HD28
.M414
DEWEY
he liOij'
^'6
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Probabilistic Analysis of the 1-Tree Relaxation for
the Euclidean Traveling
Salesman Problem
Michel X. Goemans
Operations Research Center
Massachusetts Institute of Technology
Dimitris Bertsimas
Sloan School of
Management
Massachusetts Institute of Technology
MIT Sloan School Working Paper No.
2104-8
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Probabilistic Analysis of the l-Tree Relaxation for
the Euclidean Traveling Salesman Problem
Michel X. Goemans
Operations Research Center
Massachusetts Institute of Technology
Dimitris Bertsimas
Sloan School of Management
Massachusetts Institute of Technology
MIT Sloan
School Working Paper No. 2104-88
MASSACHUSETTS INSTITUTE
OF TPCHNOI.OGY
NOV
1 4 2000
LIBRARIES
Probabilistic Anah'sis of the 1-Tree Relaxation for the
Euclidean Traveling Salesman Problem
Michel X. Goemans
'
Dimitris
December
13,
J.
Bertsimas
^
1988
Abstract
We
analyze probabilistically the classical Held-Karp lower bound derived
from the
We
1-tree relaxation for the Euclidean traveling
prove that,
if
salesman problem (ETSP).
n points are uniformly and independently distributed over the
d-dimensional unit cube, the Held-Karp lower bound on these n points
surely asymptotic to
of n.
The
ETSP
where
/?//a'(^) is a
almost
constant independent
result suggests a probabilistic explanation of the observation that
the lower bound
the
/?//a-(c?) ji''^"^^^'',
is
is
is
very close to the length of the optimal tour in practice since
almost surely asymptotic to I3rsp{d)
n''^"'^^''.
The techniques we
use exploit the polyhedral description of the Held-Karp lower
bound and the
theory of subadditive Euclidean functionals.
Key words.
Probabilistic analysis, traveling salesman problem, linear relaxation,
1-trees, subadditive
Euclidean functionals.
'Michel Goemans, Operations Research Center, MIT, Cambridge,
'Dimitris Bertsimas, Sloan School of Management,
The
MIT,
Rm
MA
02139.
E53-359, Cambridge,
Ma
02139.
research of the second author was partially supported by the National Science Foundation
under grant ECS-8717970.
1
Introduction
During the
two decades combinatorial optimization has been one of the fastest
last,
growing areas
the field of mathematical programming.
in
Some
of the major contri-
butions were Lagrangian relajcation, polyhedral theory and probabilistic analysis.
The landmark
or Fisher
the development of Lagrangian relaxation (see Geoffrion
in
combinatorial optimization problems were the two papers for the
[5]) for
(TSP) by Held and Karp
traveling salesman problem
Held and Karp
tree for the
TSP. Using
In the
now known under
[11]), to solve
a
Edmonds
[-1]
for matroids. they
second paper. Held and Karp
tlie
name
showed that
The
1-tree relaxation has
bound procedures
Hansen and Krarup
[20] or, for a survey,
the length of the optimal tour.
On
l9{i.
[-3]
and Johnson
[8],
[1]).
triangle inequality
is
the efficiency of the
paper
is
The
aimed
at
never
is
the relative gap
to solve the
bound
in
practice.
shedding new
[17],
(see
Held
Volgenant
These computational studies
is
is
extremely close to
often less or
due to Wolsey
[21] states
light
this worst-case analysis
The
much
less
than
that the Held-
Beardwood, Halton and Ilammersley
TSP
does not capture
probabilistic analysis developed in this
on the behavior of the Held-Karp lower bound.
area of probabilistic analysis has
the asymptotic behavior of the
TSP
than 2/3 of the length of the optimal tour when the
However,
satisfied.
Crowder
According to most of the above authors (see also
[12])
less
is
been extensively and
Smith and Thompson
Balas and Toth
a theoretical ground, a result
Karp lower bound
1-
Lagrangian
introduced a method, which
[10]
have shown that, on the average, the Held-Karp lower bound
Christofides
this
of subgradient optimization (Held, Wolfe and
the Lagrangian dual.
[10], Ilelbig
and Jonker
In the first paper,
as the linear relaj^ation of a classical formulation
successfully used to devise branch and
and Karp
[10].
complete characterization of the 1-tree polytope, which
same bound
relaxation gives the
TSP.
[9],
proposed a Lagrangian relaxation based on the notion of
[9]
follows from a result of
of the
[6]
if
[2].
its
origin
in
the pioneering paper by
Tlie authors characterize very sharply
the points are uniformly and independently
distributed in the Euclidean plane or,
more
generally, in R"^
portance of this early work was demonstrated
Karp
in
[13].
The
.
potential im-
Steele [18] analyzed
probabilistically a general class of combinatorial optimization problems by develop-
ing the notion of subadditive Euclidean functionals.
original proof of
Beardwood
tingale inequalities were
first
offered using martingale inequalities.
is
Mar-
applied to the probabilistic analysis of combinatorial
optimization problems by Rhee and Talagrand
In this paper,
Steele [14] the
using the Efron-Stein inequality.
et al. [2] is simplified
In Steele [19] an even simpler proof
Karp and
In
[16].
we combine the combinatorial interpretation
lower bound with the probabilistic techniques of Steele
[18].
Hekl-Karp
of the
We
prove that,
if
n points are uiiifornily and independently distributed over the d-dimensional unit
cube, the Held-Karp lower bound on these n points divided by n'
surely asymptotic to a constant 0HK{d)-
^^
d
li*"!'
=
convergence of the Ileld-Karp lower bound divided by
tlie fact
that the
bound can be viewed
2,
''
is
almost
we prove the complete
s/ti.
as the cost of the best
We
exploit extensively
convex combination of
1-trees such that each vertex has degree 2 on the average. Relying on computational
studies for the
TSP
and the matching problem
that the asymptotic gap {Ptsp
this
is
is
the
first
~ 0H !<)/ pTSP
is
the Euclidean plane,
in
less
we estimate
than 3%. To our best knowledge
time that a linear relaxation of a combinatorial optimization problem
analyzed probabilistically using subadditivity techniques.
The remaining
of this paper
the main results of the Held and
the Held-Karp lower bound
theorem. In section
for the
4
is
is
structured as follows. Section 2 reviews briefly
Karp
[9]
paper.
In section 3
we
first
prove that
monotone and subadditive and then prove the main
we use a martingale inequality
Held-Karp lower bound and we establish
its
to derive
some sharp bound
complete convergence.
Held-Karp lower bound
2
we summarize the main resuUs of Held and Karp
In this section
a lower
bound on
tiie
set V.
can be expressed as the optimal objective function value
it
linear relaxation of the following standard formulation of the
^^'"
This bound can be
equivalent ways.
in several
First,
They presented
length of the optimal tour to the symmetric traveling salesman
problem on the complete undirected graph with vertex
described
[9].
Y.
Z
HK
of the
TSP:
^u-Tu
(1)
subject to
^
+ Y,
x-„
^-n
=
V.
2
Ctj
integer
program,
Vf,j € V,j
i,j
indicates whether cities
i
We now
>
(4)
i
(5)
and j are adjacent
represents the cost of traveling from city
city j to city
(3)
yijevj>i
o<x-,,<i
tour;
(2)
V0^5cr
EE-'-'. <l'?|-l
In this
V
J«
}><
Xij
€
i
to city
_;'
or,
in
the optimal
by symmetry, from
i.
give
two alternative definitions of
a 1-tree
which constitutes the core of
the other formulations.
Definition
tree
on
V
1
T=
(V,
E)
ts a
1-iree (rooted at vertex 1) if
\ {!}, together with two edges incident to vertex
From now on we
shall always
T
consists of a spanning
1.
assume, unless otherwise stated, that the root node
identical for any 1-tree, say vertex
1.
is
T=
Definition 2
1.
T
2.
\V\
3.
T
4-
(he degree in
(V', £')
1-iret if
ts a
coniKcttd
IS
=
\E\
has
containing vertex
a cycle
Held and Karp
T
of vertex
1
1
is 2.
highlighted the relation between the linear program (l)-(4) and
[9]
the class of l-trees.
More
precisely, they
showed that the
feasible solutions to (2)-(4)
can be equivalentiy characterized as convex combinations of l-trees
vertex has degree 2 on the average. Hence, we
may
sucli that
each
rewrite (l)-(4) in the following
way:
k
UK
^ MinY^KdTr)
r
(6)
=\
subject to
^
A.
=
1
(7)
k
Y,^rdjiTr) =
Vje\'\{l}
2
(8)
r=l
A,
>
r=l....,t,
(9)
where
•
•
{Tr}j._j
c{C)
=
I.
constitutes the class of l-trees defined on the vertex set
IIe=(i,j)€E^u
'®
^^^ total cost of the subgraph
• dj(T) denotes the degree in
Finally, tlif
most
T
C=
{V,
\',
E) and
of vertex j.
common approach
to find the
Held-Karp lower hound
take the Lagrangian dual of (6)-(9) with respect to (8).
We
then obtain:
is
to
.
HK
= maxL{^i)
(10)
mm
(H)
subject to
Li^j.)=
where c^{Tr)
is tlie
c,{Tr)-2Y,^^J
cost of
tlie
1-tree T^ with respect to the costs
c,j
+
/j,
/ij
The main theorem
3
Let
tlie
tributed
=
n points A"'"'
in
the d-cube
(A'l, ,Y2,
[0, 1]
.
•
•
•
,
and independently
A'„) be uniformly
behavior, as n tends to infinity, of //A'(A"'"'). Steele
2.
[18]
We
are interested in the
proved that the asymptotic
behavior of a particular class of Euclidean functionals L defined on
to
R
],
z)
=
finite
subsets of
can be characterized very sharply as follows:
Theorem
R
dis-
Let //A'(A'''^') denote the Held-Karp lower bound on
by any of the formulations of section
A''"' as defined
R
+
1
(Steele [18]) Lei L
Euclidean [L{ai'i,ai-2
be a
ai-,,)
L(ji, X2, ..., x„)] /unc/i07)a/ d/
=
monotone
[L{Ali{i'})
> L{A)
aZ,(j-i, 2-2, ..., Xn). ^(-''i
finite
+
variance [l'aj-[L(A''"']
Vj-
-Ti -^2
<
e
+
2-,
W^^A C
...,!„
+
oc] which saiis-
fies Ihe subadditivity hypothesis;
If {Qi
:
1
<
'
<
n?
}
parfition of the d-cvhe
IS a
[0, 1]
into
with edges parallel to the axes then there exists a constant
A' \ {0},V/
>
0,
Li{xi
x„
}
n
[0, /]^)
<
^
L({xi
,
.
.
.
,
x„} n tQ,)
there exists a constant pL{d) such that
I(A'("))
imi
„l"i;;(^rT]77
n—»oo
almost surely
C >
identical subcnhes
such that V'n G
we have that
1=1
Then
t7?
=
/?^(^)
+ Ctm d-\
We
empliasize thai
can easily be seen that
It
esis.
tlie critical
property
K
II
is
in
theorem
1
the subaddilivity liypotli-
is
a Euclidean functional with finite variance.
Proposition 2 proves that the subadditivity hypothesis holds for the functional
The monotonicity
of
HK
useful formulation of the
is
proved
in
proposition
we denote by P(A) the program
clarity
UK
Proposition 2
subadditive,
is
in<{\r,
For these propositions the most
3.
Held-Karp lower bound
is
For convenience and
(6)-(9).
(6)-(9) corresponding to the set
3C >
i.e.
R
>
O.Vni e A' \ {0},V<
x„}n[o,/]'')<^//A-({ji,...,x„}nf(?,)
for any finttf suhstt of
UK.
+
A
of cities.
:
c/m''-i
.
Proof:
Using the
the case
z
=
{T,i
We
1,
.
.
fact that
=
<
.
,
in
Let p
T,2, .... T,k,
,
Let
1.
.
JI
K
is
V =
a Euclidean functional,
{j,
— m". We
,
.
.
,
a-^}
n
[0, 1]''
and
be the class of 1-trees defined on
}
V,
=
restrict ourselves to
{xi,
arbitrarily choose a root vertex
consider the optimal solution
^A,> =
.
we may
{A,>}r=i,...,it,
\',
.
.
.
,
i,,} Pi
every
Q, for
\].
Let
(with respect to the root
1,).
to P{Vt),
i.e.
1,
in
{A,r}r=i
k,
satisfies:
(12)
1
r=\
*-,
^A,,fO(T.>) =
r
VjG\;\{l,}
2
(13)
=l
A,>
>
r=\,...,ki
(14)
k,
//A-(V^)
From
cost
is
= ^A,>c(T,>)
these optimal solutions
less
than
(15)
we
shall construct a feasible solution to P{\')
whose
^HK{\]) +
where
C =
2\/d
+
Cm'^-'^
3.
(IG)
For this purpose, we consider every possible combination of
subcube Q,. There are (nr=i
selecting one 1-tree in each
us focus on one of them, say {Tir, }i=i
corresponding
spanning
li,
1-trees.
V'
d.in) =
From
Let
p.
we
these p 1-trees
A
^"i)
such combinations. Let
be the indices
(ri, r2,
.
shall construct a 1-tree
.
.
,
Tp) of the
T^ rooted
at
and satisfying the following conditions:
ifjeQ,
dj{T,r,)
c(TA)<^c(r,.,) +
(17)
Cr7i''-i
(18)
1=1
We
= nf=i
claim that, by assigning a weight of Aa
feasible solution to
1.
P(V) whose
is
to each 1-tree
T\ we get a
than (16). Indeed,
less
Using (12) recursively, we have;
A = (ri,...,rp)
A
=
zJ
7-j
=
2.
cost
-^iV,
1
=1
zZ
''^I'-l
=l
r2
=2
•^^rj
Zl Vp
'rp=P
(19)
1-
Consider any vertex j 6
E^A^^(^a)
=
A
V'.
Assume
that j G Qt- ^Ve have that:
E^Af/,(7^>,)
A
'=1
je{l
p}\{l}
rjirl
k,
=
^
r,
=
A,>,dj(T,^,)
=l
2
using (17), (12) and (13) respectively.
(20)
3.
1,
2
X.\
and
>
3
follows
from
(14).
imply that the solution
feasible in
is
P{V)- The cost of
this solution
is
given by:
A
A=(ri
1=1
Ti
Tp)
1
=
A
1
=l
P
=
Y^HK{Vi) + Cm^-'
(21)
1=1
using (18), (12), (19) and (15) respectively.
The
last point left in this
construction of the 1-tree T\ satisfying (17) and (18).
Figure
1:
Step
1
in
We
proceed
the construction of 7^.
proof
in 2 steps:
is
the
1.
(Figure
1)
In each 1-tree T,^, {i
to tiie root
1,,
say
(1,, 2,).
Figure
2.
(Figure 2)
Assume
cubes Qi and Qi+i
exists for every d.
Figure
3.
We now
2:
=
I.
.
.
,p)
.
we
delete one of the 2 edges incident
Note that typically
Step 2
in
2,
depends on
the construction of Ta.
that the numbering of the subcubes
{i
A
= l,....p—
possible
r,.
is
such that the sub-
adjacent. Such a numbering clearly
1) are
numbering
for the case
add the edges (2i,l,+i)
(i
cf
=
2
= l,...,p-
is
1)
represented in
and the edge
(2p,li)-
We
first
li-
This follows from definition
claim that the resulting subgraph T\
2.
Indeed T\
10
=
(V, £'a)
is
is
a 1-tree rooted at vertex
clearly connected, the
number
of
«,
.
We now
prove the monotonicity of the functional
Proposition
>
3 If n
3 //(en
HK
e W^
Va-i,...,2-„+i
monotone,
ts
i.e.
//A'(xi,...,a-n+i)
:
HK
>
H K{xi,
.
,Xn).
.
.
Proof:
Let {Ti,
.
.
we may assume
loss of generality
greatest rational that divides A, for
get a multiset
5 =
{T",},-!
to
remove vertex
less
is
V=
show that the optimal solution
to
\'\{n+
HK{V).
than
'
+
(lij)'!!'"
Figure
4.
Claim
1
l).(j-"
Without
Indeed,
=
dn+i{T)
+
1)
€ T}.
A
that
j'l
that
i\
in
^
^
n
+
We
1}-
For this purpose, we
T G
:
assumed
in
to be
5a
1
4,
the degree of vertex j in
sucli that dj{T')
1
in
T'
.
=
ij
^
j.
If
Let
1.
Without
Moreover, since T'
is
T
12
is
—
2,
the
ij
and
i2
T
e
1)
is
T
and
at least 3.
be the two vertices
we may assume
a 1-tree with dj{T')
(l,j) in
T
that
=
1,
we have
by (1,J]) and (l,»i)
and T' which are not
need
in
S
3j €
represented in
2 on the average, there exists
loss of generality,
we replace
T' by (l,j), we get two 1-trees
is
first
empty.
>
j and
is
such a way that
S,dnj^i{T)
As n +
I.
A
is
shall construct a feasible
such that (l,j),(l,n+ l),(j,n+
adjacent to vertex
i.e.
Now assume
set.
T 6 5a
5
A^.),
,
.
.
possible candidate for
Therefore, since the degree of each vertex
a 1-tree T' G
.
P{V)
to
duplicating T, (Aj/A) times
5a = {T
loss of generality, iSa can be
let
2.
,
P{V) can be decomposed
does not contain some particular 1-trees. Let
y
By
{A,},=i_..._fc
of 1-trees such that each 1-tree has weight A
Let
{n-\- 1).
solution to P{V') whose cost
to
1,...,^-.
gcd(Ai
and therefore every multiset can be seen as a
differentiated
we want
=
=
Without
points.
For clarity we assume that two identical 1-trees can be
the optimal solution.
in
..._/
i
Let A
n+1
of
that the optimal solution
a basic feasible solution and thus rational.
we
V
be the class of 1-trees defined on a set
,rfc}
.
5a- This
.
Figure
basically follows
and that
if
from the
+
(l.n
1-tree in
fact that (l,j)
and
1)
4:
n
(j,
+
1)
But 5
\
T while
^
were both
then T' would not be a 1-tree since T'
disconnected.
^a-
\
= {T e S
T
S^ =
in
5
\ (Si
U
X/[dn+i{T)
dn+i{T)
:
S2) {dn + i{T)
—
—
2) in order to
2)
{(l.zi),(l.n
Claim
2
\Si\
=
1^3
T
in
+
^3
1)}
and T' have the same weight
n-f-1
2
would be
is
A.
Hence, by
0.
=
i},
i
=
1.2.
We
duplicate every 1-tree
T
times and we associate to each copy a weight of
to the 1-trees in
X/{dn+i[T)
—
»5i
or S2
J>3
is still
2).
1
Since vertex
=
may assume, without
keep the solution unchanged. Call
Note that the weight associated
associated to a 1-tree
T' and dn+\{T')
in
applying this procedure repeatedly, we see that we
Let 5,
n+1) G T
{T,T'} U {T,T'} represents the same optimal
solution as previously since
loss of generality, that
(1. n4-l), (j,
has degree 2 on the average, we have
13
the resulting set.
A while the weight
^
^
+
A
Now
+ X:
2A
TeS2
TeSi
d„^,{T)
^
Un+i{^
TeSi
=
2.
(22)
^
)
the claim follows by substracting the equality X)t65
^Tes,
j„^.(T)-2
=
^^^''ce
1
from
-^
+ 117652
^"^
(22).
This means that we can regroup Si and S3 into a set S13 of pairs (Tj.Ts) of
l-trees of
and S3
(!^i
feasible solution to
=
(|t>i3|
P{\
'}
T
associate to each 1-tree
=
\S\\
whose
E S2
|«^3|)-
total cost
From
^2 and S13 we shall construct a
less
than HI\{V). More precisely, we
is
T' (a pair (T[,T^) of l-trees, respectively) defined on
XciT')
<
j^r—^ciT^)<\c{Ti) + -
we
=
XdjiT)
(23)
and (24) we
get a feasible solution to P(\'')
solution to P(\') which
1.
-^^—-c{T3),
Vj e
+ —-^—-d,iT^) =
Combining
The
such that;
is
resp.)
d„ + i(i3)-J
Xd,{Ti)
+
"n+n-'a)-^
hold.
\''
(23)
dn + li-ls)-^
{^d,{T{)
S13, respectively) a 1-tree
Xc{T)
(Ac(TO+Xdj{T')
6
(to each pair {Ti^Ts)
V
is
M
t.
Ts), resp.)
"n+U^3)-^
clearly see that,
whose cost
(24)
less
by keeping the old weights,
than the cost of the optimal
HI\'{\').
construction of T' and [Ti.T^)
is
as follows:
T eS2
+
Let {i,n
Let T'
1)
and {j,n
= T\
{{i,
n
+
+
1)
1), (>,
be the two edges incident to vertex n
n
+
U {ij}. The
1)}
fact that T'
is
+
1
in
T.
a 1-tree on
V' follows from definition 2 and the fact that we can assume without loss of
generality that
S^ =
(claim
2).
Clearly (24)
is
satisfied
and the triangle
inequality implies that (23) holds.
2.
(Ti,T3)e.s'i3
Let
i
^
1
be the unique vertex adjacent to (n
14
+ l)
in Ti
.
Let v
=
dn+i{T3)
>
3.
Let
+
j^ be the vertices adjacent to n
ji
loss of generality that
remove the
vertices
vertex n
1,
-j-
1
is
i
in the
+
and n
We may assume
in T3.
same connected component
Moreover,
in Ts.
1
1
we may assume that
J2
=
1
if
if
vertex
and only
if
as ji
1
is
(l,Ji)
without
when we
adjacent to
^
The
T3.
\/.
-^
T,
T,
t;
Figure
transformation
is
Construction o( T[ and T^.
5:
the following (see Figure
—
T{
A
The
fact that
added
IT3I
T'3
\ {{ji,n
Tj'
is
+
1),
.
.
a 1-tree
.
,
T,\{{i,n+l)}
{j^,
n+l)){J {U1J2),
obvious.
is
to T3 were already present in T3.
=
di{T^)
IT3I
=
2.
—
1
—
\V\
—
1
=
T3
2,
We
We
+
- Xc{T[)
-^cin)
u — 2
i),
,
(j:.,
0}
notice that none of the edges
then check that T3
is
is
connected,
-
15
1
and
a 1-tree. Using the triangle inequality,
we have
Xc{T,)
Ua,
T3 has a cycle containing vertex
\V'\,
Hence, by definition
5):
-^c{n)
u —
2.
"A
''A
A
A
"
A
A:=3
and therefore (23) holds. Moreover, since
-1
[
if;
i
otherwise
I
and
=
{V -1
lU =
i
otherwise
we
see that
Hence, (24)
is
satisfied.
This completes the proof of proposition
We may now
1
deduce the asymptotic behavior of
and propositions
Theorem
4 Lei
3.
2
and
HK
as a corollary to
theorem
3.
ilu n points A"'"'
=
(A'l,
.
distributed in the d-dimensional unit cube.
.
.
,
A'n) be uniformly
Then
and independenlly
there exists a constant
PHK{d)
such that
almost surely.
A number
of combinatorial optimization problems, like the Euclidean traveling
salesman problem, the Euclidean
minimum
minimum spanning
tree
problem and the Euclidean
weight matching problem, have a similar asymptotic behavior although
16
witli
a (iifT^^rPiit
padimitriou
constant
It
[15]).
^
Beard wood, Halton and Hammersley
(see
compare
therefore interesting to
is
/?//a(c') to
for closely related combinatorial optimization problems.
0HK{d) <
that
on n points
is
of n-1 points,
never
less tiian
where P\i{d)
problem,
>
/?//a-(c/)
is
the cost of the
/?r(f/)
where I3r{d)
tree problem.
the value of
In particular,
minimum spanning
is
is
it
(3
clear
tree
on a subset
the corresponding constant for the
The
relationship between
the constant for the Euclidean
a little less obvious.
is
and Pa-
Moreover, since the value of the Held-Karp lower bound
minimum spanning
Euclidean
[3\f{d).
l3TSp{d).
[2]
minimum
BuKi^)
arifi
weight matching
Using a complete characterization of the perfect
matching polytope, we may express the
cost
M
of the
minimum
weight matching
as:
il/
= A//n^^c„y„
(25)
subject to
(27)
(28)
Substituting
M
<
^
y,j
by
J:,j/2
we get a relaxation of the
which implies that l3}u<{d)
>
2/?A/((f).
linear
We
program
(l)-(4).
Hence
thus obtain the following
proposition:
Proposition 5
When
d
=
max(2/?A/(c/),/?r(d))
2, ,SA/(rf), /?7(cf)
by Papadimitriou
[1.5],
Gilbert
< 0HK{d) <
0TSP{d).
and 0TSP{d) were estimated to be
[7]
and Johnson
17
[12],
0.35, 0.68
and 0.72
respectively. Using proposition
5,
we may therefore deduce
imately
the asymptotic gap {0TSP
— 0.70)/0.70 ^ 3%.
than (0.72
less
tliat
is
approx-
This suggests a probabilistic explanation
of the observation that the Held-Karp lower
optimal tour
— Puk)/ i^TSP
bound
is
very close to the length of the
in practice.
Martingale inequality and the Held-Karp lower bound
4
In this section
we use a martingale inequality
-
Pr{|//A-(A-'"^)
for the case d
=
to
£'[FA-(A''"))]|
the Euclidean plane.
2, i.e. in
deduce a sharp bound on
As
>
i)
a consequence,
we
shall
be able
to establish the finiteness of
//A'(A'("';
for all
is
f
>
0, i.e.
- Phk >
€
s/Vr
n=l
the complete convergence of the Held-Karp lower bound. This result
stronger than the almost sure convergence of theorem
rests
4.
This section basically
upon the martingale arguments developed by Rhee and Talagrand
the
[16] for
TSP.
For each
Let IIK,
If
A,
=
E[Ai\A,]
1
=
<
J
<
we
let .4,
be the sigma
field
i/A'(A'i,...,A',_i,A', + i,...,A'„). Clearly
//A-(A-<"))
-
n,
-
//A',,
is
6
HK{X^''^) - ^[//A'CA'*"))]
a martingale difference sequence.
There erisfs
a coiisiant
We
18
<
J
<
i-
><}
= E?=id,
=
and
prove the following
7 svch ihai for every n
Pr{|//A-(A'"')-r[//A-(A'("')]|
1
£'[//A-(A'("')|.4,_i]
theorem.
Theorem
A'^,
E[H K,\A,] = E[H K,\A^_i].
^ E[HK{X^^^)\A,] -
d,
In this way,
E[A,\A,-i].
the sequence (£/,),<n
then
generated by
<2e-'^'.
Proof:
We
apply the martingale inequality (see Rhee and Talagrand
1
B =
where
^'h^^-^p^-c?^)
=1
1
max;-
||£'
[Hr=/c
[16])
^?l-'^^-]
/9
^i
^^^'^
Hoc
^ numerical constant.
'^
to prove that, for the Held-Karp lower bound,
D <
C2
The
goal
some constant €2-
for
is
We
need the following lemma.
first
Lemma
7
< A, <
7.
2\/2 for all
i
There exist constanis C3 and C4 such
2-
that,
E[A,\A,]<
^]
\/n
<
for k
—
—
A
i
<
n and k
<
n
—
\,
1
and
E[A]\A,]<—^.
n — V —
1
Proof:
The Held-Karp
lower
bound on
(A'l.
.
.
.
A',_i, A'.+j,
,
convex combination of 1-trees (Ti,...,T,), rooted
ing multipliers (Ai,...,A,).
A'j
and
(A'l,
2.
the 1-tree T.
in
(A',,(T(Tr))
.
.
The
.
,
For every Tr (r
multiplier associated to T/
<
2IA',-
—
,
.
A'„) can be viewed as a
at vertex A'j, with correspond-
we add the two edges
1,...,^)
This, produces a
Xy.
is still
new
1-tree T/
unchanged and the degree of
(A'j,A',)
spanning
A',
being
Because the degrees remain unchanged,
a feasible solution to P({A'i,
.
,
.
.
A'„}).
As a
result,
r=\
r=l
Thus, A,
=
A'„) with the degree of each vertex
=
.
Let cr(T) denote one of the two vertices adjacent to
and delete [Xj,a{Tr)).
we have constructed
.
HK, + 2\X,-Xj\
A'jl for
every j
^
i.
Hence A, < 2v2
with the monotonicity of the bound, proves the
19
first
for
any
i,
wliich, together
part of the lemma. Moreover,
.
when
k
<
<
fc
<
j
i
<
n,j
E[A]\Ak] <
7^
E[v]
we have that A, < u where u
I,
independent of
is
^
As Pr{u >
E[u'^].
(see
Karp and
Cs
-
(1
£[11"^]
'
we
2min{|,Y,
get that £'[A,|.4fc]
ai^)"-/:-!
Steele [14]),
and
VrT^nr^
<
we
.4^,
=
< exp(-a(n -
<
k
—
Xj\
and
-£"[11]
-
:
l)t^) for
easily get that
C4
<
-
n-
k
-\
some constants C3 and C4, which proves the second part of the lemma.
for
As
in
<
Since u
?'}.
some constant q
n—
<
n and k
a corollary of the
Rhee and Talagrand
1.
E[dj\Ak]
< Cs
2.
E[d^\A,,]
<
for
^^
above lemma and following exactly the same techniques as
we can
[16],
any k
easily prove that
<i <
k<i<n-h
for
where C5 and Ce are constants. As
=
E-^'i-^/c
t=k
B =
max;.
a result,
E[dl\A,]+
we can bound B
E[dJ\Ak]
Yl
+
as follows:
E[dl_,\A,]
3C5
+
Ce
n
-
k
-I
1/2
H^
[E"=;-
(n
<
^?l--iA-] ill,
_
C2.
t
-
2)
Letting 7
= j^,
the theorem
fol-
D
Applying theorem
Pr||//A"(A"("')
6 with
-E
The complete convergence
the fact that
We
E[dl\A,]
< 3C5 + Ce = C2
lows.
5
+
k<t<n-l
<
Hence,
n and
t
=
e-y/n,
we
[//A'(A"<"')]|
of the
E[H K{X''^')]/\/n.
find that
> €/"} < ie''"^
(29)
Held-Karp lower bound now follows from (29) and
tends to /?hA' as n tends to
infinity.
Concluding remarks
analyzed probabilistically the Held-Karp lower bound for the TSP. Our result
corroborates the observation that the lower bound
20
is
very close to the length of the
optimal tour
in practice.
We
would
like to
emphasize that we exploited the combi-
natorial interpretation of the Held-Karp lower
Euclidean functionals.
We
bound and the theory
of subadditive
believe that the idea of combining polyhedral charac-
terizations with probabilistic analysis has the potential to lead to very interesting
results.
21
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24
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