An FPTAS for the parametric knapsack problem

0.13 MB·

by  Michael Holzhauser

✩
An FPTAS for the Parametric Knapsack Problem
MichaelHolzhausera,∗,SvenO.Krumkea
aUniversityofKaiserslautern,DepartmentofMathematics
Paul-Ehrlich-Str.14,D-67663Kaiserslautern,Germany
Abstract
Inthispaper,weinvestigatetheparametricknapsackproblem,inwhichtheitemprofitsareaffinefunctions
7
dependingonareal-valued parameter. Theaim istoprovide asolutionforall values oftheparameter. Itis
1
0well-knownthatanyexactalgorithmfortheproblemmayneedtooutputanexponentialnumberofknapsack
2
solutions.

nWe present a fully polynomial-time approximation scheme (FPTAS) for the problem that, for any desired
a
precision ε ∈ (0,1), computes(1−ε)-approximate solutions for all values of the parameter. This is the first
J
7 FPTAS for the parametric knapsack problem that does not require the slopes and intercepts of the affine
2functions to be non-negative but works for arbitrary integral values. Our FPTAS outputs O(n2) knapsack
ε
] solutionsandrunsinstronglypolynomial-timeO(n4). Evenforthespecialcaseofpositiveinputdata,thisis
S ε2
thefirstFPTASwitha stronglypolynomialrunning time. Wealso showthatthis timeboundcan befurther
D
improved to O(n2 ·A(n,ε)), whereA(n,ε) denotesthe running time ofany FPTASfor the traditional (non-
. ε
s
parametric)knapsackproblem.
c
[
Keywords: knapsackproblems,parametricoptimization,approximationalgorithms

2
v
2
21. Introduction with weight wi ∈ N>0, the profit pi is now of the
8 formp (λ) := a +λ·b witha ,b ∈ Z. Theresult-
i i i i i
7 The knapsack problem is one of the most funda-
ingoptimizationproblemcanbestatedasfollows:
0
mentalcombinatorialoptimizationproblems: Given
. n
1asetofnitemswithweightsandprofitsandaknap- ∗
0 p (λ) = max (ai+λ·bi)·xi
sack capacity, the task is to choose a subset of the
7 i=1
1items with a maximum profit such that the weight Xn
v:of these items does not exceed the knapsack capac- wi·xi 6W
iity. TheproblemisknowntobeweaklyNP-hardand
X Xi=1
rsolvableinpseudo-polynomialtime. Moreover,sev- xi ∈ {0,1} ∀i ∈{1,...,n}
aeral constant factor approximation algorithms and
The aim of this parametric knapsack problem is
approximationschemeshavebeendevelopedforthe
to return a partition of the real line into in-
problem[1–5](cf. [6]foranoverview).
tervals (− ,λ ],[λ ,λ ],...,[λ ,λ ],[λ ,+ ) to-
In this paper, we investigate a generalization of 1 1 2 k−1 k k
∗
getherwith a solution x for each interval such that
the problem in which the profits are no longer con-
thissolutio∞nisoptimalforallvaluesofλinthe∞inter-
stant but affine functions depending on a parame-
ter λ ∈ R. More precisely, for a knapsack with ca- val. The function mapping each λ ∈ R to the profit
of the corresponding optimal solution is called the
pacity W and for each item i in the item set {1,...,n}
∗
optimal profit function and will be denoted by p (λ)
∗
in the following. It is easy to see that p is con-
∗
Correspondingauthor. Fax:[phone]. Phone: +49 tinuous, convex, and piecewise linear with break-
[phone]
pointsλ ,...,λ [6].
Emailaddresses:[email protected] 1 k
(MichaelHolzhauser),[email protected](Sven Clearly, since theparametric knapsackproblemis
O.Krumke) a generalization of the traditional (non-parametric)
knapsack problem, it is at least as hard to solve as etal. [19]forthebicriteria knapsackproblemcan be
the knapsack problem. In fact, it was shown that, used to obtain an FPTAS for the parametric knap-
eveninthecaseofintegralinputdata,theminimum sack problem running in O(n3 ·log2UB ) time,
ε2 max
number of breakpoints of the optimal profit func- whereUB denotesanupperboundonthemaxi-
max
tion can be exponentially large, so any exact algo- mumpossibleprofitwithrespecttobothoftheprofit
rithmmay needtoreturnan exponentialnumber of functions n a ·x and n b ·x .
i=1 i i i=1 i i
knapsack solutions [7]. In this paper, we are inter-
P P
ested in a fully polynomial time approximation scheme
1.2. Ourcontribution
fortheparametricknapsackproblem. Wewillshow
that, for any desired precision ε ∈ (0,1), a polyno- We present the first FPTAS for the parametric
knapsack problem without the restriction to non-
mial number ofintervals suffices in orderto be able
to provide a (1−ε)-approximate solution for each negative input data. In particular, we show that we
λ ∈ R. only need a total number of O(n2) intervals to ap-
ε
Without loss of generality, we may assume that proximate the problem (which, itself, may need an
w 6 Wforeachi∈ {1,...,n}sinceotherwiseweare exponentialnumberofintervalsasdescribedabove)
i
notabletochoseitemiatall. However,notethatwe and that we can compute an approximate solution
donotsetanyfurtherrestrictionsontheprofits,i.e., for each interval in O(n2) time, yielding an FPTAS
ε
the parameters ai and bi. In particular, each profit with a strongly polynomial running time of O(n4).
ε2
maybecomenegativeforsomespecificvalueofλ. It Our algorithm is the first FPTAS for the problem
isevenpossiblethatthereanoprofitableitemsatall with a strongly polynomial running time, being su-
forsomevaluesofλ. perior to the PTAS of Giudici et al. [17] for ε 6 0.5
and, in thespecial case ofpositiveinput data, supe-
1.1. Previouswork
rior to their FPTAS for large input values. In a sec-
Alargenumberofpublicationsinvestigatedpara- ond step, we improve this result to a running time
metric versions of well-known problems. This in- ofO(n2 ·A(n,ε)),whereA(n,ε)denotestherunning
ε
cludes the parametric shortest path problem [8– timeofanyFPTASforthetraditionalknapsackprob-
11],theparametricminimumspanningtreeproblem lem. UsingtheFPTASofKellererandPferschy[4,5],
[12,13],theparametricmaximumflowproblem[14– thisyieldsatimeboundofO n3 log 1 + n2 log2 1
16],andtheparametricminimumcostflowproblem ε ε ε4 ε
fortheparametricknapsackp(cid:16)roblem. (cid:17)
[7] (cf. [17] for an overview). The parametric knap-
sack problemconsideredhere was firstinvestigated
1.3. Organization
by Carstensen [7]. She showed that the number of
breakpoints of the optimal profit function can be- The results of this paper are divided into three
come exponentially large in general. If the parame- main parts. In Section 2, we show how we can gen-
tersare integral, thenumber ofbreakpointscan still eralizethewell-knowngreedy-like 1-approximation
2
attain a pseudo-polynomial size. The first special- algorithm for the traditional knapsack problem to
ized exact algorithm for the problem was presented the parametric setting and how the resulting profit
byEben-Chaime[18],whoshowedthattheproblem function can be “smoothened”such that it becomes
can be solved in O(knW) time, wherek denotesthe convex and continuous without losing the approx-
number of breakpoints of the optimal profit func- imation guarantee. This will be the key ingredient
∗
tionp . fortheparametricFPTAS,whichwillbepresentedin
The first approximation scheme for the prob- Section3. WewillfirstrecapitulateabasicFPTASfor
lem was recently published by Giudici et al. [17]. the traditional knapsack problem in Section 3.1 and
The authors presented a generalization of the stan- then extend it to the parametric case in Section 3.2,
dardpolynomial-timeapproximationschemeforthe presenting a first time bound for the resulting FP-
knapsackproblem,resultinginaPTASfortheprob- TAS.InSection3.3,asamainresultofthepaper,we
lem with a running time in O( 1 ·n1ε+2). In the present an improved analysis yielding the claimed
ε2
special case of positive values of λ as well as non- running time of the FPTAS. Finally, in Section 4, we
negative values of a and b for each i ∈ {1,...,n}, showthatissufficestosolvethecorrespondingsub-
i i
the authors show that an algorithm of Erlebach problems only approximately so that we can incor-
2
′
poratetraditionalFPTASs,which improves therun- interval I , the algorithm computes the same solu-
j
ningtimeofouralgorithmtotheclaimedone. tion x′, which has a profit of the form p(j)(λ) :=
α(j)+λ·β(j). For each λ ∈ I′, the 1-approximation
j 2
2. Obtainingaparametric 1-approximation algorithm eitherreturnsx′ with profit p(j)(λ) or the
2
mostvaluable itemonly. One possibility to obtain a
The parametric FPTAS will rely on a convex and parametric 1-approximation algorithm would be to
2
continuous 1-approximation of the optimal profit consider each interval I′ separately and to divide it
2 j
function p∗(λ), i.e., a parametric 1-approximation into subintervals, depending on whether p(j)(λ) or
2
algorithmfortheparametricknapsackproblem. We p(max)(λ)islarger,wherep(max)denotestheprofitof
will therefore present such an algorithm in this sec- themostvaluable item(which, again, now depends
tionanddescribehowwecanguaranteetheseprop- onλ). However,theresultingpiecewiselinear func-
ertiesofthefunction. tion pA(λ) is not necessarily continuous or convex,
whichwillberequiredlater,though(seeFigure1).
2.1. Traditional 1-approximation algorithm ′
2 Instead, we ignore the intervals I and only con-
j
Thebasic(non-parametric) 1-approximationalgo- sider the above affine functions p(j)(λ) = α(j)+λ·
2
rithm proceeds as follows: In a first step, the algo- β(j). Let S denote the set of all such functions to-
rithmsortstheitemsindecreasingorderoftheirra- getherwiththefunctionp(0)(λ) := 0andeachprofit
tios pi,whichcanbedoneinO(nlogn)time. Itthen function p (λ). By computing the upper envelope
wi i
packstheitemsinthisorderinguntilthenextitemk of the O(n2) functions in S, we obtain a function ϕ,
with k > 2 would violate the knapsack capacity (or which is based on feasible solutions whose profit is
untiltherearenoitemsleft),yieldingafeasiblesolu- notsmallerthanpA(λ)ateachλ∈ R(seethedotted
′ ′
tion x . The algorithm either returns x or, if better, curveinFigure1). Bystandardarguments,itfollows
thesolutioncontainingonlyanitemwiththelargest thatϕisconvex,piecewiselinear,andcontinuousas
profitp(max). Ifx∗denotesanoptimalsolutiontothe itisthepointwisemaximumofaffinefunctions. For
given knapsack instance, xA the solution returned m functions, the upper envelope can be computed
by the above algorithm, and xLP a solution to the in O(mlogm) time as shown1 by Hershberger [20].
LP-relaxationoftheproblem,wegetthat Within thesametime bound,wecan sorttheresult-
ingintervalsbyincreasingvaluesoftheirleftbound-
n k−1
ary. Hence,weobtainapiecewiselinear,continuous,
pA := p(xA) := pi·xAi = max p(max), pi and convex 1-approximation ϕ with O(n2) break-
(cid:14) (cid:15) 2
Xi=1 Xi=1 pointsin O(n2logn)time. Inthefollowing, wewill
k
> 1 · p > 1 ·p(xLP)> 1 ·p(x∗), refertotheintervalsbetweenthebreakpointsofϕas
2 i 2 2 I ,...,I .
1 q
i=1
X
so xA is a 1-approximation. We refer to [6] for fur-
2
therdetailsonthisstandardalgorithm. 3. ObtainingaparametricFPTAS
2.2. Parametric 1-approximation algorithm
2 BeforeweexplaintheparametricFPTASindetail,
In the parametric knapsack problem, the profits we first recapitulate the basic FPTAS for the tradi-
are affine functions of the form pi(λ) = ai +λ·bi tional (non-parametric) knapsack problem as intro-
∗
suchthattheoptimalprofitp changeswithλ. How- duced by Lawler [21] since its way of proceeding is
ever, note that the solution x′ of the traditional 1- crucial for the understandingof the parametric ver-
2
approximation algorithm only depends on the or- sion.
dering of the items and, thus, remains constant as
long as this ordering does not change. Moreover,
two items can only change their relative ordering if 1Strictlyspeaking,theauthorprovestheresultforfiniteline
their profitfunctions intersect,yieldingO(n2) inter- segmentsandnotforstraightlines. However,itiseasytocom-
′ puteupperandlowerboundsforthesmallestandlargestpos-
vals I , within which the ordering of the items re-
j sibleintersectionpointoftwooftheinvolvedfunctions,respec-
mains unchanged. For all values of λ in such an tively,andtoreducetheproblemtotheresultinginterval.
3
p sibleprofitPisthengivenby
p∗(λ)
n n ∗
n·p n p 2n
ϕ(λ) P 6 e pi 6 i 6 · 6 ,
ε·ϕ ε ϕ ε
i=1 i=1
X X
the proecedureeruns in polynomial time O(n2). The
ε
crucial observation is that we only lose a factor of
pA(λ) (1 − ε) by scaling down the profits, so the solu-
tion obtained by the above dynamic programming
λ
schemeappliedtothescaledprofitsyieldsa(1−ε)-
approximatesolutionfortheproblem(see[6,21]for
furtherdetailsonthealgorithm).
∗
Figure 1: The optimal solution value p (black thick line),
the profitfunctionpA ofthe 1-approximationalgorithm(blue
2 3.2. Parametricscaling
straightlines),andthe“smoothed”curveϕ(reddottedlines).
Although the parametric FPTAS is based on the
basic FPTAS, the instance parameters now depend
3.1. TraditionalFPTAS
on λ and, thus, change while λ increases. In par-
Consider the case of some fixed value for λ, such ticular, both the item profits and ϕ now depend on
that the profits have a constant, but possibly nega- λ, so the scaled profits p (λ) := n·pi(λ) have a
i ε·ϕ(λ)
tive value p . The basic FPTAS for the traditional
i highly non-linear behavior. Nevertjheless,skimilar to
knapsack problem is based on a well-known dy- the parametric 1-approxeimation considered in Sec-
2
namic programming scheme, which was originally
tion 2.2, the solution returned by the FPTAS does
designed to solve the problem exactly in pseudo- not change as long as the profit p (λ) of each item
i
polynomial time: Let P denote an upper bound on ′
remains constant. Hence, if I denotes an interval
j
the maximum profit of a solution to the given in-
such that the scaled profits pi reemain constant for
stance. For k ∈ {0,...,n} and p ∈ {0,...,P}, let ′
each λ ∈ I , we can evaluate the dynamic program-
j
w(k,p) denote the minimum weight that is neces- ming scheme with the profitsepi to obtain a (1−ε)-
sary in order to obtain a profit of exactly p with ′
approximatesolutionfortheintervalI . Theproofof
j
thoseitemsintheitemset{1,...,k}thathaveanon-
thepolynomialrunningtimeaendtheapproximation
negative2 profit. For k = 0, we set w(0,p) = 0
guaranteeremainunchanged.
for p = 0 and w(0,p) = W +1 for p > 0. For
Itremainstoshowhowwecandividetherealline
k ∈ {1,...,n}andforthecasethatp ∈ {0,...,p},we
k into a polynomial number of intervals such that the
computethevaluesw(k,p)recursivelybyw(k,p) =
profits remain constant in each interval. The basic
min{w(k−1,p),w(k−1,p−p )+w },representing
k k FPTAS will then be evaluated for each such inter-
the choice to either not pack the item or to pack it,
val subsequently(using the correspondingconstant
respectively. Else,ifpk ∈/ {0,...,p},wesetw(k,p) = scaledprofits)inordertoobtain(1−ε)-approximate
w(k −1,p) since we can omit negative item- and
solutionsforthewholerealline.
knapsack-profits. The largest value of p such that
Onenaturalideawouldbetobuildonthoseinter-
w(n,p) 6 W thenyields theoptimal solution to the valsdescribedinSection2.2forwhichϕ(λ)behaves
problem. Theprocedurerunsinpseudo-polynomial likeanaffinefunction: LetI ,...,I withq ∈ O(n2)
1 q
timeO(nP).
denote the affine segments of ϕ such that, for each
The idea of the basic FPTAS is to scale down the j ∈ {1,...,q}, the function ϕ takes on some affine
iMtem:=pεr·pofiftosrpsoimtoevnaelwuevpalfuuelfisllpinig:=1 ·p∗Mpi6,pw6hepr∗e. forNmowϕ(λco)n=siαd(ejr)+onλe·sβu(cjh) fionrteλr∈vaIlj.I . If ϕ(λ) = 0
n 2 (cid:4) (cid:5) j
Instead of setting p := pA as it ies done in the tradi- for λ ∈ I , it also holds that p∗(λ) = 0 since ϕ(λ) >
j
tionalFPTAS,wecanalternativelyuseourimproved 1·p∗(λ)forλ ∈R,sotheall-zerosolutionisoptimal.
2
2-approximate solutionϕ. Since themaximum pos- Otherwise,forthenon-roundedscaledprofitofeach
itemi,itholdsthat
2Notethatitemswithanegativeprofitwillnotbepresentin n·pi(λ) = n·(ai+λ·bi) =: n ·f (λ).
anoptimalsolution. ε·ϕ(λ) ε·(α(j)+λ·β(j)) ε i
4
These functions f are monotonous, since the first that each possible integral value is only attained at
i
derivativefulfills most once per interval. As it will be shown in this
Section,wearesomewhatallowedto“ignore”these
df b ·(α(j)+λ·β(j))−(a +λ·b )·β(j)
i(λ) = i i i changeswithoutlosingtheguaranteethateachpos-
dλ (α(j)+λ·β(j))2
siblevalueofthescaledprofitswillonlybeattained
= bi·α(j)−ai·β(j) aconstantnumberoftimes.
(α(j)+λ·β(j))2
Theorem1. For each item i ∈ {1,...,n}, the scaled
and,thus,doesnotchangeitssignwithinIj. Hence, profitspi(λ)attaineachvaluein{0,..., 2εn}atmostthree
within the interval Ij, each scaled profit pi has a timesasλincreasesfrom− to+ .
monotone behavior. Moreover, it holds that 0 6
e
fi(λ) 6 2 for all λ ∈ Ij since each item eiether has PROOF. In order to prov∞e the ∞claim, it suffices to
a non-negative profitwithin thewhole interval or it showthatthesignofthefirstderivativeofeachfunc-
will be ignored and since pi(λ) 6 p∗(λ) 6 2·ϕ(λ). tion fi changes at most twice while λ increases. As
These observations yield that each scaled profit pi above,letI1,...,Iq denotetheintervalsforwhichϕ
changes its (integral) value at most O(n) times takesonsomeaffineformϕ(λ) = α(j)+λ·β(j) such
ε
within Ij since we only need to consider values feor that
fpoirmbeutwlaeeeonn0lyacnhdan2εgne. OH(enn2c)et,imtheesawboitvheinreecaucrhsivIe, fi(λ) = α(aj)i++λλ··βbi(j)
ε j
ien which case we have to repeat the computation of for λ ∈ I . Since ϕ is convex and continuous, it
j
the values w(i,p). This yieldsa totalcomputational holds that β(j) 6 β(j+1) for j ∈ {1,...,q−1} and
overheadofO(n4)perintervaland,sincethereareat that there is some index h such that α(j) 6 α(j+1)
ε2
most O(n2) intervals, a total running time of O(n6) for j ∈ {1,...,h} and α(j) > α(j+1) for j ∈ {h+
ε2 1,...,q−1}. In fact, due to the construction of the
fortheparametricFPTAS.Thisrunningtimewillbe
intervals, these inequalities hold in the strict sense
significantlyimprovedinthenextsubsection.
since β(j) = β(j+1) would also imply that α(j) =
It should be noted that we need to take care of a
α(j+1) by continuity of ϕ, so both segments would
proper definition of the returned intervals: For ex-
ample,considertwoscaledprofitsoftheformsp = belong to the same interval. Hence, if we plot the
i
⌊1+λ⌋ and p = ⌊1−λ⌋. For thecritical value λ = points (β(j),α(j))T into a b-a-space, we get a pic-
j 1
1, both profits evaluate to 1. However, for λe′ := ture as shown in Figure 2. Moreover, for each j ∈
λ1+δandλe1′′ := λ1−δforasmallvalueofδ,on1eof {1,...,q−1},thereissomeλj ∈ R with
theprofitsalreadychangesitsintegralvalueandthe
α(j)+λ ·β(j) = α(j+1)+λ ·β(j+1)
dynamic programming scheme may behave differ- j j
ently. Onesimplesolutionistoassessthatweadda
dueto the continuity and construction ofϕ. Hence,
single-pointinterval[λ ,λ ]foreachcriticalvalueas
1 1 since β(j+1) = β(j)+δ for some value δ > 0, we
well as two open intervals of the forms (λ ,λ ) and j j
0 1 getthat
(λ ,λ ),whereλ andλ areadjacentcriticalvalues.
1 2 0 2
Thereturned(ordered)sequenceofintervalsthenal- α(j+1) = α(j)+λ ·β(j)−λ ·β(j+1)
j j
ternatesbetweensingle-pointintervalsandopenin-
= α(j)−λ ·δ ,
tervals. For an open interval, we can obtain an ap- j j
proximate solution by setting λ to the middle point
so the slope of the line that connects the
oftheintervalandperformingthedynamicprogram
points (β(j),α(j))T and (β(j+1),α(j+1))T evalu-
forthecorrespondingconstantscaledprofits.
atesto
3.3. Improved Analysis α(j+1)−α(j) −λ ·δ
j j
= = −λ
The major drawback of the above algorithm is β(j+1)−β(j) δ j
j
that we basically need to reset the whole proce-
dure whenever the function ϕ changes its behav- and, thus, decreases while j increases. This yields
ior. With this approach, we were able guarantee that thepiecewiselinear function g connectingeach
thateachscaledprofithasamonotonebehaviorsuch of the points (β(j),α(j))T in the order j = 1,...,q
5
is concave3 (asillustratedby thehighlightedareain though the involved functions f are rational func-
i
Figure2). tions whose denominator changes for increasing λ.
Now,forsomespecificitemi ∈ {1,...,n},consider Hence, each item only creates O(n) subintervals as
ε
thefirstderivateoff ,whichaswehaveseenevalu- opposedtotheO(n2· n)subintervalsproveninSec-
i ε
atesto tion 3.2. It remains to show that we can determine
dfi(λ) = bi·α(j)−ai·β(j) thesesubintervalsefficiently.
dλ (α(j)+λ·β(j))2 As shown in the proof of Theorem 1, the slope
of each function f changes at most twice, yielding
as shown above. Since the denominator is always i
for each item up to three partitions of the set of in-
positive, we need to bound the number of times
tervals I ,...,I such that f is monotonous within
the sign of the numerator changes. The value b · 1 q i
i
α(j)−a ·β(j) can be interpreted as the inner prod- eachpartition. Byscanningthroughthesequenceof
i
uct ofthevectors(−a ,b ) and (β(j),α(j))T. Hence, intervals of ϕ, we can determine these three parti-
i i
since(−a ,b )·(b ,a )T = 0,thesignofthederivate tionsforallitemsintotaltimeO(n·n2) = O(n3). For
i i i i
each item, each partition, and each possible scaled
changeswheneverthefunctiongcrossesthelinego-
ing through the origin and the point (bi,ai)T (see profitin{0,..., 2εn}(whichcanbeattainedonlyonce
in the partition), we perform a binary search on the
the dotted line in Figure 2). Since g is concave as
intervalsinthepartitioninordertofindavalueforλ
shown above, this can happen at most two times
whileλincreases,whichyieldstheclaim. (cid:3) atwhichthecorrespondingprofitisattained,ifsuch
a λ exists. This can be donein O(n·3· n ·logn2) =
ε
O(n2 ·logn) time in total. Finally, we need to sort
ε
a this list of critical values of λ in order to determine
thesubintervalsoftheFPTAS,whichcanbedonein
O(n· n ·log(n· n)) = O(n2 ·logn)time.
ε ε ε ε
(bi,ai)T In summary, we need O(n3 + nε2 ·log nε) time to
determinethe O(n2) subintervals ofthe FPTAS.For
ε
b each of these subintervals, we need to perform the
traditionalFPTASwiththecorresponding(constant)
scaledprofits,whichcanbedoneinO(n2)timeeach.
ε
Hence,weobtainanFPTASfortheparametricknap-
(β(j+1),α(j+1))T
sack problem running in O(n4) time in total. This
ε2
(β(j),α(j))T yieldsthemainresultofthispaper:
Theorem2. There is an FPTAS for the parametric
Figure2:Plottingtheslopeβ(j)andintersectα(j)ofeachaffine
knapsack problem running in strongly polynomial time
segment of ϕ. The piecewiselinear function connecting these
pointsisconcaveandintersectswitheachstraightlinethrough O(nε24). (cid:3)
theoriginatmosttwice.
4. CombiningFPTASs
Theorem1 showsthat each possible profitis only
attained a constant number of times per item al-
In the previous section, we have seen that the
parametric knapsack problem can be divided into
3Thiscanalsobeseenbyargumentsusedinthefieldofcom- O(n2) subproblems, for which we need to provide
ε
putational geometry: Itisknownthattheupperenvelopeofa (1−ε)-approximate solutions. These subproblems
set of affine functions of the form c·λ−d correspondsto the
were created in a way such that the scaled profits
lowersurfaceofaconvexhullinthedualspace,whichisclearly
are constantforeach subproblem. Each ofthemcan
convex.Suchadualspacecontainsapoint(c,d)T foreachaffine
functionoftheaboveformintheprimalspaceand,conversely, be seen as a new, independent, and non-parametric
an affine function λ·c−µforeach point(λ,µ)T inthe primal knapsack instance (albeit a special one, since the
space. Hence,eachlinesegment(breakpoint)ofourupperen- profits are now of polynomial size). In Section 3,
velopecorrespondstoacornerpoint(linesegment)ofthelower
wesimply solvedeach oftheresultingknapsack in-
surfaceofaconvexhullinthedualspace(cf. [22]). Infact,Fig-
ure2showsthedualspacemirroredattheb-axis. stancesexactlyinO(n2)time. Themainobservation
ε
6
of this section is that we actually do not necessarily A(n,ε) denotes the running time of an FPTAS for the
needtosolvethesubproblemsexactly–itsufficesto traditional knapsackproblem. (cid:3)
solvethemuptoafactorof(1−ε)usinganyFPTAS
forthetraditionalknapsackproblem. NotethatitclearlyholdsthatA(n,ε) ∈ Ω(n),sothe
Consider onefixedinterval I′ ofthe O(n2)subin- running time of the main procedure will dominate
ε
tervals of the problem. For each λ ∈ I′, the scaled theoverheadstocomputeϕandthesetofsubinter-
profitsp takeonconstantvalues. Moreover,itholds vals.
i
that 1·p∗(λ) 6 ϕ(λ) 6 p∗(λ)foranyλ ∈ I′asshown At present, the best FPTAS for the traditional
2
in Sectieon 2.2. Let x denote a solution returned by knapsack problem is given by Kellerer and Pfer-
some FPTAS for the traditional knapsack problem schy[4,5]andachievesarunningtimeof
that is called on an instance with the scaled profits
1
and let x denote an exact solution to the scaled in- O n·min logn,log +
ε
stance (which, e.g., can be obtained by the dynamic (cid:18) (cid:12) (cid:13)
programming scheme as above). Clearly, it holds 1 1 1 1
log ·min n, log .
that ε2 ε ε ε
(cid:12) (cid:13)(cid:19)
n n
Under the commonly used assumption that n is
p(x):= p ·x > (1−ε)· p ·x =: p(x).
i i i i muchlargerthan 1 inpractice[21],thisrunningtime
ε
i=1 i=1
X X evaluatesto
e e ′ e e ∗
For any fixed λ ∈ I and an optimal solution x for
1 1 1
theunscaledproblematλ,wethengetthefollowing O nlog + log2 ,
ε ε3 ε
approximationguaranteeforthesolutionx: (cid:18) (cid:19)
n n yielding an FPTAS for the parametric knapsack
p
p(x) = p ·x > M· i ·x problemwithastronglypolynomialrunningtimeof
i i i
M
Xi=1 Xi=1 j k n3 1 n2 1
= M·p(x) O log + log2 .
ε ε ε4 ε
> (1−ε)·M·p(x) (cid:18) (cid:19)
> (1−eε)·M·p(x∗)
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