Pipage Rounding: a New Method of
Construting Algorithms with Proven
Performane Guarantee
y
A. A. Ageev
M. I. Sviridenko
z
Abstrat
The paper presents a general method of designing onstant-fator
approximation algorithms for some disrete optimization problems
with assignment-type onstraints. The ore of the method is a simple
deterministi proedure of rounding of linear relaxations (the pipage
rounding). With the help of the method we design approximations
algorithms with better performane guarantees for some well-known
problems inluding maximum overage, max ut with given sizes
of parts and some of their generalizations.
approximation algorithm, performane guarantee, linear relaxation, rounding tehnique, maximum overage, max ut
Keywords:
1
Introdution
Rounding of linear relaxations is one of the most eetive tehniques in designing approximation algorithms with proven performane guarantees. Two
points are most important for those who aim at suess when applying this
tehnique: a good hoie of integer program formulation and a good hoie of
rounding sheme for its linear relaxation. Seleting a good integer program
formulation highly depends on the nature of the problem and and is often a
matter of intuition. As for rounding methods, there are two basi dierent
Parts
of this paper appeared in preliminary form in Proeedings of IPCO'99 [AS99℄
and ESA'00 [AS00℄.
y Sobolev
z IBM T.
Institute of Mathematis, Novosibirsk, Russia, email: ageevmath.ns.ru
J. Watson Researh Center, P.O. Box 218, Yorktown Heights, NY, email:
svirius.ibm.om
1
approahes - deterministi and random. When applying random rounding
methods one enounters the additional problem of derandomization. This
problem may prove to be extremely diÆult or quite intratable. For example, a diÆult point in implementation of random roundings is that of
produing feasible solutions in the ase of some simple equality onstraints
like a ardinality or a budget one. Though by using some triks this diÆulty
ould be overome, the resulting algorithm will be too sophistiated to admit
derandomization.
The main result of this paper is a simple deterministi rounding method
| we all it the pipage rounding | espeially oriented to takle some problems with assignment- and budget-type onstraints.
The paper is organized as follows. In Setion 2 we give a general desription of the pipage method.
In Setion 3 we demonstrate relatively simple but eetive appliations
of the method. We apply the method to the maximum overage problem.
The result is the (1 (1 1=k)k )-approximation algorithm for the maximum
overage problem where k is the maximum size of the subsets. The performane guarantee of (1 (1 1=k)k ) improves on that of 1 e in eah ase
of bounded k, established by Cornuejols, Fisher, and Nemhauser [CFN77℄
for the greedy algorithm.
In Setion 4 we onsider the maximum k-ut problem in hypergraphs with
the additional onstraint that the sizes of the parts (sides) of the partition
must be equal to given numbers (hypergraph max k-ut with given sizes of
parts or, for short, hyp max k-ut with gsp). By applying the pipage
rounding method, we prove that hyp max k-ut with gsp an be approximated within a fator of minfjSj : S 2 E g of the optimum, where E is the
edge set of the hypergraph and r = 1 (1 1=r)r (1=r)r . This gives a
0:5-approximation for the ase of graphs and a (1 e )-approximation for
the ase of hypergraphs with edges of size at least 3. We also show that
the performane guarantee of our algorithm in the later ase is best possible
unless P = NP .
In Setion 5 we present an essentially more sophistiated appliation of
our method. We ombine the pipage rounding with the partial enumeration
method of Sahni [Sa75℄ to design an algorithm with a better approximation
ratio for a knapsak generalization of maximum overage.
In the last setion we apply the pipage rounding to a sheduling problem.
In this problem, for a set of unrelated parallel mahines, it is required to nd
a shedule minimizing the total weighted ompletion time provided that for
eah mahine the number of jobs that it an proess is bounded by a given
number. The version without restritions on the numbers of jobs was studied
earlier by Skutella [Sk98, Sk99℄ who proved that it an be approximated
1
1
2
within a fator of 3=2 of the optimum. Applying pipage rounding we obtain
that the same approximation bound also holds for the (more general) problem
with the restritions on the numbers of jobs. A distintive feature of this
appliation is that it deals with a minimization problem while the previous
appliations treated maximization ones.
2
Pipage rounding: a general desription
In this setion we present a general desription of the rounding proedure,
that underlies all the algorithms presented in this paper. The desription,
being far from most general, still overs the most important partiular ases
treated so far. We will also dwell on spei features of the sheme orresponding to some important speial ases.
We start with formulating a general problem. Suppose we are given a
bipartite graph G = (U; W ; E ), a funtion F (x) dened on the rational
points x = (xe : e 2 E ) of the jE j-dimensional ube [0; 1℄jEj and omputable
in polynomial time, and p : U [ W ! Z . Consider the binary program
max F (x)
(1)
X
s. t.
xe pv ; v 2 U [ W;
(2)
+
e2Æ(v)
0 xe 1; e 2 E;
(3)
xe 2 f0; 1g; e 2 E:
(4)
Denote by F the optimal value of (1){(4). We all a solution x satisfying
(2){(3)
if some of its omponents are non-integral. Notie that the
frational relaxation (1){(3) is not assumed to be polynomially solvable.
Let x be a frational solution to (1){(3). We further desribe a rounding
algorithm, alled PIPAGE, that onverts x into an integral vetor x satisfying (2){(4). After that we will formulate some suÆient onditions on F
guaranteeing F (x) F (x).
Algorithm PIPAGE. PIPAGE onsists of uniform steps at eah of
whih a urrent frational solution x transforms into a new solution x0 with
smaller number of non-integral omponents. We proeed to desription of
the general step. Let x be a urrent vetor satisfying (2){(4). If x is integral, then the algorithm terminates and outputs x. Suppose now that x is
frational. Construt the subgraph Hx of G with the same vertex set and
the edge set Ex dened by the ondition that e 2 Ex if and only if xe is
non-integral. If Hx ontains yles, then we set R to be suh a yle. If Hx
frational
3
is a forest, then we set R to be a path of Hx whose endpoints have degree 1.
Sine Hx is bipartite, in both ases R an be uniquely represented as the
union of two mathings. Let M and M denote those mathings. Dene a
new solution x("; R) by the following rule: if e 2 E n R, then xe("; R) oinides with xe, otherwise xe ("; R) = xe + " when e 2 M , and xe("; R) = xe "
when e 2 M . Set
" = minf min xe ; min (1 xe )g
e2M
e2M
1
2
1
2
1
and
1
2
"2 = minf min (1 xe ); min xe g:
e2M1
e2M2
"1 ; R), x2 = x("2 ; R). Set x0 = x1 if F (x2 ) < F (x1 ) and x0
Let x = x(
=x
otherwise.
We all the above transformation the
x
R.
Dene '("; x; R) = F (x("; R)).
Corretness and analysis. Consider an arbitrary step of PIPAGE.
Note rst that Ex is a proper subset of Ex and hene x0 has smaller number of
frational omponents than x. This means that PIPAGE terminates after at
most jE j steps, and its output, x, is integral. Reall now that by assumption
eah pv is an integer and by desription, if R is a path, the endpoints of R have
degree 1 in H . It follows thatPx(") satises (2){(3) for every " 2 [ " ; " ℄.
Let v 2 P
U [ W and qv = e2Æ v xe . Observe that if v is not an endpoint
of R, then e2Æ v xe(") = qv for all ". Assume that qv is an integer. Then
v annot have degree 1 in Hx andPtherefore it annot be an endpoint of R
when R is a path. Consequently, e2Æ v xe(") = qv whenever " 2 [ " ; " ℄.
Assume P
now that qv is non-integral. Then, by the denition of " and " ,
bqv e2Æ v xe (") bqv + 1 whenever " 2 [ " ; " ℄. Thus we have
established the following relations between x and x.
P
P
Property 1.
e
e = Pe2Æ v xe
e2Æ v x
e2Æ v x
P
Property 2.
e
e2Æ v x
1
rounding of
2
over
0
1
2
( )
( )
1
( )
1
1
( )
If
( )
is integral, then
If
( )
is non-integral, then
b
X
e2Æ(v)
xe X
e2Æ(v)
( )
xe b
X
e2Æ(v)
2
2
2
( )
.
xe + 1:
Sine all pv are integers, Properties 1 and 2 imply that x satises (2).
Thus, after performing at most jE j steps PIPAGE transforms x into a solution x obeying (1){(4). Sine eah step an be learly implemented in
polynomial time, it follows that the overall running time of the algorithm is
polynomially bounded.
4
Assume now that there is a polynomialtime algorithm for nding a frational feasible solution x to (1){(3) suh that
F (
x) CF ;
(5)
where C is a positive onstant. If, in addition, x|the output of PIPAGE|
satised
F (
x) F (
x);
(6)
we would obtain a C -approximation algorithm for solving (1){(4). The
inequality (6) learly holds if at eah step of PIPAGE F (x0) F (x), whih in
turn is the ase if '("; x; R), treated as a funtion of ", attains its maximum
at an endpoint of the interval [ " ; " ℄. Thus we arrive at the following
ondition that guarantees (6).
Denition 1. We say that the program (1){(4) satises the "
if the funtion '("; x; R) is onvex in " for every feasible solution
x to (1){(3) and for all paths and yles R of the graph Hx .
We desribe now a general way of onstruting algorithms satisfying (5).
Assume that one an assoiate with F (x) another funtion L(x) that is dened and polynomially omputable on the same set, oinides with F (x) on
binary x satisfying (2), and suh that the program
max L(x)
(7)
X
s. t.
xe pv ; v 2 U [ W;
(8)
The pipage rounding sheme.
1
2
-onvexity
ondition
e2Æ(v)
0 xe 1; e 2 E
(9)
(heneforth alled the
) is polynomially solvable.
Denition 2. We say that the funtions F and L satisfy the F=L
if
F (
x) CL(
x);
(10)
where x is an optimal solution to (7){(9) and C is a positive onstant.
Sine L(x) F , this ondition implies (5). Thus if both of the above
onditions hold (we will further refer to them as the
) one
an onstrut a C -approximation algorithm for solving (1){(4).
Remark 1. The pipage rounding is essentially based on Properties 1 and
2. It is easy to observe that both properties hold for some modiations of
the problem (1){(4), whih expose the atual range of appliability of our
nie relaxation
lower
bound ondition
main onditions
5
method. In partiular, in (2) an be replaed by =. Moreover, after a few
obvious alterations in the desription (namely, the "and F=L
bound onditions should be replaed by the "and F=L
bound
onditions, respetively) the method is appliable to the problem obtained
from the above by onverting max in (1) into min and in (2) into or =.
onvexity
onavity
3
lower
upper
Maximum overage
In the maximum overage problem (maximum overage for short), given
a family F = fSj : j 2 J g of subsets of a set I = f1; : : : ; ng with assoiated
nonnegative weights wj and a positive integer p, it is required to nd a
subset X I with jX j = p so as to maximize the total weight of the sets in
F having nonempty intersetions with X . The polynomial-time solvability of
maximum overage learly implies that of the set over problem and so it
is NP -hard. In a sense maximum overage an be treated as an inverse of
the set over problem and like the latter has numerous appliations (see, e. g.
[Ho97℄). It is well known that a simple greedy algorithm solves maximum
overage approximately within a fator of 1
(1 1=p)p of the optimum
(Cornuejols, Fisher and Nemhauser [CFN77℄). Feige [Fe98℄ proves that no
polynomial algorithm an have better performane guarantee provided that
P6=NP. Another result onerning maximum overage is due to Cornuejols,
Nemhauser and Wolsey [CNW80℄ who prove that the greedy algorithm almost
always nds an optimal solution to maximum overage in the ase of
two-element sets. We show below that maximum overage an be solved
in polynomial time approximately within a fator of 1 (1 1=k)k of the
optimum, where k = maxfjSj j : j 2 J g. Although 1 (1 1=k)k like
1 (1 1=p)p an be arbitrary lose to 1 e , the parameter k looks more
interesting: for eah xed k (k = 2; 3; : : :) maximum overage still remains
NP-hard. E. g., in the ase when k = 2, whih is the onverse of the vertex
over problem, the performane guarantee of the greedy algorithm has the
same value of 1 e [CNW80℄, whereas our algorithm nds a solution within
a fator of 3=4 of the optimum. Ultimately, the performane guarantee of our
algorithm beats the performane guarantee of the greedy algorithm in eah
ase of bounded k and oinides with that when k is unbounded. Note also
that our result is similar in a sense to the well-known result [BE81, Ho82℄
that the set over problem an be approximated in polynomial time within a
fator of r of the optimum, where r is the maximum number of sets ontaining
an element.
Let J = f1; : : : mg. maximum overage an be equivalently reformulated as a onstrained version of max sat over variables y ; : : : ; yn with m
1
1
1
6
lauses C ; : : : ; Cm suh that Cj is the olletion of yi with i 2 Sj and has
weight wj . It is required to assign \true" values to exatly p variables so as
to maximize the total weight of satised lauses. Furthermore, analogously
to max sat (see, e. g. [GW94℄), maximum overage an be stated as the
following integer program:
1
max
m
X
j =1
X
s. t.
i2Sj
n
X
i=1
(11)
wj zj
xi zj ;
j
= 1; : : : ; m;
(12)
xi = p;
(13)
xi 2 f0; 1g; i = 1; : : : ; n;
0 zi 1; i = 1; : : : ; m:
(14)
(15)
It is easy to see that the relation \xi = 1 if i 2 X , and xi = 0 otherwise"
establishes a 1-1 orrespondene between the optimal sets in maximum overage and the optimal solutions to (11){(15). Note that the variables xi
determine the optimal values of zj in any optimal solution. Moreover, it is
lear that
overage is equivalent to maximizing the funtion
P maximumQ
F (x) = m
w
1
j
j
i2S (1 xi ) over all binary vetors x satisfying (13).
To see that that this non-linear problem is a speial ase of of the problem
(2)-(4) we dene U = fug and W = f1; : : : ; ng where u is a dummy element
orresponding to the ardinality onstraint. Observe also that the objetive
funtion (11) an be replaed by the funtion
=1
j
L(x) =
m
X
j =1
wj minf1;
X
i2Sj
xi g
of the variables x ; : : : ; xn , thus providing a nie relaxation of the form (7){
(9).
We now show that the funtions F and L just dened satisfy the "onvexity and F=L lower bound onditions.
The F=L lower bound ondition holds with C = (1 (1 1=k)k ) where
k = maxfjSj j : j 2 J g, whih is implied by the following inequality (used
rst by Goemans and Williamson [GW94℄ in a similar ontext):
1
1
k
Y
i=1
(1
yi ) (1
(1 1=k)k ) minf1;
7
k
X
i=1
yi g;
(16)
valid for all 0 yi 1, i = 1; : : : ; k. To make the paper self-ontained we
derive it below. By using the arithmeti-geometri mean inequality we have
that
k
k
Y
1
(1 yi) 1 1 kz ;
Pk
i=1
where z = minf1; i yig. Sine the funtion g(z) = 1 (1 z=k)k is
onave on the segment [0; 1℄ and g(0) = 0, g(1) = 1 (1 1=k)k , we nally
obtain
g (z ) 1 (1 1=k)k z;
as desired.
To hek the "-onvexity ondition it suÆes to observe that in this ase
' as a funtion in " is onvex beause it is a quadrati polynomial in ",
whose main oeÆient is nonnegative for eah pair of indies i and j and
eah x 2 [0; 1℄n. Thus by onretizing the general sheme desribed in the
introdution we obtain a (1 (1 1=k)k )-approximation algorithm for the
maximum overage problem.
We now demonstrate that the integrality gap of (11){(15) an be arbitrarily lose to (1 (1 1=k)k) and thus the rounding sheme desribed above
is best possible for the integer program (11){(15). Set n = kp, wj = 1 for all
j and let F be the olletion of all subsets of f1; : : : ; ng with ardinality k.
Then, by symmetry, any binary vetor with exatly p units maximizes L(x)
subjet to (13){(14) and so the optimal value of this problem is equal to
L = Cnk Cnk p. On the other hand, the vetor with all omponents equal to
1=k provides an optimal solution of weight L0 = Cnk to the linear relaxation
in whih the objetive is to maximize L(x) subjet to (13) and 0 xi 1
for all i. Now it is easy to derive an upper bound on the ratio
=1
L
L0
=
Cnk
= 1
= 1
1
Cnk
Cnk
p
(n
p)! k!(n k)!
k!(n p k)!
n!
n p n p 1 n p k + 1
:::
n
n 1
n k+1
n p n p 1 n p k + 1
:::
n
n
n
1 k1 1 k1 n1 1 k1 n2 : : : 1 k1
k
1 1 k1 k +n 1 ;
= 1
8
k + 1
n
whih tends to (1 (1 1=k)k ) when k is xed and n ! 1.
Remark 2. The algorithm and the argument above an be easily adopted
to yield the same performane guarantee in the ase of the more general
problem in whih the onstraint (13) is replaed by the onstraints
X
i2It
xi = pt ;
t = 1; : : : ; r
where fIt : t = 1; : : : ; rg is a partition of the ground set I and pt are positive
integers. It an be shown, on the other hand, that the worst-ase ratio of the
straightforward extension of the greedy algorithm annot be lower bounded
by any absolute onstant. So, our algorithm is the only algorithm with
onstant performane guarantee among those known for this generalization.
Remark 3. It an be easily observed that from the very beginning (and
with the same ultimate result) we ould onsider objetive funtions of the
following more general form:
F (x) =
m
X
j =1
wj
1
Y
i2Sj
(1
xi )
+
l
X
t=1
ut
1
Y
i2Rt
xi ;
where Sj and Rt are arbitrary subsets of f1; : : : ; ng, and ut, wj are nonnegative weights. The problem with suh objetive funtions an be reformulated
as the onstrained max sat in whih eah lause either ontains no negations
or ontains nothing but negations.
4
Hypergraph Max
k -Cut
with given sizes of
parts
In this setion we onsider Hypergraph Max k-Cut with given sizes of parts
or, for short, hyp max k-ut with gsp. An instane of hyp max k-ut
with gsp onsists of a hypergraph H = (V; E ), nonnegative weights wS on
its edges S , and k positive integers p ; : : : ; pk suh that Pki pi = jV j. It is
required to partition the vertex set V into k parts X ; X ; : : : ; Xk , with eah
part Xi having size pi , so as to maximize the total weight of the edges of
H , not lying entirely in any part of the partition (i.e., to maximize the total
weight of S 2 E satisfying S 6 Xi for all i).
Several losely related versions of hyp max k-ut with gsp were studied in the literature but few results have been obtained. Andersson and
1
=1
1
9
2
Engebretsen [AE98℄ presented an 0:72-approximation algorithm for the ordinary Hyp Max Cut problem (i.e., for the version without any restritions on
the sizes of parts). Arora, Karger, and Karpinski [AKK99℄ designed a PTAS
for dense instanes of this problem or, more preisely, for the ase when the
hypergraph H is restrited to have (jV jd) edges, under the assumption that
jS j d for eah edge S and some onstant d. Another known speial ase of
hyp max k -ut with gsp is MAX BISECTION. In this problem we should
split a regular graph into two equal parts and maximize a total weight of the
ut between them. Frieze and Jerrum [FJ97℄ obtained 0.64-approximation
algorithm based on semidenite programming relaxation.
By applying the pipage rounding method, we prove that hyp max kut with gsp an be approximated within a fator of minfjS j : S 2 E g
of the optimum, where r = 1 (1 1=r)r (1=r)r . By diret alulations
it easy to get some spei values of r : = 1=2 = 0:5, = 2=3 0:666,
= 87=128 0:679, = 84=125 = 0:672, 0:665 and so on. It is
lear that r tends to 1 e 0:632 as r ! 1. A bit less trivial fat is
that r > 1 e for eah r 3 (Lemma 2 in this paper). Summing up we
arrive at the following onlusion: our algorithm nds a feasible ut of weight
within a fator of 0:5 on general hypergraphs, i.e., in the ase when eah edge
of the hypergraph has size at least 2, and within a fator of 1 e 0:632
in the ase when eah edge has size at least 3. Finally, we show that in the
ase of hypergraphs with eah edge of size at least 3 the bound of 1 e
annot be improved, unless P=NP.
It is easy to see that an instane of hyp max k-ut with gsp an be
equivalently formulated as the following (nonlinear) integer program:
2
4
5
3
6
1
1
1
1
max
s. t.
F (x) =
k
X
t=1
n
X
i=1
X
S 2E
wS
1
k Y
X
t=1 i2S
xit
(17)
xit = 1 for all i;
(18)
xit = pt
(19)
for all t;
2 f0; 1g for all i and t:
(20)
The equivalene is shown by the one-to-one orrespondene between optimal
solutions to the above program and optimal k-uts fX ; : : : ; Xk g of instane
of hyp max k-ut with gsp dened by the relation \xit = 1 if and only if
i 2 Xt ".
By dening U = f1; : : : ; kg and W = f1; : : : ; ng we an see that the
xit
1
10
problem (17)-(20) is a speial ase of the general non-linear problem (2)-(4).
As a nie relaxation we onsider the following linear program:
max
s. t.
X
S 2E
zS
jS j
k
X
t=1
n
X
i=1
(21)
wS zS
X
i2S
xit
for all S 2 E;
(22)
xit = 1
for all i;
(23)
xit = pt
for all t;
(24)
0 xit 1 for all i and t;
(25)
0 zS 1 for all S 2 E:
(26)
It is easy to see that, given a feasible matrix x, the optimal values of zS in
the above program an be uniquely determined by simple formulas. Using
this observation we an exlude the variables zS and rewrite (21){(26) in the
following equivalent way:
max
L(x) =
X
S 2E
wS minf1; min(jS j
t
X
i2S
xit )g
(27)
subjet to (23){(25). Note that F (x) = L(x) for eah x satisfying (18){(20).
We laim that, for every feasible x and every yle D in the graph Hx
(for denitions, see Setion 2), the funtion '(") = F (x(")) is a quadrati
polynomial
Q with a nonnegative leading oeÆient. Indeed, observe that eah
produt Qi2S xit (") ontains at most two modied variables. Assume that
a produt i2S xit (") ontains exatly two suh variables xi t (") and xi t (").
Then they an have only one of the following forms: either xi t + " and xi t "
or xi t " and xi t + ", respetively. In either ase " has a nonnegative
oeÆient in the term orresponding to the produt. This proves that the
"-onvexity ondition does hold.
For any r 1, set r = 1 (1 1=r)r (1=r)r :
Lemma 1.
x = (xit )
(27) (23) (25) S 2 E
1
Then
1
be a feasible solution to
k Y
X
t=1 i2S
2
1
2
2
2
Let
1
xit
jSj minf1; min
(jS j
t
11
,
X
i2S
{
xit )g:
and
.
Proof.
ities
P
Let zS = minf1; mint(jS j
= max
t
qS
Note that
i2S xit
X
i2S
)g. Dene qS and t0 by the equal-
xit =
X
xit :
0
i2S
= minf1; jS j qS g:
Using the arithmeti-geometri mean inequality and the fat that
zS
k X
X
t=1 i2S
we obtain that
1
k Y
X
t=1 i2S
xit = 1
1
1
=1
=1
jS j
Y
xit
= jS j
XY
xit
0
i2S
P
i2S
xit
t6=t0 i2S
P
xit0 jS j X i2S xit
jS j
q jS j
S
P
q jS j
S
S
q jS j
S
jS j
jS j
(28)
jS j
jS j
t6=t0
j j
1
P
t6=t0
jS j
i2S xit
jS j
i2S xit0
jS j
jS j
P
jS j
qS jS j
jS j :
jS j
(29)
y
Let (y) = 1 1 jSy j
jS j .
jS j 1 qS jS j. Then by (28), zS = jS j qS , and hene by
(29),
k Y
jS j z jS j
X
S
1
xit 1
1 jzSSj
jS j = (zS ):
Case 1:
t=1 i2S
Sine the funtion is onave and (0) = 0, (1) = jSj, it follows that
1
k Y
X
t=1 i2S
xit jS jzS :
1 qS jS j 1. Here zS = 1. Sine (y) is onave and
(1) = (jS j 1) = jSj,
Case 2:
1
k Y
X
t=1 i2S
xit jS j:
12
0 qS 1. Again, zS = 1. For every t, set t = Pi2S xit . Note
that, by the assumption of the ase,
0 t 1;
(30)
and, moreover,
k
X
t = jS j:
(31)
Case 3:
t=1
By the arithmeti-geometri mean inequality it follows that
k Y
X
t=1 i2S
xit
k X
t jS j
t=1
jS j
k
(by (30))
X
jS j jSj t
(by (31))
= jS j jSjjS j:
t=1
Consequently,
1
Corollary 1.
jS j
1
xit 1 jS j
jS j
i2S
jS j
jS j
= 1 jS1 j
(jS j 1) jS1 j
1 jS j
jS j
j
Sj 1
1 jS j
(jS j 1) jS j
= jSj:
k Y
X
t=1
Let
x = (xit )
be a feasible solution to
(27) (23) (25)
,
{
. Then
F (x) (min jS j)L(x):
S 2E
The orollary states that the F=L lower bound ondition holds with
C = min jS j:
S 2E
Hene the pipage rounding provides an algorithm that nds a feasible k-ut
whose weight is within a fator of minS2E jSj of the optimum.
Note that = 1=2. We now establish a lower bound on r for all r 3.
2
13
Lemma 2.
For any
r 3,
r > 1
e 1:
We rst dedue it from the following stronger inequality:
r
1 1r < e 1 21r for all r 1:
Indeed, for any r 3,
1 1 1 r
=1
Proof.
1
r
rr
1
>1
rr
r
1 21r
+ 1r e2 rr1
e
1
(32)
1
=1 e
>1 e :
To prove (32), by taking natural logarithm of both sides of (32) rewrite it in
the following equivalent form:
1 + r ln(1 1r ) < ln(1 21r ) for all r 1:
Using the Taylor series expansion
1
1
ln(1
r
1 i
X
) =
we obtain that for eah r = 1; 2; : : : ;
1 + r ln 1 1 = 1 + r
= 21r
1
<
2r
= ln(1
1
i=1
1
1
2r
1
4r
r
1
3r
i
1
3r
2
2
1
2(2r)
1 );
2r
2
3
:::
3
:::
1
3(2r)
3
:::
as required.
We now show that in the ase of r-uniform hypergraphs the integrality
gap for the relaxation (21){(26) an be arbitrarily lose to r . It follows that
14
no other rounding of this relaxation an provide an algorithm with a better
performane guarantee.
Indeed, onsider the following instane: the omplete r-uniform hypergraph on n = rq verties, k = 2, wS = 1 for all S 2 E , p = q and p = n q.
It is lear that any feasible ut in this hypergraph has weight
1
Cnr
Cqr
2
Cnr q :
Consider the feasible solution to (23){(26) in whih
xi = 1=r and xi = 1 1=r for eah i:
The weight of this solution is equal to Cnr , sine for eah edge S we have
1
2
r
X
i2S
xi1
X
r
i2S
xi2 = 1
and therefore zS = 1 for all S 2 E . Thus the integrality gap for this instane
is at most
Cnr Cqr Cnr q
q !(n r)! (n q )!(n r)!
=
1
r
Cn
(q r)!n! (n q r)!n!
1 (q qr! )!nr (n (nq qr)!)!nr
r
r
1 (q r) (n q r)
=1
r
n
1 1
r
q
r
r
n
r
1
1
1 r q ;
whih tends to r as q ! 1.
We onlude the paper with a proof that the performane bound of 1 e ,
our algorithm provides on hypergraphs with eah edge of size at least 3,
annot be improved, unless P = NP .
In the Maximum Coverage problem (maximum overage for short),
given a family F = fSj : j 2 J g of subsets of a set I = f1; : : : ; ng with
assoiated nonnegative weights wj and a positive integer p, it is required
to nd a subset X I (alled
) with jX j = p so as to maximize
the total weight of the sets in F having nonempty intersetions with X . It
is well known that a simple greedy algorithm solves maximum overage
approximately within a fator of 1 e of the optimum (Cornuejols, Fisher
and Nemhauser [CFN77℄). Feige [Fe98℄ proved that no polynomial algorithm
an have better performane guarantee, unless P=NP.
1
overage
1
15
Our proof onsists in onstruting an approximation preserving redution
from maximum overage to hyp max k-ut with gsp. Let a set I , a
olletion S ; : : : ; Sm I , nonnegative weights (wj ), and a positive number
p form an instane A of maximum overage. Construt an instane B
of hyp max k-ut with gsp as follows: I 0 = I [ fu ; : : : ; umg (assuming
that I \ fu ; : : : ; umg = ;), (S 0 = S [ fu g; : : : ; Sm0 = Sm [ fumg), the same
weights wj , and p = p, p = jI 0j p. Let (X; I 0 n X ) be a maximum weight
ut in B with the sizes of parts p and p . It is lear that its weight is at
least the weight of a maximum overage in A. Thus it remains to transform
(X; I 0 n X ) into a overage of A with the same weight. If X I , we are done.
Assume that X ontains uj for some j . Then suessively, for eah suh j ,
replae uj in X by an arbitrary element in Sj that is not a member of X , or
if Sj X , by an arbitrary element of I that is not a member of X . After
this transformation and after possibly inluding a few more elements from I
to get exatly p elements, we arrive at a overage Y I in A whose weight
is at least the weight of the ut (X; I 0 n X ) in
1
1
1
1
1
1
1
2
1
5
2
Maximum overage with knapsak onstraint
In this setion we show that the pipage rounding in onjuntion with the
partial enumeration method by Sahni [Sa75℄ an produe good onstantfator approximations even in the ase of a knapsak onstraint.
The maximum overage problem with a knapsak onstraint (MCKP) is,
given a family F = fSj : j 2 J g of subsets of a set I = f1; : : : ; ng with assoiated nonnegative weights
wj and osts j , and a positive integer B , to nd
P
a subset X I with j2X j B so as to maximize the total weight of the
sets in F having nonempty intersetions with X . MCKP inludes both maximum overage and the knapsak problem as speial ases. Wolsey [Wo82℄
appears to be the rst who sueeded in onstruting a onstant-fator approximation algorithm for MCKP even in a more general setting with an
arbitrary nondereasing submodular objetive funtion. His algorithm is of
greedy type and has performane guarantee of 1 e 0:35 where is the
root of the equation e = 2 . Reently, Khuller, Moss and Naor [KMN99℄
have designed a (1 e )-approximation algorithm for MCKP by ombining
the partial enumeration method of Sahni for the knapsak problem [Sa75℄
with a simple greedy proedure. In this setion we present an 1 (1 1=k)k approximation algorithm for MCKP where k is the maximum size of sets in
the instane. Our algorithm exploits the same idea of partial enumeration
but instead of nding a greedy solution, solves a linear relaxation and then
rounds the frational solution by a bit more general \pipage" proedure.
1
16
Generalizing (11){(15) rewrite MCKP as the following integer program:
max
s. t.
m
X
j =1
X
i2Sj
n
X
i=1
(33)
wj zj
xi zj ;
j
2 J;
(34)
i xi B;
(35)
0 xi; zj 1; i 2 I; j 2 J
(36)
xi 2 f0; 1g; i 2 I:
(37)
Note that one an exlude variables zj by rewriting (33){(37) as the following equivalent nonlinear program:
max
F (x) =
m
X
j =1
wj
Y
1
i2Sj
(1
xi )
(38)
subjet to (35){(37).
Set k = maxfjSj j : j 2 J g. Denote by IP [I ; I ℄ and LP [I ; I ℄ the integer
program (33){(37) and its linear relaxation (33){(36) respetively, subjet to
the additional onstraints: xi = 0 for i 2 I and xi = 1 for i 2 I where I
and I are disjoint subsets of I . By a solution to IP [I ; I ℄ and LP [I ; I ℄
we shall mean only a vetor x, sine the optimal values of zj are trivially
omputed if x is xed.
We rst desribe an auxiliary algorithm A whih nds a feasible solution
xA to the linear program LP [I ; I ℄. The algorithm is divided into two phases.
The rst phase onsists in nding an optimal solution xLP to LP (I ; I ) by
appliation one of the known polynomial-time algorithms. The seond phase
of A transforms xLP into xA through a series of \pipage" steps. We now
desribe the general step. Set xA xLP . If at most one omponent of xA is
frational, stop. Otherwise, hoose two indies i0 and i00 suh that 0 < xAi < 1
and 0 < xAi < 1. Set xAi (") xAi + ", xAi (") xAi "i =i and xAk (") xAk
for all k 6= i0 ; i00. Find an endpoint " of the interval
[ minfxi ; (1 xi ) i g; minf1 xi ; xi i g℄:
0
1
0
0
1
1
0
0
1
0
1
0
1
1
0
1
0
00
0
0
00
0
00
00
00
0
00
i0
00
0
00
i0
suh that F (x(")) F (xA). Set xA x("). Go to the next step.
The orretness of the seond phase follows from the fat that the vetor x(") is feasible for eah " in the above interval, and from the earlier
observation (see Setion 3) that the funtion F (x(")) is onvex.
17
Eah "pipage" step of the seond phase of A redues the number of frational omponents of the urrent vetor xA at least by one. So, nally, A
outputs an \almost integral" feasible vetor xA having at most one frational
omponent.
By onstrution, F (xA) F (xLP ). By (16), it follows that
F (xA ) F (xLP )
X
X
X
wj + 1 (1 1=k)k
wj minf1;
xLP
i g (39)
j 2J1
j 2J nJ1
i2Sj
where and heneforth J = fj : Sj \ I 6= ;g.
We now present a desription of the whole algorithm.
Input: An instane of the integer program (33){(37);
Output: A feasible solution x to the instane;
Among all feasible solutions x to the instane satisfying Pi2I xi 3, by
omplete enumeration nd a solution x of maximum weight;
x x;
P
for all I I suh that jI j = 4 and i2I i B do
1
1
0
0
1
begin
I0
t
1
1
;;
0;
while t = 0 do
begin
apply A to LP [I ; I ℄;
if all xAi are integral
then begin t 1; x^
0
1
xA end
otherwise
begin
nd i0 suh that xAi is frational;
x^i
0;
for all i 6= i0 do x^i xAi ;
I
I [ fi0 g
0
0
0
0
end
if F (^x) > F (x) then x
end
end
x^
We now prove that the performane guarantee of the desribed algorithm is
indeed (1 (1 1=k)k .
Let X be an optimal set of the given instane of MCKP. Denote by
x the inidene vetor of X . Reall that x is an optimal solution to the
18
equivalent nonlinear program (38), (35){(37). If jX j 3, then the output
of the algorithm is an optimal solution. So we may assume that jX j 4.
W.l.o.g. we may also assume that the set I is ordered in suh a way that
X = f1; : : : ; jX jg and for eah i 2 X , the element i overs the sets in F
not overed by the elements 1; : : : ; i 1 of the maximum total weight among
j 2 X n f1; : : : ; i 1g.
Consider now that iteration of the algorithm at whih I = f1; 2; 3; 4g.
At this iteration the algorithm runs through q steps, q n 4. At step t it
alls the algorithm A to nd an\almost integral" feasible solution xt = xA
to IP [I t ; I ℄ where I t = fi ; : : : ; it g. If all omponents of xt are integral
then t = q and the iteration ends up. Otherwise the vetor xt is replaed by
the integral vetor x^t whih oinides with xt in its integral omponents and
equals zero in its single frational omponent indexed by it . If the weight
of x^t exeeds that of the reord solution, the latter is updated. Then the
algorithm sets I t = I t [ fit g and goes to the next step. Thus at the
iteration the algorithm nds a series of feasible integral solutions x^ ; : : : ; x^q
to (38), (35){(37) or, equivalently, subsets X^ ; : : : ; X^q I satisfying X^t I
and X^t \ I t = ; where I t =q fi ; : : : ; it g, t = 1; : : : ; q.
Assume rst that X \I = ;. Then x is an optimal solution to IP [I q ; I ℄.
Reallq that x^q = xq and the latter is obtained from the optimal solution to
LP [I ; I ℄ by the \pipage" proess. By (39) it follows that the solution x^q ,
being not better than the output solution, has weight within a fator of
(1 (1 1=k)k ) of the optimum.
Thus, in this ase we are done.
q
Assume now
that
X \ I 6= ; and let I s be the rst set in the series
I = ;; : : : ; I q , having nonempty intersetion with X . In other words, is is
the rst index in the series i ; : : : ; iq lying in X . We laim then that
F (^
xs ) 1 (1 1=k)k F (x ):
(40)
Sine the weight of x^s does not exeed the weight of the output vetor of the
algorithm this would prove that (1 (1 1=k)k) is the performane guarantee
of the algorithm.
In the following argument for brevity we shall use alternately the sets and
their inidene vetors as the arguments of F .
Indeed, the funtion F an be also treated as the set funtion F (X )
dened on all subsets X I . It is well known that F (X ) is submodular and,
onsequently, have the property that
F (X [ fig) F (X ) F (X [ Y [ fig) F (X [ Y );
(41)
for all X; Y I and i 2 I . Let i 2 I and Y I . Then
1=4F (I ) = 1=4F (f1; 2; 3; 4g) = 1=4 F (f1; 2; 3; 4g) F (f1; 2; 3g +
1
0
1
1
0
+1
0
1
0
1
1
0
1
0
1
1
0
0
0
1
0
1
0
0
+1
0
1
1
1
1
19
1
F (f1; 2; 3g) F (f1; 2g) +
F (f1; 2g) F (f1g) +
F (f1g) F (;)
by the hoie of I
1
1=4 F (f1; 2; 3; ig) F (f1; 2; 3g +
F (f1; 2; ig) F (f1; 2g) +
F (f1; ig) F (f1g) +
F (fig) F (;)
by (41)
F (Y [ fig) F (Y ) :
Thus, we have proved that for any Y I and any i 2 I ,
1=4F (I ) F (Y [ fig) F (Y ):
(42)
Reall that the integral vetor x^s is obtained from an \almost integral"
vetor xs returned by the algorithm A, by the replaement with zero its
single frational omponent xsi . It follows that
F (X^ s [ fis g) F (xs ):
(43)
Let xLP , zLP denote an optimal solution to LP [I s; I ℄. Using (37), (42) and
(43) we nally obtain
F (^
xs ) = F (X^ s)
=
F (X^ s [ fis g)
F (X^ s [ fis g) F (X^ s)
by (42)
F (X^s [ fisg) 1=4F (I )
by (43)
X
F (xs) 1=4F (I )
X
Y
X
=
wj +
wj 1
(1 xsi) 1=4 wj
1
1
s
0
1
1
1
(by (39))
j 2J1
3=4
i2Sj
j 2J nJ1
X
j 2J1
wj +
j 2J1
1 (1 1=k)k
X
X
k
X
1 (1 1=k)
j 2J1
(by the hoie of s)
20
wj +
j 2J nJ1
j 2J nJ1
wj minf1;
wj minf1;
X
i2Sj
X
i2Sj
xLP
i g
xLP
i
g
1 (1 1=k)
k
X
j 2J1
wj +
= 1 (1 1=k)k F (x ):
6
X
j 2J nJ1
wj minf1;
X
i2Sj
x g
i
Sheduling unrelated parallel mahines with
presribed numbers of jobs to be proessed
on eah mahine
We onsider the problem of sheduling unrelated parallel mahines with the
total weighted ompletion time as the objetive funtion and bounded numbers of jobs to be proessed on eah mahine. An instane of the problem
onsists of a set J = fJ ; : : : ; Jng of n jobs and a set M = fM : : : ; Mm g of
m mahines, positive integers pij , Ri , and rational weights wj for all Mi 2 M,
Jj 2 J . Eah positive integer pij is a proessing requirement assoiated with
the job Jj and depending on the mahine Mi . Eah job Jj must be proessed
without interruption for the respetive amount of time on one of the m mahines. Every mahine an proess at most one job at a time. The mahine
Mi an proess at most Ri jobs, i. e., in any feasible shedule at most Ri jobs
must be assigned to the mahine Mi (the ardinality onstraints). We denote
the ompletion time of the job Jj in a feasiblePshedule by Cj . The objetive
is to nd a feasible shedule that minimizes J 2J wj Cj where wj 0 is a
weight assoiated with eah job Jj .
Polynomial algorithms for some speial ases of the problem with general
Ri were developed by Granot, Skorin-Kapov and Tamir [GST97℄. For the
problem of sheduling unrelated parallel mahines in the absene of ardinality onstraints (or when all Ri = n) Skutella [Sk98, Sk99℄ and Sethuraman
and Squillante [SS99℄ independently suggested a onvex programming relaxation that leads to a 3/2-approximation algorithm. In the following by
using essentially the same onvex relaxation in onjuntion with the pipage
rounding we obtain an approximation algorithm with the same performane
guarantee for the problem with ardinality onstraints.
Notie that the problem an be reformulated as an assignment problem
of jobs to mahines sine for a given assignment of jobs to mahines the
sequening of the assigned jobs an be done optimally on eah mahine Mi
by applying the lassial Smith's ratio rule [Sm56℄: shedule the jobs in order
of noninreasing ratios wj =pij . For eah mahine Mi , dene the orresponding
total order i on the set of jobs by setting Jj i Jk if either wj =pij > wk =pik
or wj =pij = wk =pik and j < k. Now the problem an be further reformulated
1
1
j
21
as an integer quadrati program in nm assignment variables:
X
min
Jj 2J
s:t:
Cj
=
X
Mi 2M
X
Jj 2J
(44)
wj Cj
X
Mi 2M
xij
xij
xij pij +
= 1;
ki j
xik pik ;
Jj
2 J;
(45)
2 J;
(46)
Mi 2 M;
(47)
Jj
Ri;
X
2 f0; 1g; Jj 2 J ; Mi 2 M;
(48)
where xij = 1 means that job Jj is proessed on the mahine Mi . Plugging
onstraints (45) into the objetive funtion we obtain the following quadrati
integer problem:
min F (x) = T x + 21 xT Dx
(49)
X
s:t:
xij = 1; Jj 2 J ;
(50)
xij
Mi 2M
X
Jj 2J
xij
Ri;
Mi 2 M;
(51)
2 f0; 1g; Jj 2 J ; Mi 2 M;
(52)
where x is a vetor of length mn onsisting of all variables xij lexiographially ordered with respet to the order 1; : : : ; m of the mahines and then,
for eah mahine Mi, the jobs ordered aording
to i . The vetor 2 Rmn
is given by ij = wj pij and D = d ij hk is a symmetri mn mn matrix
given by
8
9
if i 6= h or j = k; =
< 0
d ij hk = wj pik if i = h and k i j;
:
wk pij if i = h and j i k: ;
Observe that the funtion F (x) satises the "-onavity ondition. Indeed, x any feasible frational solution to the program (49){(52) and any
path or yle R in the graph Hx obtained by substituting U = J and W =
M. The funtion '("; x; R) is onave sine for any mahine Mi 2 M = W
there are at most two edges in R inindent to Mi and if there are exatly
two suh edges in R then one edge is dereased by " and another is inreased by the same amount. Sine d ij hk is positive only if i = h (i.e.
xij
(
(
)(
)(
)
)
(
22
)(
)
xij ("; R)xhk ("; R) is multiplied by a positive number only if i = h) the funtion '("; x; R) is a quadrati polynomial with nonpositive main oeÆient.
Skutella [Sk98, Sk99℄ observed that for any binary vetor x 2 f0; 1gmn,
the linear form T x an be rewritten as xT diag()x where diag() denotes
the diagonal matrix whose diagonal entries oinide with the entries of the
vetor . It follows that the following nonlinear program is a relaxation of
the problem (49){(52):
min
s. t.
(53)
(54)
(55)
(56)
Z
12 T x + 21 xT (D + diag())x;
Z T x;
X
xij = 1; Jj 2 J ;
Z
Mi 2M
X
Jj 2J
xij
Ri ;
Mi 2 M;
(57)
0 xij 1; Jj 2 J ; Mi 2 M:
(58)
Skutella shows (Lemma 2.4, [Sk99℄) that the matrix D + diag() is positive
semidenitive. Therefore (53){(58) is a onvex program and an be solved
within an additive error " in polynomial time [GLS88℄ (notie that the running time of this algorithm is polynomial on log " ). The objetive funtion
(53) an be replaed by
1
1
L(x) = maxfT x; T x + xT (D + diag ())xg:
2
2
We onsider the problem of minimizing L(x) subjet to (56){(58) as a
nie relaxation.
To get a 3=2-approximation using the pipage rounding we only need to
show that F (x) 3=2L(x) for all feasible vetors x. Indeed,
1
1
2
1
2
1
2
1
2
3
2
F (x) = T x + xT Dx T x + T x + xT (D + diag ())x L(x):
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23
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