Parameterized complexity of configuration integer programs

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Operations Research Letters

www.elsevier.com/locate/orl

Parameterized complexity of configuration integer programs

Dušan Knop

^a

^,

^∗

^,

¹

, Martin Koutecký

^b

^,

²

, Asaf Levin

^c

^,

³

, Matthias Mnich

^d

^,

⁴

, Shmuel Onn

^c

^,

⁵

aDepartmentofTheoreticalComputerScience,FacultyofInformationTechnology,CzechTechnicalUniversityinPrague,Prague,CzechRepublic bCharlesUniversity,Prague,CzechRepublic

cTechnion–IsraelInstituteofTechnology,Haifa,Israel

dHamburgUniversityofTechnology,InstituteforAlgorithmsandComplexity,Hamburg,Germany

a rt i c l e i n f o a b s t r a c t

Articlehistory:

Received6August2021

Receivedinrevisedform31October2021 Accepted2November2021

Availableonline15November2021

Keywords:

Parameterizedalgorithms ConfigurationIP Surfing

Configuration integer programs (IP) have been key in the design of algorithms for NP-hard high- multiplicityproblems.First,wedevelopfastexact(exponential-time)algorithmsforConfigurationIPand matchinghardnessresults.Second, weshowcase the implications oftheseresults tobin-packing and facility-location-likeproblems.

©^{2021TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-ND} license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

In 1961, Gilmore and Gomory [11] introduced the fundamen- tal and widely influential notion of Configuration IP (ConfIP for short), and applied it to the BinPacking problem. In BinPack- ing, one is given a set J ^{ofn items with sizes p}1

, . . . ,

pn

∈

N, which need to be assigned (or packed) to a set of m identical bins of a common capacity T

∈

N such that the total size of itemspackedper binisnotlarger thanitscapacity.Thisclassical problem has long been investigated from the perspective of ap- proximationalgorithms [13,20],andinrecentyearsalsofromthe perspective ofexactalgorithms [3,12,16,18].Gilmore andGomory usedtheConfIPtodescribeaBinPackingsolutionbyalistoftu- ples“(packingofonebins,multiplicity

μ

ofbinswithpackings)”.

In this paper,we consider a more generalform ofConfiguration IP.Formotivation,observethatthenaturalinputencoding forBin

*

Correspondingauthor.

E-mailaddresses:dusan.knop@fit.cvut.cz(D. Knop),koutecky@iuuk.mff.cuni.cz (M. Koutecký),levinas@ie.technion.ac.il(A. Levin),matthias.mnich@tuhh.de (M. Mnich),onn@ie.technion.ac.il(S. Onn).

1 Partially supported by the OP VVV MEYS funded project CZ.02.1.01/0.0/0.0/16_019/0000765“ResearchCenterforInformatics”.

2 Partially supported by Charles University project UNCE/SCI/004, and bythe project19-27871XofGAˇCR.

3 PartiallysupportedbyIsraelScienceFoundationgrant308/18.

4 SupportedbyDFGgrantMN59/4-1.

5 PartiallysupportedbytheDresnerchairandIsraelScienceFoundationgrant 308/18.

Packing does not list the item sizes one by one; rather, the n itemsare classified intod

^{n itemtypes andthe input specifies} the size pj

∈

N andthe number nj of items oftype j. The so- calledhigh-multiplicityencoding(atermfirstcoinedbyHochbaum andShamir [14]) thus givesa size vector p

= (

p1

, . . . ,

pd

)

anda multiplicityvector n

= (

n₁

, . . . ,

n_d

)

with

ⁿ

1

=

ⁿ1

+ · · · +

ⁿd

=

^n.

Observenowthat, foranypacking

σ

,each bini definesa vector x

= (

x1

, . . . ,

x_d

)

,withxj beingthenumberofitemsoftype j as- signed to i by

σ

; such a vector x is called a configuration. This high-multiplicitybinpackingproblem(andmanyother optimiza- tionproblems)canberephrasedinthefollowingway,aswasfirst observedbyGilmoreandGomory [11].LetNbetheset

{

⁰

,

1

, . . . }

^. LetCT

=

x

∈

N^d

|

^px

≤

^T

denotethesetofconfigurationsofsize at most T. Then, to decide if the given instance admits a feasi- ble solution amounts to deciding if n can be written as a sum ofm configurationsfromCT,i.e.,n

=

m

i=¹xⁱ,withxⁱ

∈

CT forall i

=

¹

, . . . ,

m.Thistaskleads toa relativelysimpleformof config- urationIP,which hasan integralvariable

λ

x

∈

N foreach config- uration x

∈

CT, and asks for a solution with

x∈CT

λ

x

=

^{m and}

x∈CT

λ

x

·

^x

=

^n.

However, other optimization problems lead to more complex formsofConfigurationIP.Maybetherearebinsofdifferenttypes, inwhich case

τ

denotes the number ofbintypes with

μ

ⁱ being thenumberofbinsoftype i,wherebinsofacommontypehave commoncharacteristics.Itemsmayalsohaveseveralothercharac- teristics,andthenan item typeisthe setofitemswithcommon characteristics. We may also incur a cost for each configuration.

Thus,ourConfigurationIPisdefinedasfollows.

https://doi.org/10.1016/j.orl.2021.11.005

0167-6377/©^{2021TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense} (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Configuration IP(ConfIP)

Input: Dimensiond

∈

N,finitesetsX¹

, . . . ,

Xτ

⊆

Z^d, objectivefunctions f¹

, . . . ,

fτ

:

Z^d

→

Z,numbers

μ

¹

, . . . , μ

^τ

_∈

N,andatargetvectorn

∈

Z^d. Find: min τ

i=¹

x∈^Xi fⁱ

(

x

) · λ

ⁱ_x

|

_τ

i=¹

x∈^Xix

· λ

ⁱ_x

=

ⁿ

,

x∈^Xi

λ

_xⁱ

= μ

ⁱ

, λ

ⁱ_x

∈

N

∀

^x

∈

^Xi

∀

ⁱ

∈ {

¹

, . . . , τ }

.

WerefertovandenAkkeretal. [26] forpracticalinvestigations ofConfIP,andtoHochbaumandShmoys [15],Alonetal. [1],Fer- nandez de laVega andLueker [9] andKarmarker andKarp [20]

forstudiesofapproximationalgorithmsbasedonConfIP. Previous studiesofthebinpackingprobleminthehigh-multiplicitysettings weredonebyJansenandSolis-Oba [17] andJansenandKlein [16].

Ourcontributionsandpaperoutline.Ourcontributionistwo-fold.

First, inSection 3we provideseveralfixed-parameter algorithms, andinSection4weprovehardnessresultsforConfIP,delineating thecomplexity landscapewithregardto themostnaturalparam- eters showing that some trade-offs evident in our analysis are inevitable. Second, to showcase the usefulness and versatility of ourapproach, weapply inSection 5ouralgorithms tohighmul- tiplicity problemsinbin packingandsurfing, a generalmodel of facilitylocationandmulticommodityflows.

2. Preliminaries

Forpositiveintegersm

,

nweset

[

^m

,

n

] = {

^m

,

m

+

¹

, . . . ,

n

}

^and

[

ⁿ

] = [

¹

,

n

]

^{. Wewrite vectorsinboldface(e.g.,x}

,

y) andtheiren- tries in normal font (e.g., the i-th entry of x is x_i or x

(

i

)

). For

α _∈

R,

α

isthefloorof

α

,and

α

istheceiling of

α

.Fordata object O,wedenoteby

^O

^{itsbinaryencodinglength.Theset}N isthesetofnon-negativeintegers,thatis,N

= {

⁰

,

1

,

2

, . . . }

^.

TheinputofConfIPcanbegivenexplicitlyonlyinfairlylimited scenarios.Thus,weassumethateach(possiblyverylarge)setXⁱis definedsuccinctly.Thefollowingdefinitioncapturesthecasewhen each Xⁱ isdefined asa projection of integerpoints ofa rational polytope.

Definition1(P -representation).Fori

=

¹

, . . . , τ

,let Pⁱ

⊆

R^d+di be a polytope andlet

π

ⁱ

((

x

,

x

)) =

^x

∈

R^d bea projection discarding the last dⁱ coordinates. We call the collection P¹

, . . . ,

Pτ a P - representation of X¹

, . . . ,

Xτ if Xⁱ

= π

ⁱ

(

Pⁱ

) ∩

Z^d,foreach i

∈ [ τ _]

. Let each Pⁱ by defined as Pⁱ

=

(

x

,

x

) |

^Ai

(

x

,

x

) ≤

^bi

for some Aⁱ

∈

Z^mi×(d+di) andbⁱ

∈

Z^mi.The parametersofa P -representation arethefollowingquantities:M

=

^maxi∈[τ]mⁱ,D

=

^maxi∈[τ]dⁱ,

=

maxi∈[τ]

^Ai

∞,L

= ,

b¹

, . . . ,

bτ

^.

We consistentlyusesuperscripts torefer toobjectsandquan- titiesrelatedtothe types(e.g., Xⁱ

,

dⁱ

,

fⁱ

, . . .

).Toavoidconfusion, we always use parentheseswhen intending to expressexponen- tiation (e.g.,

(

dⁱ

)

²). With each Xⁱ given implicitly, the objective functions fⁱ alsomust haveimplicitrepresentations, orbe given by oracles. We consider the following classes of objective func- tions:

•

^{linear:givenvectorsw1}

, . . . ,

wτ

∈

Z^d,let fⁱ

(

x

) =

^wix.

•

^{convex:each fi}

(

x

)

isaconvexfunction.

•

extension-separableconvex:each fⁱ

(

x

) =

^min

x:(x,x)∈Pⁱ∩Z^d+dgⁱ

(

x

,

x

)

for gⁱ a separableconvex function. (In some ofour applica- tions theobjective is only expressible asa separableconvex functionintermsoftheauxiliaryvariablesx.)

•

^{concave:each fi}

(

x

)

isaconcavefunction.

•

fixed-charge:each fⁱ

(

x

) =

^ci

∈

N ifx

=

^{0and fi}

(

x

) =

^{0 oth-} erwise;wecallcⁱ apenalty.

ForaConfIPinstancegiveninits P-representation,set f_max

=

^max

i∈[τ] max

(x,x)∈Z^d+di:^Ai(x,x)≤^bi

^fi

(

x

) .

CONFIP asN-foldIP.Below, weconnect ConfIP withaspecial classofintegerprograms(IPs).ThebaselineIPis:minf

(

x

) :

^Ax

=

b

,

l

≤

^x

≤

^u

,

x

∈

Zⁿ where f

:

Rⁿ

→

R, A

∈

Z^m×n, b

∈

Z^m, and l

,

u

∈ (Z ∪ {±∞} )

ⁿ.Wedenote fmax

=

^max

x∈Zⁿ:^l≤^x≤^u

|

^f

(

x

) |

^.Ageneral- izedN-foldIPmatrixisdefinedas

E^(N)

=

⎛

⎜ ⎜

⎜ ⎜

⎜ ⎝

E¹₁ E²₁

· · ·

^E₁^N E¹₂ 0

· · ·

⁰

0 E²₂

· · ·

0

.. . .. . . . . .. .

0 0

· · ·

^E₂^N

⎞

⎟ ⎟

⎟ ⎟

⎟ ⎠ .

Here,r

,

s

,

t

,

N

∈

N,E^(N)isan

(

r

+

^Ns

) ×

^Nt-matrix,Ei1

∈

Z^r×t and Eⁱ₂

∈

Z^s×t,i

∈ [

^N

]

^{,are integermatrices.ProblemIPwith A}

=

^E(N) is known as generalized N-foldinteger programming (generalized N-fold IP)[7]. The structure of E^(N) allows us todivide any Nt- dimensionalobject,such asthevariablesofx,boundsl

,

u,orthe objective f, into N bricks of sizet,e.g. x

= (

x¹

, . . . ,

x^N

)

. Weuse subscriptstoindexwithin a brickandsuperscriptsto denotethe indexofthebrick,i.e., xⁱ_j is the j^th variableofthe i^th brickwith j

∈ [

^t

]

^andi

∈ [

^N

]

^{.Wecalla brickintegralifallofits}coordinates areintegral,andfractionalotherwise.

ThehugeN-foldIP problemisthe(high-multiplicity)extension ofgeneralized N-fold IP wherethere arepotentially exponentially manybricks.The inputtoa hugeN-foldIP problemwith

τ

types ofbricksis definedby matrices Eⁱ₁

∈

Z^r×t and E₂ⁱ

∈

Z^s×t,i

∈ [ τ ]

^, vectorsl¹

, . . . ,

lτ,u¹

, . . . ,

uτ

∈

Z^t,b⁰

∈

Z^r, b¹

, . . . ,

bτ

∈

Z^s,func- tions f¹

, . . . ,

fτ

:

R^t

→

R satisfying

∀

ⁱ

∈ [ τ ], ∀

^x

∈

Z^t

:

^fi

(

x

) ∈

Z andgivenbyevaluationoracles,andintegers

μ

¹

, . . . , μ

^τ

_∈

Nsuch that _τ

i=¹

μ

ⁱ

₌

N. We say that a brick is of type i if its lower and upper bounds are lⁱ and uⁱ, its right hand side is bⁱ, its objectiveis fⁱ, andthe matricesappearing at the corresponding coordinates are Eⁱ₁ and Eⁱ₂. We let E be the 2

× τ

block ma- trix E

=

E¹₁ E²₁

· · ·

^Eτ1 E¹₂ 2²₂

· · ·

^Eτ₂

. We refer toOnn [25] and Knop et al. [22,23] forstudiesofhuge N-foldIP.

3. Algorithmsfor CONFIP

Ourgoalistoprovethefollowingtheorem.

Theorem1.LetS^{beaConfIPinstancegiveninits}P -representation (if the objective f is convex or concave,we assume itis presented byanevaluationoracle),andletN

= μ

1

=

_τ

i=¹

μ

ⁱandL

ˆ =

^L

+

ⁿ

,

fmax

,

N

^.

1 ConfIPwithalinear,convex,orfixed-chargeobjectivecanbesolved in time

(

N

(

d

+

^D

))

^O(N(d+D))L

ˆ

^O(1), and is thus fixed-parameter tractableparameterizedbyN andd

+

^D.

2 ConfIPwithaconcaveobjectivecanbesolvedintime

(

M N

(

d

+

^D

) ·

log

)

^O(N(d+D))L

ˆ

^O(1),andisthusfixed-parametertractableparam- eterizedbyN,M,d

+

^D,andwith

giveninunary.

3 ConfIP with a linear or an extension-separable convex objective canbe solvedintime

(

Md

)

^O(M2d+d2M)L

ˆ

^O(1),and isthusfixed- parametertractableparameterizedbyM,d,and

.

4 ConfIPwithalinearorfixed-chargeobjectivecanbesolvedintime

( τ

dD Mlog

)

^τ(d+D)O(1)L

ˆ

^O(1),andisthusfixed-parametertractable

parameterizedby

τ

,M,d,andD if

isgiveninunary.

Part 3 andPart 4thus meanthat we can solve ConfIP either in doubly-exponential time parameterized by d, D,

τ

, m, and M andall numbershave tobe giveninunary, orsingle-exponential timeparameterizedbyd,m, M,andthelargestcoefficient

(but for polynomial

τ

). In Part 2 and Part 4 we use the fact that

(

log

α )

^β

≤

^2β2/2

+ α

^o(1) [5,Hint 3.18] to saythat havinglog

in thebaseamountstofixed-parameteralgorithms when

isgiven in unary. It is worth noting that the parameter-dependence for N-fold IP w.r.t. the “usual” parameters is

(

Md

)

^O(d2M+M2d

)

. As thegeneralruntimedependson,e.g., ellipsoidmethod,we givea boundof

(

L

+

^N

+

^D

)

^O

(

1

)

thatsufficestocomparewiththeabove Theorem1.ThenextlemmashowshowtomodelConfIPashuge N-foldIP.

Lemma2.LetaConfIP instanceS ^{begiven inits} P -representation.

Thenintime

( τ ₊

D

+

^M

+

^L

+ μ ,

n

)

,onecanconstructahugeN-fold IPwhichmodelsS^{andhasparametersr}

=

^d,s

=

^2M,t

=

^d

+

^D

+

^M,

^E

∞

=

,N

= μ

1,andwith fⁱ beingtheobjectiveforbricksof type i.

Proof. LetE1

= (

I0

) ∈

Z^d×(d+D+M)whereIisthe

(

d

×

^d

)

-identity matrix and 0 is a

(

d

× (

D

+

^M

))

-all-zero matrix, let E₁ⁱ

=

^E1 for each i

∈ [ τ _]

, and let b⁰

=

^{n. The last M} coordinates ofeach brick will play the role of slack variables in order to model in- equalities in the system Aⁱx

≤

^{bi. For each i}

∈ [ τ ]

^{, obtain Ei}₂ from Aⁱ by adding M

−

^{mi zero rows and D}

−

^{di zero columns,} and then appending from the right the M

×

^{M identity matrix,} ensuring E₂ⁱ has M rows and d

+

^D

+

^{M columns, and append} M

−

^{mizeroes tobi.Formallyextendtheobjectivefunction fi to} d

+

^D

+

^{MdimensionsbymakingitignorethelastM}

+

^D

−

^didi- mensions.Foreachi

∈ [ τ ]

^,defineli

= {−∞}

^d+di

× {

⁰

}

^{M+D−di and} uⁱ

= {+∞}

^d+di

×{

⁰

}

^D−di

×{+∞}

^M.Let

μ

ⁱbethenumberofbricks oftype i,foreachi

∈ [ τ _]

.

It is easy to check that, foreach brick j

∈ [

^N

]

^{oftype i}

∈ [ τ _]

oftheresultinghuge N-fold formulation,x^j restrictedtothefirst d

+

^di coordinates can take on exactly the values of Pⁱ

∩

Z^d+di. Moreover, theobjectivevalueofthe brickis exactlytheobjective value of corresponding point of Pⁱ

∩

Z^d+di in the ConfIP prob- lem.Finally,bythedefinitionofE1,thesumoftherestrictionsof all bricks to the firstd coordinates is exactly n.This showsthat wehavereducedtheConfIPinstancetoahugeN-foldIPinstance in a way which allows us to recover the ConfIP optimum from thehuge N-foldIP optimum.Moreover,theboundsare clearlyas statedinthelemma.

ThefirstthreepartsofTheorem1areestablishednextvia ap- plication ofprevious results,as we describe next: Use Lemma 2 to obtainan N-fold IP instance.Part 1forconvexor linearfunc- tionsfollowsbyapplyinganalgorithmbyDadushandVempala [6]

for solving convex IPs, which runs in time p^O(p)

ˆ

L^O(1), where p

=

^N

(

d

+

^D

)

is the dimension aswe can deletethe slack vari- ables and use the system of inequalities instead of the N-fold IP. For afixed-charge objective,we guess, for each i

∈ [ τ ]

^where 0

∈

^{Xi, a number}

μ ¯

ⁱ

≤ μ

ⁱ such that an optimal solution

λ

has

λ

₀ⁱ

= μ

ⁱ

_{− ¯} μ

ⁱ.With thisguess athand, the objectiveis fullyde- terminedto be _τ

i=1

μ ¯

ⁱcⁱ andit remainsto verifywhetherthere existsacorrespondingdecompositionofnbysolvingConfIPwith thevector

μ ¯

insteadof

μ

andwithoutanyobjective.Finally,pick the bestamongall guesses whosecorresponding ConfIP is feasi- ble.There areatmostNτ

≤

^{NN guesses. ToprovePart2,we use}

analgorithmbyCooketal. [4] toenumerateallverticesofthecor- respondingpolyhedronintime

(

log

·

^{M N}

(

D

+

^d

))

^O(N(D+d))

ˆ

L^O(1). Sinceaminimumofaconcavefunctionisalwaysattainedataver- tex,itsufficestoevaluatetheobjectiveoneach vertexandreturn thebest asthe output.Part 3is byapplying thefixed-parameter algorithmforhuge N-foldIPbyKnopetal. [23].

Thus,itremainstoprovePart4ofTheorem1thatisournext goal. Our proof builds on a Structure Theorem of Goemans and Rothvoß and the idea of the proof of their main theorem [12, Theorem 2.2]. The Structure Theorem applies to the single-type settingandimpliesthatforanysolution

λ

correspondingtoade- compositionofn,thereexists asolution

λ ˆ

whosesupportmostly lies within a precomputable and not-too-large set Y of “impor- tant” configurations. We first extend the Structure Theorem into themultitypesetting,andthenuseit asfollows.Foreachtype i, we compute the set of “important” configurations Yⁱ, and then guessfromit a smallsubset ofconfigurations whichwill appear inthesolution.Usingthis,weconstructanILPinsmalldimension, solveit,andderivefromitanoptimalsolution

λ

.Wetakespecial caretoenforcethemultiplicityconstraint(i.e.,

λ

ⁱ

1

= μ

ⁱ,foreach i

∈ [ τ ]

)andarguehow toencode alinearanda fixed-chargeob- jective.LetusbeginwiththeStructureTheoremofGoemans and Rothvoß [12]:

Proposition3(StructureTheorem [12]).Let P

= {

^x

|

^Ax

≤

^b

} ⊆

R^d be a polytope with A

∈

Z^m×d and b

∈

Z^m such that all coeffi- cientsare bounded by

in absolute value. Thenthere exists a set Y

⊆

^P

∩

Z^d ofsize

|

^Y

| ≤

^S

:=

^mddO(d)

(

log

)

^d thatcan becom- puted in time S^O(1) with the following property. For every vector n

=

x∈^P∩Z^d

λ

xxwith

λ ∈

N^P∩Zd,thereexistsavector

λ ˆ ∈

N^P∩Zd suchthatn

=

x∈^P∩Z^d

λ ˆ

xx,satisfyingthefollowingthreeconditions

λ ˆ

x

∈ {

⁰

,

1

} ∀

^x

∈ /

^{Y ,}

|

^supp

(ˆ λ) ∩

^Y

| ≤

^22d,

|

^supp

(ˆ λ) \

^Y

| ≤

^22d.

Next,weextendProposition3tothemultitypesetting.

Lemma4.LetP¹

, . . . ,

Pτ bea P -representationofX¹

, . . . ,

Xτ.Then, foreach i

∈ [ τ _]

,there exists a set Yⁱ

⊆

^Pi

∩

Z^d+di of size

|

^Yi

| ≤

Sⁱ

:= (

mⁱ

)

^(d+di)

(

d

+

^di

)

^O(d+di)

(

log

)

^d+dithatcanbecomputedintime

(

Sⁱ

)

^O(1)withthefollowingproperty.Foreveryvector

n

=

τ i=1

(x,x)∈Pⁱ∩Z^d+di

λ

ⁱ_(x,x)x

withnon-negativeintegral

λ

,thereexistsanon-negativeintegralvector

λ ˆ

suchthatn

=

^τ

i=¹

(x,x)∈^Pi∩Z^d+di

λ ˆ

ⁱ_(x,x)x,and,foreachi

∈ [ τ ]

^, a)

λ ˆ

ⁱ_(x,x)

∈ {

⁰

,

1

} ∀ (

x

,

x

) / ∈

^Yi

,

b)

|

^supp

( λ ˆ

ⁱ

) ∩

^Yi

| ≤

^22(d+di)

,

c)

|

supp

(ˆ λ

ⁱ

) \

Yⁱ

| ≤

2^2(d+di)

,

d)

ˆ λ

ⁱ

1

= λ

ⁱ

1

.

Proof. First,extendeach Pⁱbyacoordinatewhichisalways1,i.e., replace Pⁱ by

{(

¹

,

x

,

x

) | (

x

,

x

) ∈

^Pi

}

^{.This only increases the di-} mensionby1 andrequiresanadditionalequalityconstraint.Then, applyProposition3toeachPⁱindividually.Let

μ

ⁱ

₌ λ

ⁱ

1andN

= λ

1.Observethatif

(

N

,

n

) =

_τ

i=¹

(1,x,x)∈^Pi∩Z^d+di

λ

ⁱ_(1,x,x)

(

1

,

x

)

, thenthereisadecompositionof

(

N

,

n

)

into

τ

summands

( μ

ⁱ

,

nⁱ

) =

(1,x,x)∈^Pi∩Z^d+di

λ

ⁱ_(1,x,x)

(

1

,

x

)

to which Proposition 3 applies di- rectly, and we obtain all points except for point d). To argue thislastpoint,notethatbothdecompositions

λ

ⁱ

, λ ˆ

ⁱdecompose nⁱ into

μ

ⁱ points of Pⁱ

∩

Z^d+di andtheclaim holds. Thus, foreach i

∈ [ τ _]

, Yⁱ isobtained byapplying Proposition 3to theextended polytope Pⁱ and then projecting out the first coordinate of each elementofthecomputedset.

WearenowreadytofinishtheproofofPart4ofthetheorem.

First,computethesetsY¹

, . . . ,

Yτ fromLemma4.Ourgoalnowis to setup an ILP insmall dimension,whose solutioncorresponds toanoptimalsolution

λ

withthepropertiesofLemma4.Fixsuch an optimal

λ

. Foreachi

∈ [ τ _]

,guessa subset Zⁱ

⊆

^{Yi whichsat-} isfies

|

^Zi

| ≤

^{22(d+di) and}

|

^supp

(λ

ⁱ

) \

^Zi

| ≤

^{22(d+di).Alsoguess the} number

μ ¯

ⁱ

_{= |}

supp

(λ

ⁱ

\

^Zi

) |

^{.There are}_τ

i=¹

(

Sⁱ

)

^O(22(d+D))choices.

Foreach guess, we apply the followingprocedure, andthen pick thebestobtainedvalue acrossallguesses andtransformitintoa solutionofConfIP.Introduceavariable

λ

ⁱ_(z,z)foreachi

∈ [ τ _]

and each

(

z

,

z

) ∈

^Zi.Additionally,foreachi

∈ [ τ _]

,introduce

μ ¯

ⁱvectors ofvariables

(

x

,

x

)

ⁱ_j.Then,considerthefollowingconstraints:

Aⁱ

(

x

,

x

)

ⁱ_j

≤

^bi

∀

ⁱ

∈ [ τ _] ,

j

∈ [ ¯ μ

ⁱ

_]

(1)

τ

i=¹

⎡

⎣

(z,z)∈Zⁱ

λ

₍ⁱ_z,z)z

+

¯ μⁱ

j=¹ xⁱ_j

⎤

⎦ =

^{n (2)}

(z,z)∈^Zi

λ

ⁱ_(z,z)

= μ

ⁱ

− ¯ μ

ⁱ

∀

ⁱ

∈ [ τ ]

^with0

∈ /

^{Xi (3)}

(z,z)∈^Zi

λ

ⁱ_(z,z)

≤ μ

ⁱ

_{− ¯} μ

ⁱ

_∀

i

∈ [ τ _]

with0

∈

^{Xi (4)}

λ

ⁱ_(z,z)

∈ N ∀

i

∈ [ τ ], ∀(

z

,

z

) ∈

Zⁱ (5)

(

x

,

x

)

ⁱ_j

∈ Z

^d+di

∀

ⁱ

∈ [ τ ], ∀

^j

∈ [ ¯ μ

ⁱ

],

⁽⁶⁾

and,dependingontheobjectiveoftheConfIPinstance,solvewith oneoftheobjectives

linear

(λ,

x

,

x

) =

τ i=1

⎡

⎣

(z,z)∈^Zi

λ

ⁱ_(z,z)

(

wⁱz

) +

¯ μⁱ

j=1

wⁱxⁱ_j

⎤

⎦ ,

or

,

fixed-charge

(λ,

x

,

x

) =

τ i=¹

⎡

⎣

(z,z)∈Zⁱ

λ

ⁱ_(z,z)cⁱ

+ ¯ μ

ⁱcⁱ

⎤

⎦ .

Constraints (1) and (6) ensurethatthevariablevectors

(

x

,

x

)

ⁱ_jas- sumevaluesfrom Pⁱ

∩

Z^d+di,foreachi

∈ [ τ _]

,enforcingthemean- ing that these variables represent the part of solution

λ ˆ

whose support does not lie in Yⁱ.Constraint (2) ensures that the solu- tionindeedcorrespondstoadecompositionofnintopointsfrom

(

Pⁱ

∩

Z^d+di

)

.Theconstraints (3)–(4) ensurethatthenumberof non-zeroconfigurationsoftype iisatmost

μ

ⁱ.

Let

(λ,

x

,

x

)

be an optimum of the ILP above, computed using an algorithm for ILP in small dimension (cf. [10,19]).

We construct a solution

λ

^∗ of ConfIP as follows. For each i

∈ [ τ _]

and each z

∈

Z^d such that z

∈ π

ⁱ

(

Pⁱ

)

, let

λ

^∗

(

i

,

z

) =

z∈Z^di:(^z,z)∈^Pi

λ(

i

,

z

,

z

) +

j∈[ ¯μⁱ],(x,x)ⁱ_j=(z,z)1

,andlet

λ

^∗

(

i

,

0

)

= μ

ⁱ

₋

_z∈(πi(Pⁱ)\⁰)∩Z^d

λ

^∗

(

i

,

z

)

. We argue that

λ

^∗ is an optimal solution.

First, consider a linearobjective.Observe that anysolution of ConfIPinducesadecompositionn

=

_τ

i=¹nⁱ,andthatthisdecom- position fully determines the objective function, which becomes τ

i=1wⁱnⁱ.Furthermore, Part d) ofLemma 4guarantees that we can (almost) restrict our attentionto thespecial sets Yⁱ without ruling out anydecompositionof ninto nⁱ.Second, considering a fixed-chargeobjective,observethattheconstraint (4) andoursep- aratehandlingof

λ

^∗

(

i

,

0

)

encodesthisobjectiveappropriately.

Regardingruntime,thenumberoftimeswesolvetheILPcon- structed above is equalto the number of guesses ofthe sets Zⁱ andthenumbers

μ ¯

ⁱ,whichisboundedby

τ i=¹

(

Sⁱ

)

^O(22(d+D))

≤

(

M

)

^(d+dmax)

(

d

+

^D

)

^O(d+D)

(

log

)

^(d+D)

)

_τ2O(^d+D)

≤

(

M

+

^d

+

^D

+

^log

)

^d+dmax

_τ2^O(d+D)

≤ (

M

+

d

+

D

+

log

)

^τ(d+D)O(

1)

.

TheILPwehaveconstructedhasdimensionatmostp

=

_τ

i=¹

( |

^Zi

| + ¯ μ

ⁱ

(

d

+

^di

)) ≤ τ

2^2(d+D)

+

^22(d+D)

(

d

+

^D

) ≤ ( τ +

^d

+

^D

)

2^2(d+D), andcanbe solvedintime p^O(p)L

ˆ

^O(1) byKannan’salgorithm [19]

forsolvingILPs(recallthat

ˆ

L

=

^N

,

n

,

b¹

, . . . ,

bτ

, ,

fmax

^).Hence, the total runtime is bounded by

(

dD Mlog

)

^τ(d+D)O

(1)

ˆ

L^O(1). This concludestheproofofthelastpartofthetheorem;thus,wehave establishedTheorem1.

Remark.Goemans and Rothvoß prove a similar statement [12, Corollary5.1] to Part4ofTheorem 1,wheretheinputn andthe coefficients w have to be given in unary if one desires an FPT algorithm,whereas inour case they can be given inbinary. The difference is that they invoke the Structure Theorem on a poly- topeP whichisadisjunctiveformulationoftheunionofpolyhe- draP¹

∪· · ·∪

^Pτ.Thisdisjunctiveconstruction,however,introduces alargecoefficient,increasing

.Similarly,alinearobjectivecould be handledintheir setting byintroducing an extra variablex_d+1 andsettingx_d+1

=

^{wx,butthisconstraintwouldagainincrease}

. We circumvent both of these limitations by using the Structure Theoremdirectly.

4. Hardnessof CONFIP

Proposition5.SolvingConfIP

1 isW

[

¹

]

^-hardparameterizedbyd only,evenif

isgiveninunaryand noobjectivefunction;

2 withafixed-chargeobjectiveisNP-hard evenwithd

=

^1and

=

¹ (butwithlargepenalties);

3 withaseparableconcavequadraticobjectiveisa)NP-hard evenwith d

=

^2and

=

^1,andb)W

[

¹

]

^-hardparameterizedbyd evenwhen thelargestcoefficientoftheobjectiveisgiveninunaryand

=

^1.

Proof. Part1. Consider the UnaryBinPacking problem, which takesas inputn items ofsizes o₁

, . . . ,

o_n

∈

N withmax_i∈[n]o_i

=

O

≤

^poly

(

n

)

,acapacity B

∈

N,andanintegerk

∈

N,andweask whethertheitemscanbepackedintokbinsofcapacityB.Jansen etal. [18] have shown that UnaryBinPacking is W

[

¹

]

^{-hard pa-} rameterizedbyk,evenfortightinstanceswheren

i=¹o_i

=

^kB. Weshallconstructa ConfIP instancewithntypes.We let Pⁱ, foreachi

∈ [

ⁿ

]

^{,bedefinedbythesystemof}inequalitiesk

j=1xj

=

oi

,

k

j=¹yj

=

¹

,

xj

≤

^{O y}j

, ∀

^j

∈ [

^k

] ,

x

≥

^0.Weletd

=

^k, dⁱ

=

^{k, and}

μ

ⁱ

=

^{1, for each i}

∈ [

ⁿ

]

^{, and define the y variables} to be the one which are discarded by the projection

π

ⁱ. Finally, welet nbeak-dimensionalvectorofall B.Itiseasy toseethat

π

ⁱ

(

Pⁱ

) ∩

Z^k

= {(

^oi

,

0

, . . . ,

0

), (

0

,

oⁱ

,

0

, . . . ,

0

), . . . , (

0

, . . . ,

0

,

o_i

)}

^and eachelementencodesthebinintowhichitemi isassigned.Thus, theConfIPinstanceisfeasibleifandonlyifthereexistsanassign- mentofitemstobinssuchthatthesumofitemsizesofeach bin isB.

Part3b). We will continue working withthe ConfIP instance constructed above. Recall that minimizing a concave function is equivalenttomaximizingaconvexfunction,whichistheperspec- tive we shalltake here. Ourgoal nowis to modeltheconstraint