The Price of Information in Combinatorial Optimization

25 Pages·2017·0.28 MB·English

The Price of Information in
Combinatorial Optimization
∗
Sahil Singla†
7
1 November2,2017
0
2

v
o
N Abstract

1 Consideranetworkdesignapplicationwherewewishtolaydownaminimum-costspanningtreein

agivengraph;however,weonlyhavestochasticinformationabouttheedgecosts. Tolearntheprecise
]
S cost of any edge, we have to conduct a study that incurs a price. Our goal is to find a spanning tree
D while minimizing the disutility, which is the sum of the tree cost and the total price that we spend on
the studies. In adifferentapplication,eachedgegivesa stochastic rewardvalue. Ourgoalisto finda
.
s
spanningtreewhilemaximizingtheutility,whichisthetreerewardminusthepricesthatwepay.
c
[ Situations such as the above two often arise in practice where we wish to find a good solution to

an optimization problem, but we start with only some partial knowledge about the parameters of the
1
problem. The missing informationcan be foundonly after paying a probing price, which we call the
v
5 priceofinformation.Whatstrategyshouldweadopttooptimizeourexpectedutility/disutility?
0 AclassicalexampleoftheabovesettingisWeitzman’s“Pandora’sbox”problemwherewearegiven
4 probability distributionson values of n independentrandomvariables. The goal is to choose a single
0
variablewithalargevalue,butwecanfindtheactualoutcomesonlyafterpayingaprice. Ourworkisa
0
generalizationofthismodeltoothercombinatorialoptimizationproblemssuchasmatching,setcover,
.
1
facility location, and prize-collecting Steiner tree. We give a technique that reduces such problems
1
to their non-price counterparts, and use it to design exact/approximation algorithms to optimize our
7
1 utility/disutility. Our techniques extend to situations where there are additional constraints on what
: parameterscanbeprobedorwhenwecansimultaneouslyprobeasubsetoftheparameters.
v
i
X
r
a
∗PartofthisworkwasdonewhiletheauthorswerevisitingtheSimonsInstitutefortheTheoryofComputing. Wearegrateful
toAnupamGuptaandViswanathNagarajanforseveraldiscussionsonthisproject.
†ComputerScienceDepartment,CarnegieMellonUniversity,Pittsburgh,PA15213,USA.Email: [email protected]. Re-
search supported inpart by aCMU Presidential Fellowshipand NSFawards CCF-1319811, CCF-1536002, CCF-1540541, and
CCF-1617790.
1 Introduction
Suppose we want to purchase a house. We have some idea about the value of every available house in the
market, say based on its location, size, and photographs. However, to find the exact value of a house we
havetohireahouseinspectorandpayheraprice. Ourutilityisthedifferenceinthevalueofthebesthouse
thatwefindandthetotalinspectionpricesthatwepay. Wewanttodesignastrategytomaximizeourutility.
TheaboveproblemcanbemodeledasWeitzman’s“Pandora’sbox”problem[Wei79]. Givenprobability
distributions of n independent random variables X and given their probing prices π , the problem is to
i i
adaptively probeasubsetProbed [n]tomaximizetheexpectedutility:
⊆
E max X π .
i i
"i∈Probed{ }−i∈Probed #
X
Weitzman gave an optimal adaptive strategy that maximizes the expected utility (na¨ıve greedy algorithms
can behave arbitrarily bad: see Section A.1). However, suppose instead of probing values of elements, we
probeweightsofedgesinagraph. Ourutilityisthemaximum-weightmatchingthatwefindminusthetotal
probing pricesthatwepay. Whatstrategyshouldweadopttomaximizeourexpectedutility?
Inadifferent scenario, consider anetwork design minimization problem. Suppose wewishto laydown
a minimum-cost spanning tree in a given graph; however, we only have stochastic information about the
edge costs. Tofindthe precise cost X ofany edge, wehave to conduct a study that incurs aprice π . Our
i i
disutility is the sum of the tree cost and the total price that we spend on the studies. We want to design a
strategytominimizeourexpected disutility. Animportantdifference betweenthesetwoscenarios isthatof
maximizingutilityvsminimizingdisutility.
Situationsliketheaboveoftenarisewherewewishtofinda“good”solutiontoanoptimizationproblem;
however,westartwithonlysomepartialknowledgeabouttheparametersoftheproblem. Themissinginfor-
mationcanbefoundonlyafterpayingaprobingprice,whichwecallthepriceofinformation. Whatstrategy
should weadopt to optimize our expected utility/disutility? In this work we design optimal/approximation
algorithms for several combinatorial optimization problems in an uncertain environment where we jointly
optimizethevalueofthesolution andthepriceofinformation.
1.1 Utility/DisutilityOptimization
Tobegin, theabovemaximum-weight matchingproblemcanbeformallymodeledasfollows.
Max-Weight Matching Given a graph G with edges E, suppose each edge i E takes some random
∈
weight X independently from a known probability distribution. We can find the exact outcome X only
i i
after paying a probing price π . The goal is to adaptively probe a set of edges Probed E and select a
i
matchingI Probedtomaximizetheexpected utility, ⊆
⊆
E X π ,
i i
−
"i∈I i∈Probed #
X X
where the expectation is over random variables X = (X ,...,X ) and any internal randomness of the
1 n
algorithm. We observe that we can only select an edge if it has been probed and we might select only a
subset of the probed edges. This matching problem can be used to model kidney exchanges where testing
compatibility ofdonor-receiver pairshasanassociated price.
1
Tocapturevaluefunctionsofmoregeneralcombinatorialproblemsinasingleframework,wedefinethe
notionofsemiadditive functions.
Definition 1.1 (Semiadditive function). We say a function f(I,X) : 2V R|V| R is semiadditive if
× ≥0 → ≥0
thereexistsafunction h :2V R suchthat
≥0
→
f(I,X) = X +h(I).
i
i∈I
X
For example, in the case of max-weight matching our value function f(I,X) = X is additive,
i∈I i
i.e. h(I) = 0. We call these functions semiadditive because the second term h(I) is allowed to effect the
P
function ina“non-additive” way;however,notdepending onX. Herearesomeotherexamples.
Uncapacitated Facility Location: Given a graph G = (V,E) with metric (V,d), CLIENTS V,
• ⊆
and facility opening costs X : V R , we wish to open facilities at some locations I V. The
≥0
→ ⊆
function isthesumoffacilityopening costsandtheconnection coststo CLIENTS. Hence,
f(I,X) = X + mind(j,i). (1)
i
i∈I
Xi∈I j∈CXLIENTS
Hereh(I) = j∈CLIENTSmini∈Id(j,i)onlydepends onI,andnotonfacilityopening costsX.
Prize-CollectPing Steiner Tree: Givenagraph G = (V,E)withsomeedgecosts c : E R ,aroot
≥0
• →
noder V,andpenaltiesX : V R . Thegoalistofindatreethatconnectsasubsetofnodesto
≥0
∈ →
r,whiletrying tominimizethecost ofthetreeandthesum ofthepenalties ofnodes Inotconnected
tor. Hence,
f(I,X) = X +Min-Steiner-Tree(V I),
i
\
i∈I
X
whereMin-Steiner-Tree(V I)denotes theminimumcosttreeconnecting allnodesinV Itor.
\ \
We can now describe an abstract utility-maximization model that captures problems such as Pandora’s
box,max-weightmatching, andmax-spanning tree,inasingleunifyingframework,
Utility-Maximization Suppose we are given a downward-closed (packing)1 constraint 2V and a
F ⊆
semiadditive function val. Each element i V takes a value X independently from a known probability
i
∈
distribution. To find the outcome X we have to pay a known probing price π . The goal is to adaptively
i i
probe a set of elements Probed V and select I Probed that is feasible (i.e., I ) to maximize the
⊆ ⊆ ∈ F
expectedutility,
E val(I,X) π ,
i
−
" i∈Probed #
X
wheretheexpectation isoverrandomvariables Xandanyinternal randomness ofthealgorithm.
Forexample, in the max-weight matching problem valisan additive function and asubset ofedges Iis
feasibleiftheyformamatching. Similarly,whenvalisadditiveand isamatroid,thisframeworkcaptures
F
max-weightmatroidrankfunction, whichcontains Pandora’sboxandmax-spanning treeasspecialcases.
Thefollowingisourmainresultfortheutility-maximization problem:
1AnindependencefamilyF ⊆2V iscalleddownward-closedifA∈F impliesB∈F foranyB⊆A.Aset-systemiscalled
upward-closedifitscomplementisdownward-closed.
2
Theorem 1.2. For the utility-maximization problem for additive value functions and various packing con-
straints ,weobtainthefollowingefficient algorithms.
F
k-system2: Forstochastic elementvalues, wegetak-approximation.
•
Knapsack: Forstochastic itemvaluesandknownitemsizes, wegeta2-approximation.
•
Some important corollaries of Theorem 1.2 are an optimal algorithm for the max-weight matroid rank
problem3 and a2-approximation algorithm for themax-weight matching problem. Theorem 1.2isparticu-
larlyinterestingbecauseitgivesapproximationresultsformixed-signobjectives,whichareusuallydifficult
tohandle.
Wealsoshowthatifvalisallowedtobeanymonotonesubmodularfunctionthenonecannotobtaingood
approximation results: thereisanΩ˜(√n)hardnesseveninadeterministic setting(seeSectionA.3).
Next,wedescribeadisutility-minimizationmodelthatcapturesproblemslikethemin-costspanningtree.
Disutility-Minimization Suppose we are given an upward-closed (covering) constraints ′ 2V and a
F ⊆
semiadditive function cost. Each element i V takes a value X independently from a known probability
i
∈
distribution. To find the outcome X we have to pay a known probing price π . The goal is to adaptively
i i
probe a set of elements Probed V and select I Probed that is feasible (i.e., I ′) to minimize the
⊆ ⊆ ∈ F
expecteddisutility,
E cost(I,X)+ π ,
i
" i∈Probed #
X
wheretheexpectation isoverrandomvariables Xandanyinternal randomness ofthealgorithm.
Forexample, inthemin-cost spanning treeproblem, costisanadditive function andasubset ofedges I
are in ′ if they contain a spanning tree. Similarly, when val is the semiadditive facility location function
F
as defined in Eq. (1) and every non-empty subset of V is feasible in ′, this captures the min-cost facility
F
location problem.
Remark: The disutility-minimization problem can be also modeled as a utility-maximization problem by
allowing itemvalues tobenegativeandworking withtheinfeasibility constraints (ifA ′ thenV A
∈ F \ ∈
),butsuchatransformation isnotapproximation factorpreserving.
F
Wenowmentionourresultsinthismodel. (SeeSection4forformaldescriptions oftheseproblems).
Theorem 1.3. For the disutility-minimization problem for various covering constraints ′, we obtain the
F
followingefficient algorithms.
MatroidBasis: Forstochastic elementcosts, wegettheoptimaladaptivealgorithm.
•
Set Cover: For stochastic costs of the sets, we get a min O(log V ),f -approximation, where V is
• { | | }
theuniverse andf isthemaximumnumberofsetsinwhichanelementcanoccur.
Uncapacitated Facility Location: For stochastic facility opening costs in a given metric, we get a
•
1.861-approximation.
2Anindependence familyF ⊆ 2V isak-systemifforanyY ⊆ V wehave maxA∈B(Y)(|A|) ≤ k, whereB(Y)denotesthe
minA∈B(Y)(|A|)
setofmaximalindependent setsofF includedinY [CCPV11]. Thesearemoregeneralthanintersectionofk matroids: e.g.,a
2-systemcapturesmatchingingeneralgraphsandak-systemcapturesmatchinginahypergraphwithedgesofsizeatmostk.
3Forweightedmatroidrankfunctions,Kleinbergetal.[KWW16]independentlyobtainasimilarresult.
3
Prize-Collecting SteinerTree: Forstochasticpenalties inagivengraphwithgivenedgecosts,weget
•
a3-approximation.
FeedbackVertexSet: ForstochasticvertexcostsinagivengraphwegetanO(logn)-approximation.
•
1.2 Constrained Utility-Maximization
Ourtechniquescanextendtosettingswhereweimposerestrictionsonthesetofelementsthatwecanprobe.
In particular, we are given a downward-closed set system and the constraints allow us to only probe a
J
subset of elements Probed . This is different from the model discussed in Section 1.1 as earlier we
∈ J
could probe any set of elements but could get value for only a subset elements that belong to . As an
F
example, consider a generalization of the Pandora’s box problem where besides paying probing prices, we
canonlyprobeatmostk elements. Wenowformallydefineourproblem.
ConstrainedUtility-Maximization Supposewearegivendownward-closedprobingconstraints 2V
J ⊆
andprobabilitydistributionsofindependentnon-negativevariablesX fori V. TofindX wehavetopay
i i
∈
aprobing priceπ . ThegoalistoprobeasetofelementsProbed tomaximizetheexpected utility,
i
∈ J
E max X π .
i i
"i∈Probed{ }−i∈Probed #
X
Remark: Onecandefineanevenmoregeneral versionofthisproblem wherewesimultaneously haveboth
downward-closedsetsystems and ,andthegoalistomaximizeasemiadditivefunctionvalcorrespond-
F J
ing to , while probing a set feasible in . Forease of exposition, we do not discuss it here and consider
F J
ourvaluefunction tobethemaxfunction, asintheoriginalPandora’sboxproblem.
Depending on the family of constraints , we design efficient approximation algorithms for some set-
J
tingsoftheaboveproblem. Thefollowingisourmainresultforthisproblem (proofinSection5.1).
Theorem1.4. Iftheconstraints formanℓ-systemthentheconstrained utility-maximization problem has
J
a3(ℓ+1)-approximation algorithm.
Since the cardinality (or any matroid) constraint forms a 1-system, an application of Theorem 1.4 gives a
6-approximation algorithm forthePandora’sboxproblemunderacardinality probingconstraint.
Theaboveconstrainedutility-maximizationproblemispowerfulandcanbeusedasaframeworktostudy
variants ofPandora’s box. Forexample, consider the Pandora’s box problem wherewealso allow toselect
anunprobed boxiandgetvalueE[X ],withoutevenpayingitsprobingpriceπ . Thiscanbemodeledusing
i i
a partition matroid constraint where each box has two copies and the constraints allow us to probe at most
oneofthem. ThefirstcopyhasadeterministicvalueE[X ]withzeroprobingpriceandthesecondcopyhas
i
arandom valueX withpriceπ . UsingTheorem1.4,wegeta6-approximation forthisvariant.
i i
Asanon-trivialapplicationofthisconstrainedutility-maximizationframework,inSection5.3wediscuss
a set-probing utility-maximization problem where the costs are on subsets of random variables, instead of
individual variables. Thus for a subset S V, we pay price π to simultaneously probe all the random
S
⊆
variables X for i S. This complicates the problem because to find X , we can probe a “small” or a
i i
∈
“large” setcontaining i,butatdifferentprices. Formally,wedefinetheproblem asfollows.
Set-ProbingUtility-Maximization Givenprobability distributions ofindependent non-negative variables
X for i V and given set family = S ,S ,...,S , where S V for j [m] has a probing price
i 1 2 m j
∈ S { } ⊆ ∈
4
π 0. Theproblem istoprobesomeofthesetsin withindicesinProbed [m]tomaximize
j
≥ S ⊆
E max X π .
i j
∃j∈Probeds.t.Sj∋i{ }−j∈Probed 
X
 
Note that when weprobe multiple sets containing an element i, we find the same value X and not a fresh
i
samplefromthedistribution.
Remark: IfthesetsS arepairwisedisjointthenonecansolvetheaboveproblemoptimally: replacingeach
j
setS withanewrandomvariableX′ = max X havingprobingpriceπ reducesittoPandora’sbox.
j j i∈Sj{ i} j
Weusetheconstrained utility-maximization problemframeworktoshowthefollowingresult.
Theorem 1.5. The set-probing utility-maximization problem has a 3(ℓ + 1)-approximation efficient al-
gorithm, where ℓ is the size of the largest set in . Moreover, no efficient algorithm can be o(ℓ/logℓ)-
S
approximation, unlessP = NP.
1.3 OurTechniques
Howdoweboundtheutility/disutility oftheoptimaladaptivestrategy? Theusualtechniquesinapproxima-
tionalgorithmsforstochastic problems(seerelatedworkinSection1.4)eitherusealinearprogram(LP)to
bound the optimal strategy, or directly argue about the adaptivity gap of the optimal decision tree. Neither
of these techniques is helpful because the natural LPs fail to capture a mixed-sign objective—they wildly
overestimatethevalueoftheoptimalstrategy. Ontheotherhand,theadaptivitygapofourproblemsislarge
evenforthespecialcaseofthePandora’sboxproblem—see anexampleinSectionA.2.
Weneedtwocrucialideasforbothourutility-maximization anddisutility-minimization results. Ourfirst
ideaistoshowthatforsemiadditivefunctions, onecanboundtheutility/disutility oftheoptimalstrategyin
the price-of-information world (hereafter, the PoI world) using arelated instance in a world where there is
no price to finding the parameters, i.e., π = 0 (hereafter, the Free-Info world). This new instance still has
i
independent random variables, however, the distributions are modified based on the original probing price
π (seeDefn2.2). Thisproofcrucially reliesonthesemiadditive natureofourvalue/cost function.
i
Oursecond ideaistoshow thatanyalgorithm with“nice” properties intheFree-Info worldcanbeused
togetanalgorithm withasimilarexpected utility/disutility inthePoIworld. Wecallsuchanicealgorithm
FRUGAL (anddefineitformally inSection3.1). Forintuition, imaginea FRUGAL algorithm tobeagreedy
algorithm, or an algorithm that is not “wasteful”—it picks elements irrevocably. This also includes simple
primal-dual algorithms thatdonothavethereverse-deletion step.
Theorem1.6. IfthereexistsaFRUGALα-approximation Algorithm tomaximize(minimize)asemiaddi-
A
tive function over some packing constraints (covering constraints ′) in the Free-Info world then there
F F
exists an α-approximation algorithm for the corresponding utility-maximization (disutility-minimization)
problem inthePoIworld.
Finally,toproveourresultsfromSection1.1,inSection4weshowwhymanyclassicalalgorithms, ortheir
suitable modifications, are FRUGAL.
We remark that although Theorem 1.6 gives good guarantees for several combinatorial problems in the
PoIworld, therearesomenaturalproblems wheretherearenogood FRUGAL algorithms. Onesuchimpor-
tantproblemistofindtheshortests tpathinagivengraphwithstochasticedgelengths. It’saninteresting
−
5
openquestion tofindsomeapproximation guarantee forthisproblem. Anotherinteresting open question is
toshowthatfindingtheoptimalpolicyforthemax-weightmatchinginthePoIworldishard.
Ourtechniquesfortheconstrainedutility-maximizationprobleminSection5againusetheideaofbound-
ingthisprobleminthePoIworldwithasimilarproblemintheFree-Infoworld. Thislatterproblemturnsout
tobethesameasthestochastic probing problem studied in[GN13,ASW16,GNS16,GNS17]. Byproving
an extension of the adaptivity gap result of Gupta et al. [GNS17], we show that one can further simplify
theseFree-Info problems tonon-adaptive utility-maximization problems (bylosing aconstant factor). This
wecannow(approximately) solveusingourtechniques fortheutility-maximization problem.
1.4 Related Work
An influential work of Dean et al. [DGV04] considered the stochastic knapsack problem where we have
stochasticknowledgeaboutthesizesoftheitems. Chenetal.[CIK+09]studiedstochasticmatchingswhere
we find about an edge’s existence only after probing, and Asadpour et al. [ANS08] studied stochastic sub-
modularmaximizationwheretheitemsmayormaynotbepresent. Severalfollowuppapershaveappeared,
e.g. forknapsack[BGK11,Ma14],packingintegerprograms[DGV05,CIK+09,BGL+12],budgetedmulti-
armed bandits [GM12, GM07, GKMR11, LY13, Ma14], orienteering [GM09, GKNR12, BN14], match-
ing [Ada11, BGL+12, BCN+15, AGM15], and submodular objectives [GN13, ASW16]. Most of these
resultsproceedbyshowingthatthestochasticproblemhasasmalladaptivitygapandthenfocusonthenon-
adaptiveproblem. Infact,Guptaetal.[GNS17,GNS16]showthatadaptivity gapforsubmodular functions
overanypacking constraints isO(1).
Mostoftheaboveworksdonotcapture mixed-sign objective ofmaximizing thevalueminustheprices.
Some of them instead model this as a knapsack constraint on the prices. Moreover, most of them are
for maximization problems as for the minimization setting even the non-adaptive problem of probing k
elements to minimize the expected minimum value has no polynomial approximation [GGM10]. This is
also the reason we do not consider constrained (covering) disutility-minimization in Section 1.2. There is
also a large body of work in related models where information has a price. We refer the readers to the
followingpapersandthereferences therein[GK01,CFG+02,KK03,GMS07,CJK+15,AH15,CHKK15].
ThePandora’sboxsolutioncanbewrittenasaspecialcaseoftheGittinsindextheorem[Git74]. Dumitriu
etal. [DTW03]consider aminimization variant of theGittins index theorem when there isno discounting.
Anotherveryrelevant paper isthatofKleinberg etal.[KWW16],whiletheir resultsaretodesign auctions.
TheirproofofthePandora’sboxprobleminspired thiswork.
Organization In Section 2 we show how to bound the optimal strategy in the PoI world using a corre-
sponding problem intheFree-Info world. InSection3weintroduce theideaofusing a FRUGAL algorithm
to design a strategy with a good expected utility/disutility in the PoI world. In Section 4 we show why
manyclassicalalgorithms, ortheirsuitablemodifications, areFRUGAL. Finally,inSection5wediscussthe
settings wherewehaveprobing constraints, anditsapplication totheset-probing problem.
2 Bounding the Optimal Strategy for Utility/Disutility Optimization
In this section we bound the expected utility/disutility of the optimal adaptive strategy for a combinatorial
optimization in the PoI world in terms of a surrogate problem in the Free-Info world. We first define the
6
gradeτ andsurrogate Y ofnon-negative randomvariables.
Definition 2.1 (Grade τ). For any non-negative random variable X , let τmax be the solution to equation
i i
E[(X τmax)+]= π andletτmin bethesolutiontoequation E[(τmin X )+]= π .
i − i i i i − i i
Definition2.2 (Surrogate Y). For any non-negative random variable X , letYmax = min X ,τmax and
i i { i i }
letYmin = max X ,τmin .
i { i i }
Note that τmax could be negative in the above definition. The following lemmas bound the optimal
i
strategyinthePoIworldintermsoftheoptimalstrategy ofasurrogate problemintheFree-Infoworld.
Lemma2.3. Theexpectedutilityoftheoptimalstrategytomaximizeasemiadditivefunctionvaloverpack-
ingconstraints inthePoIworldisatmost
F
EX[max val(I,Ymax) ].
I∈F { }
Lemma 2.4. The expected disutility of the optimal strategy to minimize a semiadditive function cost over
covering constraints ′ inthePoIworldisatleast
F
EX[min cost(I,Ymin) ].
I∈F′{ }
We only prove Lemma 2.3 as the proof of Lemma 2.4 is similar. The ideas in this proof are similar to
thatofKleinbergetal.[KWW16,Lemma1]toboundtheoptimaladaptivestrategy forPandora’sbox.
ProofofLemma2.3. Consider a fixed optimal adaptive strategy. Let A denote the indicator variable that
i
element i is selected into I and let 1 denote the indicator variable that element i is probed by the optimal
i
strategy. Notethattheseindicators arecorrelated andthesetofelements withnon-zero A isfeasible in .
i
F
Now,theoptimalstrategy hasexpected utility
= E val(I,X) π
i
−
" i∈Probed #
X
= E (A X 1 π ) +E[h(I)]
i i i i
−
" #
i
X
= E A X 1 E [(X τmax)+] +E[h(I)],
i i− i Xi i − i
" #
i
X(cid:0) (cid:1)
usingthedefinitionofπ . SincevalueofX isindependent ofwhetherit’sprobedornot,wesimplifyto
i i
= E A X 1 (X τmax)+ +E[h(I)].
i i− i i − i
" #
i
X(cid:0) (cid:1)
7
Moreover, since wecanselect anelement intoIonly afterprobing, wehave 1 A . Thisimplies thatthe
i i
≥
expectedutilityoftheoptimalstrategyis
E A X A (X τmax)+ +E[h(I)]
≤ i i− i i − i
" #
i
X(cid:0) (cid:1)
= E A Ymax +E[h(I)]
i i
" #
i
X
= E[val(I,Ymax)].
Finally,sinceelementsinIformafeasibleset,thisisatmostE[maxI∈F val(I,Ymax) ].
{ }
3 Designing an Adaptive Strategy for Utility/Disutility Optimization
In this section we introduce the notion of a FRUGAL algorithm and prove Theorem 1.6. We need the
followingnotation.
Definition 3.1 ( Y ). For any vector Y with indices in V and any M V, let Y denote a vector of
M M
⊆
length V withentriesY forj M andasymbol ,otherwise.
j
| | ∈ ∗
3.1 A FRUGAL Algorithm
ThenotionofaFRUGALalgorithmissimilartothatofagreedyalgorithm,oranyotheralgorithmthatisnot
“wasteful”—itselectselementsone-by-oneandirrevocably. Itsdefinitioncaptures“non-greedy”algorithms
suchastheprimal-dual algorithm forsetcoverthatdoesnothavethereverse-deletion step.
We define a FRUGAL algorithm in the packing setting. Consider a packing problem in the Free-Info
world(i.e., i,π = 0)wherewewanttofindafeasiblesetI and 2V aresomedownward-closed
i
∀ ∈ F F ⊆
constraints, whiletryingtomaximizeasemiadditive functionval(I,Y) = Y +h(I).
i∈I i
Definition3.2(FRUGALPackingAlgorithm). ForapackingproblemwithPconstraints andvaluefunction
F
val,wesayAlgorithm isFRUGALifthereexistsamarginal-value functiong(Y,i,y) : RV V R≥0
A × × →
R that is increasing in y, and for which the pseudocode is given by Algorithm 1. We note that this
≥0
algorithm alwaysreturnsafeasiblesolution ifweassume .
∅ ∈ F
Algorithm1FRUGAL PackingAlgorithm
A
1: StartwithM = andvi = 0foreachelementi V.
∅ ∈
2: Foreachelementi 6∈M,computevi = g(YM,i,Yi). Letj = argmaxi6∈M &M∪i∈F{vi}.
3: Ifvj > 0thenaddj intoM andgotoStep2. Otherwise,returnM.
A simple example of a FRUGAL packing algorithm is the greedy algorithm to find the maximum weight
spanning tree(ortomaximizeanyweightedmatroidrankfunction), whereg(Y ,i,Y )= Y .
M i i
We similarly define a FRUGAL algorithm in the covering setting. Consider a covering problem in the
Free-Info world where we want to find a feasible set I ′, where ′ 2V is some upward-closed
∈ F F ⊆
constraint, whiletryingtominimizeasemiadditive function cost(I,Y) = Y +h(I).
i∈I i
P
8
Definition 3.3(FRUGAL Covering Algorithm). For acovering problem withconstraints ′ and cost func-
F
tioncost,wesayAlgorithm is FRUGAL ifthereexists amarginal-value function g(Y,i,y) : RV V
A ≥0× ×
R R thatisincreasing iny,andforwhichthepseudocode isgivenbyAlgorithm 2.
≥0 ≥0
→
Algorithm2FRUGAL CoverageAlgorithm
A
1: StartwithM = andvi = 0foreachelementi V.
∅ ∈
2: Foreachelementi 6∈M,computevi = g(YM,i,Yi). Letj = argmaxi6∈M{vi}.
3: Ifvj > 0thenaddj intoM andgotoStep2. Otherwise,returnM.
We note that for a covering problem it is unclear whether Algorithm 2 returns a feasible solution as we
do not appear to be looking at our covering constraints ′. To overcome this, we say the marginal-value
F
function g encodes ′ if whenever M is infeasible then there exists an element i M with v > 0. This
i
F 6∈
meansthatthealgorithm willreturnafeasible solution aslongasV ′.
∈F
A simple example of a FRUGAL covering algorithm is the greedy min-cost set cover algorithm, where
g(Y ,i,Y ) = S S /Y . Notethathereg encodes ourcoverageconstraints.
M i | j∈M∪i j|−| j∈M j| i
Remark: Obser(cid:16)veSthat a crucial dSifference b(cid:17)etween FRUGAL packing and covering algorithms is that a
FRUGALpackingalgorithmhastohandleY RV (i.e. someentriesinYcouldbenegative)butaFRUGAL
∈
covering algorithm has to only handle Y RV . The intuition behind this difference is that unlike the
∈ ≥0
disutility minimization problem,theutilitymaximization problemhasamixed-sign objective.
3.2 Using a FRUGAL Algorithmto DesignanAdaptiveStrategy
AfterdefiningthenotionofaFRUGAL algorithm, wecannowproveTheorem1.6(restated below).
Theorem1.6. IfthereexistsaFRUGALα-approximation Algorithm tomaximize(minimize)asemiaddi-
A
tive function over some packing constraints (covering constraints ′) in the Free-Info world then there
F F
exists an α-approximation algorithm for the corresponding utility-maximization (disutility-minimization)
problem inthePoIworld.
WeproveTheorem1.6onlyfortheutility-maximization settingastheotherproofissimilar. Lemma2.3
already givesusanupper bound ontheexpected utility oftheoptimal strategy fortheutility-maximization
problem in terms of the expected value of a problem in the Free-Info world. This Free-Info problem can
besolved using Algorithm . Themain idea inthe proof ofthis theorem is toshow that ifAlgorithm is
A A
FRUGAL thenwecanalsorunamodifiedversionof inthePoIworldandgetthesameexpectedutility.
A
ProofofTheorem1.6. LetAlg(Ymax, )denotethesetI returned byAlgorithm whenitrunswith
A ∈ F A
elementweightsYmax. Since isanα-approximation algorithm (whereα 1),weknow
A ≥
1
val(Alg(Ymax, ),Ymax) max val(I,Ymax) . (2)
A ≥ α · I∈F { }
The following crucial lemma shows that one can design an adaptive strategy in the PoI world with the
sameexpectedutility.
9