Dewey
)28
1414
3.
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
ORDER STATISTICS AND THE LINEAR ASSIGNMENT PROBLEM
by
J.B.G. Frenk*
M. van Houweninge***
A.H.G. Rinnooy Kan ** ***
Sloan School of Management
MIT
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
ORDER STATISTICS AND THE LINEAR ASSIGNMENT PROBLEM
by
Frenk*
van Houweninge***
A.H.G. Rinnooy Kan ** ***
J. B.C.
M.
Sloan School of Management
MIT
Working Paper #1620-85
I
FEB 2 2
i?c.
HJ>
ORDER STATISTICS AND THE LINEAR
ASSIGNMENT PROBLEM
J.B.G. Frenk*
M.
van Houweninge***
A.H.G. Rinnooy Kan ** ***
Abstract
Under mild conditions on the distribution function F, we analyze
the asymptotic behavior in expectation of the smallest order statistic,
both for the case that F is defined on
F is defined on
(0,
«')
.
(-°o,
+°°)
and for the case that
These results yield asymptotic estimates of
the expected optimal value of the linear assignment problem under the
assumption that the cost coefficients are independent random variables
with distribution function
"
**
***
F.
Department of Industrial Engineering and Operations Research,
University of California, Berkeley.
Sloan School of Management, M.I.T., Cambridge, MA.
Econometric Institute, Erasmus University, Rotterdam.
1.
INTRODUCTION
Given an n x n matrix (a
^zS
to find a permutation
^
n
)
,
the linear assignment problem (LAP),
that minimizes
problem], which has many apolications
variety of algorithms (see, e.g.,
a.
E.
,
1=1
This classical
...
lu^Ci,
.
can he solveH pfficlentlv hv
.
(Lawler 1976).
is
It can be
a
conveniently
viewed as the problem of finding a minimum weight perfect matching in a
-'.
complete bipartite graph
analysis of the value
.
Z of
r
Here we shall be concerned with a probabilistic
the LAP, under the assumption that the
are independent, identically distributed (i.i.d.)
coefficients a
ij
random variables with distribution function
We shall be particularly
F.
interested in the asymptotic behavior of
EZ =
Emin^,_3
n
(D
^i=l£i^(i)
Previous analysis of this nature have focused on several special
choices for
EZ^
= 0(1);
In the case that a., is
F.
-13
the initial upper bound of
recently improved to
distributed
,
2
(Karp 1984).
(n log n)
EZ =
3
uniformly distributed on
(0,
1),
on the constant (Walkup 1979) was
In the case that -a.,
is exponentially
(Loulou 1983)
We shall generalize the above results by showing that, under mild
conditions on
F,
EZ^
is asymptotic to nF
(1/n)
.
The interpretation of
this result is that the asymptotic behavior of EZ^/n is determined by
that of the smallest order statistic.
In Section 2, we establish lower
and upper bounds on the expected value of this statistic, that may be of
interest on their own.
in
In Section 3, we apply the technique developed
(Walkup 19 79) to these bounds to arrive at the desired result.
As we
shall see, the condition on F under which the result is valid, is in
a sense both a necessary and a sufficient one.
:
ORDER STATISTICS
2.
Suppose that X.
variables with distribution function
X.
(U.), where the U.
F
d.
random
is a sequence of i.i.d.
..., n)
(1=1,
is well known that
It
F.
are independent and uniformly distributed
The smallest order
on (0,1), and where F~ (y) = inf {v|F(v')>y}.
Y.
statistic (i.e., the minimum) of random variables —1
will
..., Y
,
—
'
be denoted by Y,
— l:n.
We first consider the cast that
lim
F"^(l/n) = --
(2)
under the additional assumption that
/|xJF(dx)
<
*
(3)
.
We start by deriving an upper bound on EX^
X
1
Lenrnia
•
n
•
(F defined on (-°°,+°°))
n
<
EX,
—1
P roof
:
:
(1-(1- -)
n
F"^(^)
n
°°
+ n (l-F(O))""^
)
/ X F(dx)
(4)
o
We observe that
EX,
—1 n
= E min {F
(U,
—
:
= EF ^(U,
l:n
Let V. = max {U^, 1/n}
)
,
...,
F
(U
—
)
(5)
)
(i=l,
..., n)
.
Clearly, EF-1(U^ .^)<EF-1(V^
Hence,
^^"'^^l:n)
^
F-l(l/n) Pr (V^^^=l/n} +
U^^V^.J
I
^
^/^)
=
—1 :n
1
F ^(1/n)
_,
>l/n}) + n / F ^(x)
(1-Pr {U,
l:n
1/n
,
-3-
_,
(l-x)""
dx
(6)
.^)
Now
(2)
and (3) imply that the latter term is bounded by
1
n
^
(1-x)"
F ^(x)
/
dx <
F(0)
^
-1
n (l-F(O))"
-1
F
/
dx =
(x)
F(0)
n (l-F(O))"
Together,
x F(dx).
/
(7)
D
and (7) imply (4).
(6)
Since l-F(O)
'
<
we obtain as an immediate consequence that
1,
—4^^^^
lim inf
>
-
1
-
(8)
(1/n)
F
of the same form (and thus an upper
To derive a lower bound on EX,
—l:n
bound on EX,
-\L :n
(1/n)), an assumption is needed on the rate of
/F
decrease of F when X
->
-
<=°)
of positive decrease at -
for some a
>
1.
°°
We shall assume that F is a function
.
,
i.e.
It can be shown
that
,
(De Haan & Resnick 1981)
that this
condition implies that
F(-x)/F(-ax)
In (lim inf
a(F) = lim
,,^.
:;
a-«°
exists and is positive."
In a
The condition is satisfied, for instance
when (F(x) decreases polynomially (0
<
-4-
oc
(F)
<
«>)
or exponentially
•
=
(ct(F)
when x
fast
«=)
equivalent with (De Haan
Resnick 1981)
—
lim sup
—
;
y
>
&
Condition (9) implies and is
°°.
F"^ (1/ay)
1
with a
-
->
^
-^
^
<
(11)
F-^ (1/y)
Again,
1.
F~^ (l/ay)/F"^ (1/y)
In (lim sup
lim
»
^—
a^
'^^^^
In a
g(F) = l/c((F)
can be shown to exist and to be equal to
Theorem
1
defined on (- ^, +
(F
EX
-1="
lim sup
<
.
°°))
°°
(13)
-1
F ^(1/n)
n^<°
if and only if F is a function of positive decrease at -
with a(F)
Proof
.
>
«=
1.
We note that
EF~^(U
= n
)
f"^(x)
/
(1-x)''"-'-
dx + n
/
f"-'-(x)
(l-x)"""^ dx.
F(0)
(14)
The latter term is bounded by
oo
n(l-F(0)
)""'-
/
X F(dx)
(15)
and hence
lim
1
n /
g(0)
F
"
(x)
(l-x)"^
''"
dx
(16)
F ^
(1/n)
-5-
.
If nF(0)
>
1,
the former term is bounded from below by
F(0)
n
_
/
F
(x)
(1-x)^ dx >
F
(x)
exp(-nx) dx =
F
(x/n)
1 - F(0)
F(0)
n
/
1 - F(0)
1
/
1
exp(-x) dx +
1 - F(0)
nF(0)
/
1
1 -
F(0)
F"-^(x/n)
exp(-x) dx
(17)
1
The monotonicity of F
Implies that, for large n, the latter term
is at least as large as
F ^(1/n)
1 -
Also,
F(0)
>
and (Frenk 1983, Theorem 1.1.7) imply that there
1
exist constants B>0 and
X
£
6 e
(0,1)
such that for sufficiently large n and
(0,1)
<
-jlh^M.
F
6f.
(18)
1
a(F)
(11),
exp(-x) dx
/
(12)),
1
/
,
B^-e
(19)
\l/n)
so that,
for sufficiently large n,
_1
F
(x/n)
exp (-x) dx
1
< B /
-6-
.g
X
exp(-x) dx
<
«=
(20)
Together,
and (18) imply (13)
(20)
Now, suppose that
(13)
is satisfied,
F(°>
i.e., that
-1
n /
F
lim sup
(1-x)
(x)
n-1^
dx
<
-,
°°
(21)
F-^ (1/n)
If a
<
nF(0)
nF
then
,
(a/n)
dx
(1-x)
/
n
>
F "^(x)
/
dx
(1-x)
(22)
and hence
lim sup
Hence (cf.
^±""1^1
"^
0(—
.-=
^)
-,
F-\l/n)
F is of strict decrease with a(F) > 1, and all
(11))
that has to be shown is that a(F)
show that a(F) =
?^
F
Thus, it is sufficient to
1.
-1
"(x/n)
-1
dx
/
lim sup
F ^(1/n)
In (De Haan & Resnick 1981)
'^(z
F
li'\-«.
(1/xn) x
F
)
> z
(1/xn
—,
F
dx
(24)
(1/n)
it is shown that there exists a sequence
(z > 1)
such that
)
—
-2
= ^
1
and a function
'
implies that
1
1
n,
(23)
l-exp(-a)
= ^(x)
>
X
25)
^'^^
F~\l/n^)
for almost every x
>
1,
i.e., except in the (countably man>) points x
where ^ is discontinuous.
X
m
,
with X
£
m
But this implies the existence of a sequence
(2m,2m+l), such that for all N
-7-
-1
F
/
1
lim sup,
'm=l
and Theorem
1
-2
X
)
dx
> Z
^
m=l
^ (X
m
)
(-i
X
x
^
,
(26)
J
which goes to + » when N
Lemma
^
;
2m+2
\^
(1/xn
^
->
D
.
imply that, under conditions (2) and (3), the
1
following statements are equivalent:
(i)
F is a function of
(ii)
1-e
<
lim inf
n
positive decrease at -
^-
< lim sup
—l:n
'^n
—^,
F ^(1/n)
«>
with a(F)
«>
^-
"°
— l:n
—^
>
1;
<
<=°.
(1/n)
F
Now let us deal with the (much simpler) case that
lim
n
F~\-^)
«
->
=
(27)
n
No additional assumption such as (3) is needed.
Lemma
2
(F defined on (0,
.
EX,
—
1
:
>
n
^—
n
F"^
=°)
(1 -
)
—
n
)"
(28)
Define
Proof:
,
1/n
if U,
>
1/n
if U.
<
1/n
—1
(29)
—1
Then
EX,
— l:n
= EF
Nu,
— l:n
)
>
EF
Sw,
— l:n
)
=
F ^(-)
n
(1
-
h""
n
(30)
D
Again,
let us assume that F
lim inf
^
xiO
^}^\
F(ax)
satisfies (11), or
.:.
1
-8-
that,
equivalently
(31)
)
^
for some a
<
Thus, F being defined on (0,
1.
°°)
the function is
,
assumed to be of positive decrease at 0.
Theorem
(F defined on
2
(0,
«>) )
EX
~^-"
lim sup
<
(32)
oo
(1/n)
F
if and only if F is a function of positive decrease at 0.
Proof
:
1
-l:n
1
_1
n / F
^
_,
= n /
Q
EX,
(x)
F
,
(1-x)""
(x)
dx
<
exp (-nx) dx =
-1
(x/n) exp
F
/
(-x)
dx
(33)
As before, we split the integral in two parts, corresponding to
x
e
(0,1)
and x
£
(0,1)
and x
e
(l,n)
respectively.
The first part is
bounded by
F
n/n)
/
exp (-x) dx
As in the proof of Theorem
1,
(34)
we can bound
-1
/
F
(x/n)
exp (-x) dx
(35)
1
_
F
\l/n)
-9-
This yields the proof of (32).
by invoking (12).
Conversely,
implies that, since for
(32)
1
F~^(a/n)
/
1
(1-x)""-^ dx < /
<
a
1
<
,
^
f"-^(x)
(l-x)"""^ dx,
(36)
a/n
we may conclude that
icroo
F"-^(a/n)
/
lim sup
T
(1-x)"
dx
a/n
.
<
(37)
<=°
;
F
\l/n)
D
which leads directly to (11).
Hence, in the case that
(2 7)
holds, we have the following two
equivalent conditions:
(i)
F is a function of positive decrease at 0;
-
EX
—
EX^
1
(ii)
< lim inf
^ -^^
lim sup
^
Lii^
-
F ^(l/n)
F -^(1/n)
We note that no condition on a(F) occurs in (i)
the case that F is defined on (c,
<
°°)
.
We also note that
for any finite
reduced to the above one.
-10-
c
can easily be
3.
THE LINEAR ASSIGNMENT PROBLEM
Our analysis of the linear assignment problem is based on a
technique developed in (Walkup 1981)
Very roughly speaking, this
.
weighted bipartite graph all edges but
randomly
if in a complete,
approach can be summarized as follows:
few of the smaller weighted
a
ones at each node are removed, then the resulting graph will still
contain a perfect matching with high probability.
In that way we
derive a probabilistic upper bound on the value Z of the LAP.
More precisely, assume that the LAP coefficients
are i.i.d.
a^.
.
(i,
j
= l,
random variables with distribution function F.
.
.
It
.
,n)
is
possible to construct two sequences b.. and c.. of i.i.d. random
variables such that
a,
.
-ij
1
- min
{b
.
.
-ij
,
c,.}
-ij
(38)
Indeed, since we desire that Pr {a.
.
^ x}
Pr {min {b.., c..} > x} = Pr {b.. > x}
-ij -ij
-ij
distribviLion function F of b
.
.
and
c.
=
Pr (c,. > x}, the common
-ij
will have to satisfy
l-F(x) = (l-F(x))^
(39)
so that
F"^(x) = F"^(l-(l-x)^)
(40)
For future reference, we again observe that b
d F~ (V
-iJ -ij
.
c. =F
-ij
.
)
and
(W..), where V.. and W.. are i.i.d. and uniformly
^ distributed
-ij
-ij
-ij
-11-
If we fix any pair of indices
on (0,1).
statistics of V
(j=l,
..., n)
are independent of and distributed
as the order statistics of W..(i=l,
order statistics bv —
V,
1 n
'
< V.
—/ n
<
..., n)
V
<
...
—n
:
:
(i,j), then the order
:
we shall denote these
;
and W,
—1
n
:
n
< W^
—2
:
<... < W
—n
n
:
respectively.
Now, let G
and T={t
(t.,
n
,
....
t
n
}
with weight b.. on arc
(s^
-ij
For anv realization b
s.).
J
be the complete directed bipartite graph on S={s^
.
.
(co)
,
iJ
1
c
.
.
t.) and c..
,
i
(w)
,
and by removing arc (t., s.) unless
smallest weights at
t
.
.
c
.
.
..., s
}
on arc
we construct G (d,w) by
n
iJ
removing arc (s., t.) unless b..(a)) is one of the
at s.
-11
J
,
(w)
d
'
smallest weights
is one of the d
Let us define P(n,d) to be the probability
that G (d) contains a (perfect) matching.
A counting argument can
now be used to prove (Walkup 1981) that
1-P(n,2)
<
1-P(n,d)
<
i
(41)
jn
^
iZ2
(d+l)(d-2)
(-^)
(d>3)
n
(42)
We use these estimates to prove two theorems about the asymptotic value
of EZ.
Again, we first deal with the case that
lim
F~-'-(l/n)=
-«>
(43)
under the additional assumption that
/|xlF(dx)
(44)
<«>
-12-
n
Theorem
3
(F defined on (-°°, -H»)
If F is a function of positive decrease at
3
^
n
)
2
^-
lim infr
.
-,
<
.
2e^
Proof.
-°°
nF
with a(F)
<
>
then
1,
^-
lim sup ^_
-,
•
nF
(1/n)
<
(1/n)
Since
EZ
— > nEa,
(46)
—1 :n
is an immediate consequence of Theorem 1.
the upper bound in (45)
For the lower bound we apply (41) and (42) as follows.
Obviously,
EZ
(2) contains a matching)
— = P(n,2) E(Z|G
—
—
+ (1-P(n,2)) E(z|g
—
—
(2)
does not contain a matching)
(47)
The second conditional expectation is bounded trivially by
aEa
-n
is
=
0(n
:n
2
(cf.
)
(44)).
The first conditional expectation
bounded by
nEF"'^(max {V_
-z :n
,
W„
})
-I :n
(48)
Hence it suffices to prove that
EF
,
.
lim
.
W
}
-Z:n-^:n
(max {¥„
mf
r
,
F~
(x
n
)
= F~
(1/n)
,,
(1 -
—
>zo
i
i
,
,„
)
2e^^^
F ^(1/n)
To this end, define x =1 - (1-1/n)
n
>
1/2
and note from (40) that
so that
-13-
(49)
°°
(45)
^^-''
EF-l(max {V„
-2:n
W
}) <
-w:n
,
< X
(1/n) Pr {V„
F
-2:n
E(F~l(max
{V.,
-2:n
,
n
< x
W„
,
-z:n
W
})
-2:n
I
n
+
}
,
max tV„
-2:n
,
W.
-2:n
)
,
}
a x
n
(50)
To bound the first term, note that
P^
=
%:n^^n' ^2:n^^n^
,^n
,n,
(I,
„
(,
k=2
k
X
)
k ,.
>n-k,
(1-x )
)
n
n
(1-x )" - nx (1-x
(1 -
n
n
which tends to (1-3/ (2e
The second term in (50)
1/2
2
))
2
=
)""^^
(51)
n
as
n->«°.
is equal to
1
/
X
F-l(x)d(Pr{V.
— z :n
<
x}^) =
n
1
_
_2
< x} x(l-x)"
2n(n-l) / F-l(x) Pr{V,
-Z :n
x
n
After a transformation x=l-(l-y)
(52)
1/2
(cf.
dx
(52)
(40)), we find that
for large n is bounded by
n(n-l) /
F
^y) (l-(l-y)^^^)
(1-y)
^""^^
^^dy
<
F(0)
n(n-l)
F"^(y)dy,
(l-F(O))^''"^^^^ /
F(0)
(53)
C
thus completing the proof of (49).
-14-
Again, the case that
liin^_^ F"^(l/n) =
(54)
is much simpler to analyze.
Theorem
4
.
defined on (0,
(F
o°)
If F is a function of positive decrease at 0,
EZ
EZ
<
(55)
^ lim sup
lim inf
nF ^(1/n)
Proof.
then
We have, for all
d >
nF
3,
(1/n)
that
EZ < (1-P(n,d)) E(Z|G (d) does not contain a matching) +
+ P(n,d) E(z|g
2
<
As in
n
(d)
does contain a matching)
2
-d-2^-d +d+4
(^19^),
^g-_i
_^
(d'^
we use constants B,
^^^
-d:n
-a:n
J
to bound
>
3
^^
-d^+d+4, -1,,,
^ -d^+d+3+3
u ^u
and choose 1d such
that^
/nF
(1/n) by Bn
.
,
,
,
,
_2 -
-
-
-d +d+3+6<0.
For this value d, we bound EF"! (max {V^
-d:n
,
W,
}) as
-d:n
before by
d (-) / F-l(x)
\d/
Q
Pr{V-d:n
<
x}
(l-x)""V"^dx
These two bounding arguments yield that lim sup
The lower bound on lim inf
EZ/nF
(1/n)
EZ/nF
(1/n) <
follows from (46).
°o.
C
The conditions of strict decrease on F turned out to be necessary as
well as sufficient to describe the asymptotic behavior of the smallest
order statistic (Theorems
above theorems.
and 2)
that play an important role in the
It can easily be seen that this condition is necessary
and sufficient in Theorem
holds for Theorem
1
3.
4
as well, and one suspects that the same
Theorems
3
and 4 capture the behavior of the expected LAP value for a
wide range of distributions.
To derive almost sure convergence results
under the same mild conditions of F, the results from [Walkup 1981]
would have to be strengthened further.
For special cases such as the
uniform distribution, however, almost sure results can indeed be
derived quite easily (see [Van Houweninge 1984]).
Acknowledgements
The research of the first author was partially supported by the
Netherlands Organization for Advancement of Pure Research (ZWO) and
by a Fulbright Scholarship.
The research of the third author was
partially supported by NSF Grant ECS-831-6224 and by a NATO Senior
Scientist Fellowship.
-16-
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