  0}
W forms a cone which can be characterised by
its extreme rays giving matrix Q such that
QB  0
Hence
QAx  Qb
This is the projection of formulation
into space of original variables x
ij
COMPARING FORMULATIONS
Project out variables by Fourier-Motzkin
elimination to reduce to space of conventional
formulation.
P (r) is polytope of LP relaxation of projection of
formulation r.
Formulation S (Sequential)
Project out around each directed cycle S by
summing
u  u  nx  n  1
i
j
ij
n  x  n  1 S
i , jS
ie
x  S 
i , jS
ij
ij
S
n
Hence PS   PC 
weaker than |S|-1 (for S a
subset of nodes)
Formulation F1 (1 Commodity Network Flow)

Projects to
ijs
Hence
xij | S | 
|S|
|S|
stronger t han | S | n 1
n
P( S )  P( F1)  P(C )
Formulation F1' (Amended 1 Commodity Network
Flow)
Projects to
Hence
1
n 1

jS {1}
jS
xij   xij | S | 
i , jS
|S|
n 1
P( S )  P( F1)  F ( F1' )  P(C )
Formulation F2 (2 Commodity Network Flow)
Projects to

i, j
Hence
xij | S | 
|S|
n 1
P(F2) = P(F1)
Formulation F3 (Multi Commodity Network
Flow)

Projects to
x ij | S | 1
i, j
S
Hence
Formulation T1
P(F3) = P(C)
(First Stage Dependant)
Projects to
 x 
iS
j S {1}
ij
|S |
n 1
x n
i , jN
ij
(Cannot convert 1st constraint to ‘  xij  ‘form since
i , j5
Assignment Constraints not present)
Formulation T2
(Second Stage Dependant)
Projects to
1
n 1
 x 
ij
iS
jS {1}
1
n 1
 x   x | S | 
i S {1}
j S
ij
ijjs
ij
|S|
n 1
+ others
Hence
P(T2)  P(F1')
Formulation T3
(Third Stage Dependant)
Projects to
1
n 1
 x 
i S
jS {1}
ij
1
|S|
xij   xij | S | 

n , j S
n  1 ijSS{1}
n 1
+ others
Can show stronger than T2
Hence
P(T3)  P(T2)
Computational Results of a 10-City TSP in order to
compare sizes and strengths of LP Relaxations
Model
Size
C
(Conventional
LP
Obj
Its
Secs
IP
Obj
Nodes
Secs
502x90
(Ass. Relax
+Subtours (5)
+Subtours (3)
+Subtours (2)
766
804
835
878
37
40
43
48
1
1
1
1
766
804
835
881
0
0
0
9
1
1
1
1
S
(Sequential)
92x99
773.6
77
3
881
665
16
F1
1(Commodity Flow
F'
(F1 Modified)
120x180
794.22
148
1
881
449
13
120x180
794.89
142
1
881
369
11
F2
(2 Commodity Flow)
140x270
794.22
229
2
881
373
12
F3
(Multi
Commodity Flow)
857x900
878
1024
7
881
9
13
T1
(1st Stage
Dependent)
90x990
(10)x(900)
364.5
63
4
881
120x990
(39) x (900)
799.46
246
18
881
483
36
193x990
(102)x(900)
804.5
307
5
881
145
27
(2nd
T2
Stage Dependent)
T3
(3rd Stage
Dependent)


Solutions obtained using NEW MAGIC and
EMSOL
P(TSP)
TSP Polytope – not fully known
P(X)
Polytope of Projected LP relaxations
Reference

AJ Orman and HP Williams,
A Survey of Different Formulations of the
Travelling Salesman Problem,
in C Gatu and E Kontoghiorghes (Eds),
Advances in Computational Management
Science 9 Optimisation, Econometric and
Financial Analysis (2006) Springer