Local-Search based Approximation Algorithms for Mobile Facility Location Problems

0.51 MB·English

by  Sara Ahmadian

Local-Search based Approximation Algorithms for Mobile Facility
Location Problems∗
SaraAhmadian† ZacharyFriggstad† ChaitanyaSwamy†
Abstract
3
1 Weconsiderthemobilefacilitylocation(MFL)problem. Wearegivenasetoffacilitiesandclients
0 locatedinacommonmetricspaceG=(V,c). Thegoalistomoveeachfacilityfromitsinitiallocation
2 toadestination(inV)andassigneachclienttothedestinationofsomefacilitysoastominimizethesum

n of the movement-costs of the facilities and the client-assignment costs. This abstracts facility-location
a settingswhereonehastheflexibilityofmovingfacilitiesfromtheircurrentlocationstootherdestinations
J
soastoserveclientsmoreefficientlybyreducingtheirassignmentcosts.

8 Wegivethefirstlocal-searchbasedapproximationalgorithmforthisproblemandachievethebest-
1
known approximation guarantee. Our main result is (3 + (cid:15))-approximation for this problem for any

constant (cid:15) > 0 using local search. The previous best guarantee for MFL was an 8-approximation al-
]
S gorithm due to [13] based on LP-rounding. Our guarantee matches the best-known approximation
D guarantee for the k-median problem. Since there is an approximation-preserving reduction from the
. k-medianproblemtoMFL,anyimprovementofourresultwouldimplyananalogousimprovementfor
s
c thek-medianproblem. Furthermore,ouranalysisistight(uptoo(1)factors)sincethetightexamplefor
[ thelocal-searchbased3-approximationalgorithmfork-mediancanbeeasilyadaptedtoshowthatour

1 local-search algorithm has a tight approximation ratio of 3. One of the chief novelties of the analysis
v isthatinordertogenerateasuitablecollectionoflocal-searchmoveswhoseresultinginequalitiesyield
8 the desired bound on the cost of a local-optimum, we define a tree-like structure that (loosely speak-
7
ing) functions as a “recursion tree”, using which we spawn off local-search moves by exploring this
4
treetoaconstantdepth. Ourresultsextendtotheweightedgeneralizationwhereineachfacilityihasa
4
. non-negativeweightwiandthemovementcostforiiswitimesthedistancetraveledbyi.
1
0
3
1 Introduction
1
:
v
Facility location problems have been widely studied in the Operations Research and Computer Science
i
X
communities (see, e.g., [25] and the survey [20]), and have a wide range of applications. In its simplest
r
a version, uncapacitated facility location (UFL), we are given a set of facilities or service-providers with
openingcosts,andasetofclientsthatrequireservice,andwewanttoopensomefacilitiesandassignclients
toopenfacilitiessoastominimizethesumofthefacility-openingandclient-assignmentcosts. Anoft-cited
prototypical example is that of a company wanting to decide where to locate its warehouses/distribution
centerssoastoserveitscustomersinacost-effectivemanner.
We consider facility-location problems that abstract settings where facilities are mobile and may be
relocated to destinations near the clients in order to serve them more efficiently by reducing the client-
assignmentcosts. Moreprecisely,weconsiderthemobilefacilitylocation(MFL)problemintroducedby[11,
∗A preliminary version [2], without the results in Section 6, appeared in the Proceedings of the 24th Annual ACM-SIAM
SymposiumonDiscreteAlgorithms,2013.
†{sahmadian,zfriggstad,cswamy}@math.uwaterloo.ca. Dept. ofCombinatoricsandOptimization,Univ. Wa-
terloo,Waterloo,ONN2L3G1.SupportedinpartbyNSERCgrant327620-09andanNSERCDiscoveryAcceleratorSupplement
award.Thesecondandthirdauthorsarealsosupportedbythethirdauthor’sOntarioEarlyResearcherAward.
1
13], which generalizes the classical k-median problem (see below). We are given a complete graph G =
(V,E ) with costs {c(u,v)} on the edges, a set D ⊆ V of clients with each client j having d units of
G j
demand, and a set F ⊆ V of k initial facility locations. We use the term facility i to denote the facility
whoseinitiallocationisi ∈ F. AsolutionS toMFLmoveseachfacilityitoafinallocations ∈ V (which
i
could be the same as i), incurring a movement cost c(i,s ), and assigns each client j to a final location
i
s ∈ S, incurring assignment cost d c(j,s). The total cost of S is the sum of all the movement costs and
j
assignment costs. More formally, noting that each client will be assigned to the location nearest to it in S,
wecanexpressthecostofS as
(cid:88) (cid:88)
MFL(S) := c(i,s )+ d c(j,σ(j))
i j
i∈F j∈D
where σ(v) (for any node v) gives the location in S nearest to v (breaking ties arbitrarily). We assume
throughoutthattheedgecostsformametric. Weusethetermsnodesandlocationsinterchangeably.
MobilefacilitylocationfallsintothegenreofmovementproblemsintroducedbyDemaineetal.[11]. In
theseproblems,wearegivenaninitialconfigurationinaweightedgraphspecifiedbyplacing“pebbles”on
thenodesand/oredges;thegoalistomovethepebblessoastoobtainadesiredfinalconfigurationwhilemin-
imizingthemaximum,ortotal,pebblemovement. MFLwasintroducedbyDemaineetal. asthemovement
problemwherefacility-andclient-pebblesareplacedrespectivelyattheinitiallocationsofthefacilitiesand
clients,andinthefinalconfigurationeveryclient-pebbleshouldbeco-locatedwithsomefacility-pebble.
Our results. We give the first local-search based approximation algorithm for this problem and achieve
the best-known approximation guarantee. Our main result is a (3 + (cid:15))-approximation for this problem
for any constant (cid:15) > 0 using a simple local-search algorithm. This improves upon the previous best 8-
approximation guarantee for MFL due to Friggstad and Salavatipour [13], which is based on LP-rounding
andisnotcombinatorial.
Thelocal-searchalgorithmweconsiderisquitenaturalandsimple. Observethatgiventhefinallocations
ofthefacilities,wecanfindtheminimum-costwayofmovingfacilitiesfromtheirinitiallocationstothefinal
locationsbysolvingaminimum-costperfect-matchingproblem(andtheclientassignmentsaredetermined
by the function σ defined above). Thus, we concentrate on determining a good set of final locations. In
our local-search algorithm, at each step, we are allowed to swap in and swap out a fixed number (say p)
of locations. Clearly, for any fixed p, we can find the best local move efficiently (since the cost of a set
of final locations can be computed in polytime). Note that we do not impose any constraints on how the
matching between the initial and final locations may change due to a local move, and a local move might
entail moving all facilities. It is important to allow this flexibility, as it is known [13] that the local-search
procedurethatmoves,ateachstep,aconstantnumberoffacilitiestochosendestinationshasanunbounded
approximationratio.
Our main contribution is a tight analysis of this local-search algorithm (Section 4). Our guarantee
matches(uptoo(1)terms)thebest-knownapproximationguaranteeforthek-medianproblem. Sincethere
isanapproximation-preservingreductionfromthek-medianproblemtoMFL[13]—choosearbitraryinitial
facility locations and give each client a huge demand D—any improvement of our result would imply an
analogousimprovementforthek-medianproblem. (Inthisrespect, ourresultisanoteworthyexceptionto
the prevalent state of affairs for various other generalizations of UFL and k-median—e.g., the data place-
mentproblem[4],{matroid-,red-blue-}median[22,16,9,6],k-facility-location[12,15]—wherethebest
approximationratiofortheproblemisworsebyanoticeablefactor(comparedtoUFLork-median);[14]is
anotherexception.) Furthermore, ouranalysisistight(uptoo(1)factors)becausebysuitablysettingD in
thereductionof[13],wecanensurethatourlocal-searchalgorithmforMFLcoincideswiththelocal-search
algorithmfork-medianin[3]whichhasatightapproximationratioof3.
2
We also consider a weighted generalization of the problem (Section 5), wherein each facility i has a
weightw indicatingthecostincurredper-unitdistancemovedandthecostformovingitos isw c(i,s ).
i i i i
(Thiscanbeusedtomodel,forexample,thesettingwheredifferentfacilitiesmoveatdifferentspeeds.) Our
analysisisversatileandextendstothisweightedgeneralizationtoyieldthesameperformanceguarantee. For
thefurthergeneralizationoftheproblem,wherethefacility-movementcostsmaybearbitraryandunrelated
totheclient-assignmentcosts(forwhicha9-approximationcanbeobtainedviaLP-rounding;see“Related
work”),weshowthatlocalsearchbasedonmultipleswapshasabadapproximationratio(Section7).
Theanalysisleadingtotheapproximationratioof3(asalsothesimpleranalysisinSection3yieldinga
5-approximation)cruciallyexploitsthefactthatwemayswapmultiplelocationsinalocal-searchmove. It
isnaturaltowonderthenifonecanproveanyperformanceguaranteesforthelocal-searchalgorithmwhere
wemayonlyswapinandswapoutasinglelocationinalocalmove. (Naturally,thesingle-swapalgorithm
iseasiertoimplementandthusmaybemorepractical). InSection6,weanalyzethissingle-swapalgorithm
andprovethatitalsohasaconstantapproximationratio.
Ourtechniques. Theanalysisofourlocal-searchprocedurerequiresvariousnovelideas. Asiscommon
intheanalysisoflocal-searchalgorithms,weidentifyasetoftestswapsanduselocaloptimalitytogenerate
suitable inequalities from these test swaps, which when combined yield the stated performance guarantee.
One of the difficulties involved in adapting standard local-search ideas to MFL is the following artifact: in
MFL,thecostof“opening”asetSoflocationsisthecostofthemin-costperfectmatchingofF toS,which,
unlikeotherfacility-locationproblems,isahighlynon-additivefunctionofS (andasmentionedabove,we
needtoallowforthematchingfromF toS tochangeinnon-localways). Inmostfacility-locationproblems
with opening costs for which local search is known to work, we may always swap in a facility used by the
globaloptimum(bypossiblyswappingoutanotherfacility)andeasilyboundtheresultingchangeinfacility
cost, and the main consideration is to decide how to reassign clients following the swap in a cost-effective
way; in MFL we do not have this flexibility and need to carefully choose how to swap facilities so as to
ensure that there is a good matching of the facilities to their new destinations after a swap and there is a
frugalreassignmentofclients.
This leads us to consider long relocation paths to re-match facilities to their new destinations after a
swap,whichareoftheform(...,s ,o ,s ,...),wheres ando arethelocationsthatfacilityiismovedto
i i i(cid:48) i i
inthelocalandglobaloptimum,SandO,respectively,ands istheS-locationclosesttoo . Byconsidering
i(cid:48) i
aswapmoveinvolvingthestartandendlocationsofsuchapathZ,wecanobtainaboundonthemovement
costofallfacilitiesi ∈ Z wheres isthestartofthepathoro servesalargenumberofclients. Toaccount
i i
for the remaining facilities, we break up Z into suitable intervals, each containing a constant number of
unaccountedlocationswhichthenparticipateinamulti-locationswap. Thisinterval-swapmovedoesnotat
firstappeartobeusefulsincewecanonlyboundthecost-changeduetothismoveintermsofasignificant
multiple of (a portion of) the cost of the local optimum! One of the novelties of our analysis is to show
howwecanamortizethecostofsuchexpensivetermsandmaketheircontributionnegligiblebyconsidering
multipledifferentwaysofcoveringZwithintervalsandaveragingtheinequalitiesobtainedfortheseinterval
swaps. Theseideasleadtotheproofofanapproximationratioof5forthelocal-searchalgorithm(Section3).
The tighter analysis leading to the 3-approximation guarantee (Section 4) features another noteworthy
idea,namelythatofusing“recursion”(uptoboundeddepth)toidentifyasuitablecollectionoftestswaps.
We consider the tree-like structure created by the paths used in the 5-approximation analysis, and (loosely
speaking) view this as a recursion tree, using which we spawn off interval-swap moves by exploring this
tree to a constant depth. To our knowledge, we do not know of any analysis of a local-search algorithm
that employs the idea of recursion to generate the set of test local moves (used to generate the inequalities
thatyieldthedesiredperformanceguarantee). Webelievethatthistechniqueisanotablecontributiontothe
analysisoflocal-searchalgorithmsthatisofindependentinterestandwillfindfurtherapplication.
3
Relatedwork. Asmentionedearlier,MFLwasintroducedbyDemaineetal.[11]inthecontextofmove-
ment problems. Friggstad and Salavatipour [13] designed the first approximation algorithm forMFL. They
gave an 8-approximation algorithm based on LP rounding by building upon the LP-rounding algorithm of
Charikar et al. [8] for the k-median problem; this algorithm works only however for the unweighted case.
Theyalsoobservedthatthereisanapproximation-preservingreductionfromk-mediantoMFL.Werecently
learned that Halper [17] proposed the same local-search algorithm that we analyze. His work focuses on
experimentalresultsandleavesopenthequestionofobtainingtheoreticalguaranteesabouttheperformance
oflocalsearch.
ChakrabartyandSwamy[6]observedthatMFL,evenwitharbitrarymovementcostsisaspecialcaseof
thematroidmedianproblem[22]. Thus,theapproximationalgorithmsdevisedformatroidmedianindepen-
dentlyby[9]and[6]yieldan8-approximationalgorithmforMFLwitharbitrarymovementcosts.
There is a wealth of literature on approximation algorithms for (metric) uncapacitated and capacitated
facility location (UFL and CFL), the k-median problem, and their variants; see [27] for a survey on UFL.
Whereas constant-factor approximation algorithms for UFL and k-median can be obtained via a variety of
techniques such as LP-rounding [28, 23, 8, 9], primal-dual methods [18, 19], local search [21, 7, 3], all
knownO(1)-approximationalgorithmsforCFL(initsfullgenerality)arebasedonlocalsearch[21,30,5].
Wenowbrieflysurveytheworkonlocal-searchalgorithmsforfacility-locationproblems.
Startingwiththeworkof[21],local-searchtechniqueshavebeenutilizedtodeviseO(1)-approximation
algorithms for various facility-location problems. Korupolu, Plaxton, and Rajaraman [21] devised O(1)-
approximation for UFL, and CFL with uniform capacities, and k-median (with a blow-up in k). Charikar
√
andGuha[7],andAryaetal.[3]bothobtaineda(1+ 2)-approximationforUFL.Thefirstconstant-factor
approximation for CFL was obtained by Pa´l, Tardos, and Wexler [26], and after some improvements, the
current-best approximation ratio now stands at 5 + (cid:15) [5]. For the special case of uniform capacities, the
analysisin[21]wasrefinedby[10],andAggarwaletal.[1]obtainthecurrent-best3-approximation. Arya
et al. [3] devised a (3+(cid:15))-approximation algorithm for k-median, which was also the first constant-factor
approximationalgorithmforthisproblembasedonlocalsearch. GuptaandTangwongsan[15](amongother
results)simplifiedtheanalysisin[3]. Webuilduponsomeoftheirideasinouranalysis.
Local-searchalgorithmswithconstantapproximationratioshavealsobeendevisedforvariousvariants
oftheabovethreecanonicalproblems. MahdianandPa´l[24],andSvitkinaandTardos[29]considersettings
wheretheopeningcostofafacilityisafunctionofthesetofclientsservedbyit. In[24],thiscostisanon-
decreasing function of the number of clients, and in [29] this cost arises from a certain tree defined on the
client set. Devanur et al. [12] and [15] consider k-facility location, which is similar to k-median except
that facilities also have opening costs. Hajiaghayi et al. [16] consider a special case of the matroid median
problemthattheycallthered-bluemedianproblem. Mostrecently,[14]consideredaproblemthattheycall
thek-medianforestproblem,whichgeneralizesk-median,andobtaineda(3+(cid:15))-approximationalgorithm.
2 The local-search algorithm
Asmentionedearlier,tocomputeasolutiontoMFL,weonlyneedtodeterminethesetoffinallocationsof
thefacilities,sincewecanthenefficientlycomputethebestmovementoffacilitiesfromtheirinitialtofinal
locations, andtheclientassignments. Thismotivatesthefollowinglocal-searchoperation. Givenacurrent
setS ofk = |F|locations,wecanmovetoanyothersetS(cid:48) ofklocationssuchthat|S\S(cid:48)| = |S(cid:48)\S| ≤ p,
where p is some fixed value. We denote this move by swap(S \ S(cid:48),S(cid:48) \ S). The local-search algorithm
starts with an arbitrary set of k final locations. At each iteration, we choose the local-search move that
yieldsthelargestreductionintotalcostandupdateourfinal-locationsetaccordingly; ifnocost-improving
move exists, then we terminate. (To obtain polynomial running time, as is standard, we modify the above
proceduresothatwechoosealocal-searchmoveonlyifthecost-reductionisatleast(cid:15)(currentcost).)
4
3 Analysis leading to a 5-approximation
(cid:0) (cid:1)
Wenowanalyzetheabovelocal-searchalgorithmandshowthatitisa 5+o(1) -approximationalgorithm.
For notational simplicity, we assume that the local-search algorithm terminates at a local optimum; the
modificationtoensurepolynomialrunningtimedegradestheapproximationbyatmosta(1+(cid:15))-factor(see
alsoRemark3.8).
Theorem3.1 LetF∗ andC∗ denoterespectivelythemovementandassignmentcostofanoptimalsolution.
(cid:16) (cid:17) (cid:16) (cid:17)
Thetotalcostofanylocaloptimumusingatmostpswapsisatmost 3+O(cid:0) 1 (cid:1) F∗+ 5+O(cid:0) 1 (cid:1) C∗.
p1/3 p1/3
Althoughthisisnotthetightestguaranteethatweobtain,wepresentthisanalysisfirstsinceitintroduces
(cid:0) (cid:1)
many of the ideas that we build upon in Section 4 to prove a tight approximation guarantee of 3+o(1)
forthelocal-searchalgorithm. Fornotationalsimplicity,weassumethatalld sare1. Allouranalysescarry
j
overtriviallytothecaseofnon-unit(integer)demandssincewecanthinkofaclientj havingd demandas
j
d co-locatedunit-demandclients.
j
Notation and preliminaries. We use S = {s ,...,s } to denote the local optimum, where facility i is
1 k
movedtofinallocations ∈ S. WeuseO = {o ,...,o }todenotethe(globally)optimalsolution, where
i 1 k
againfacilityiismovedtoo . Throughout,weusestoindexlocationsinS,andotoindexlocationsinO.
i
Recallthat,foranodev,σ(v)isthelocationinS nearesttov. Similarly,wedefineσ∗(v)tobethelocation
in O nearest to v. For notational similarity with facility location problems, we denote c(i,s ) by f , and
i i
c(i,o )byf∗. (Thus, f andf∗ arethe movementcostsofiinS andO respectively.) Also, we abbreviate
i i i i
c(cid:0)j,σ(j)(cid:1)toc ,andc(cid:0)j,σ∗(j)(cid:1)toc∗. Thus,c andc∗ aretheassignmentcostsofj inthelocalandglobal
j j j j
(cid:80) (cid:80)
optimum respectively. (So MFL(S) = f + c .) Let D(s) = {j ∈ D : σ(j) = s} be the set
i∈F i j∈D j
of clients assigned to the location s ∈ S, and D∗(o) = {j ∈ D : σ∗(j) = o}. For a set A ⊆ S, we define
D(A) = (cid:83) D(s);wedefineD∗(A)forA ⊆ Osimilarly. Definecap(s) = {o ∈ O : σ(o) = s}. Wesay
s∈A
thatscapturesallthelocationsincap(s). Thefollowingbasiclemmawillbeusedrepeatedly.
Lemma3.2 Foranyclientj,wehavec(cid:0)j,σ(σ∗(j))(cid:1)−c(cid:0)j,σ(j)(cid:1) ≤ 2c∗.
j
Proof : Let s = σ(j), o = σ∗(j), s(cid:48) = σ(o). The lemma clearly holds if s(cid:48) = s. Otherwise, c(j,s(cid:48))−
c(j,s) ≤ c(j,o)+c(o,s(cid:48))−c(j,s) ≤ c∗+c(o,s)−c(j,s) ≤ c∗+c(o,j) = 2c∗wherethesecondinequality
j j j
followssinces(cid:48) istheclosestlocationtooinS.
To prove the approximation ratio, we will specify a set of local-search moves for the local optimum,
andusethefactthatnoneofthesemovesimprovethecosttoobtainsomeinequalities,whichwilltogether
yield a bound on the cost of the local optimum. We describe these moves by using the following digraph.
(cid:0) (cid:1)
ConsiderthedigraphG(cid:98) = F∪S∪O,{(si,i),(i,oi),(oi,σ(oi))}i∈F . WedecomposeG(cid:98)intoacollection
ofnode-disjoint(simple)pathsP andcyclesCasfollows. Repeatedly,whilethereisacycleC inourcurrent
digraph, we add C to C, remove all the nodes of C and recurse on the remaining digraph. After this step,
a node v in the remaining digraph, which is acyclic, has: exactly one outgoing arc if v ∈ S; exactly one
incomingandoneoutgoingarcifv ∈ F;andexactlyoneincoming,andatmostoneoutgoingarcifv ∈ O.
Nowwerepeatedlychooseanodev ∈ S withnoincomingarcs,includethemaximalpathP startingatvin
P, remove all nodes of P and recurse on the remaining digraph. Thus, each triple (s ,i,o ) is on a unique
i i
pathorcycleinP∪C. Definecenter(s)tobeo ∈ Osuchthat(o,s)isanarcinP∪C;ifshasnoincoming
arcinP ∪C,thenletcenter(s) = nil.
We will use P and C to define our swaps. For a path P = (s ,i ,o ,...,s ,i ,o ) ∈ P, define
i1 1 i1 ir r ir
start(P)tobes andend(P)tobeo . Noticethatσ(o ) ∈/ P. Foreachs ∈ S,letP (s) = {P : end(P) ∈
i1 ir ir c
cap(s)},T(s) = {start(P) : P ∈ P (s)},andH(s) = {end(P) : P ∈ P (s)} = cap(s)\center(s). Note
c c
5
that |P (s)| = |T(s)| = |H(s)| = |cap(s)|−1 for any s ∈ S with |cap(s)| ≥ 1. For a set A ⊆ S, define
c
(cid:83) (cid:83) (cid:83)
T(A) = T(s), H(A) = H(s), P (A) = P (s).
s∈A s∈A c s∈A c
Abasicbuildingblockinouranalysis,involvesashiftalongans (cid:32) o = o sub-pathZ ofsomepathor
i(cid:48)
cycleinP ∪C. Thismeansthatweswapoutsandswapino. Weboundthecostofthematchingbetween
F andS∪{o}\{s}bymovingeachinitiallocationi ∈ Z, i (cid:54)= i(cid:48) toσ(o ) ∈ Z andmovingi(cid:48) too . Thus,
i i(cid:48)
weobtainthefollowingsimpleboundontheincreaseinmovementcostduetothisoperation:
shift(s,o) = (cid:88)(f∗−f )+ (cid:88) c(cid:0)o ,σ(o )(cid:1) ≤ 2(cid:88)f∗−c(o,σ(o)). (1)
i i i i i
i∈Z i∈Z:oi(cid:54)=o i∈Z
Thelastinequalityusesthefactthatc(cid:0)o ,σ(o )(cid:1) ≤ c(o ,s ) ≤ f∗ +f foralli. ForapathP ∈ P,weuse
i i i i i i
(cid:0) (cid:1)
shift(P)asashorthandforshift start(P),end(P) .
3.1 Theswapsused,andtheiranalysis
We now describe the local moves used in the analysis. We define a set of swaps such that each o ∈ O is
swappedintoanextentofatleastone,andatmosttwo. WeclassifyeachlocationinS asoneofthreetypes.
Definet = (cid:4)p1/3(cid:5). Weassumethatt ≥ 2.
• S : locationss ∈ S with|cap(s)| = 0.
0
• S : locationss ∈ S \S with|D∗(center(s))| ≤ tor|cap(s)| > t.
1 0
• S : locationss ∈ S with|D∗(center(s))| > tand0 < |cap(s)| ≤ t.
2
AlsodefineS := S ∪{s ∈ S : |cap(s)| ≤ t}(sos ∈ S iff|cap(s)| ≤ tand|D∗(center(s))| ≤ t}).
3 0 1 3
To gain some intuition, notice that it is easy to generate a suitable inequality for a location s ∈ S :
0
we can “delete” s (i.e., if s = s , then do swap(s,i)) and reassign each j ∈ D(s) to σ(σ∗(j)) (i.e., the
i
location in S closest to the location serving j in O). The cost increase due to this reassignment is at most
(cid:80) 2c∗, and so this yields the inequality f ≤ (cid:80) 2c∗. (We do not actually do this since we
j∈D(s) j i j∈D(s) j
take care of the S -locations along with the S -locations.) We can also generate a suitable inequality for
0 1
a location s ∈ S (see Lemma 3.4) since we can swap in cap(s) and swap out {s} ∪ T(s). The cost
2
(cid:80) (cid:0) (cid:1)
increase by this move can be bounded by shift(P) and c s,center(s) , and the latter quantity
P∈Pc(s)
canbechargedto 1 (cid:80) (c +c∗); ourdefinitionofS istailoredpreciselysoastoenablethis
t j∈D∗(center(s)) j j 2
latterchargingargument. GeneratinginequalitiesfortheS -locationsismoreinvolved,andrequiresanother
1
building block that we call an interval swap (this will also take care of the S -locations), which we define
0
afterprovingLemma3.4. WestartoutbyprovingasimpleboundthatonecanobtainusingacycleinC.
Lemma3.3 ForanycycleZ ∈ C,wehave0 ≤ (cid:80) (cid:0)−f +f∗+c(o ,σ(o ))(cid:1).
i∈Z i i i i
Proof: ConsiderthefollowingmatchingofF∩Z toS∩Z: wematchitoσ(o ). Thecostoftheresulting
i
(cid:80) (cid:80) (cid:80)
new matching is f + c(i,σ(o )) which should at least f since the latter is the min-cost
i∈/Z i i∈Z i i i
wayofmatchingF toS. Soweobtain0 ≤ (cid:80) (cid:0)−f +c(i,σ(o ))(cid:1) ≤ (cid:80) (cid:0)−f +f∗+c(o ,σ(o ))(cid:1).
i∈Z i i i∈Z i i i i
Lemma3.4 Lets ∈ S ando = center(s),andconsiderswap(X := {s}∪T(s),Y := cap(s)). Wehave
2
0 ≤ MFL(cid:0)(S\X)∪Y(cid:1)−MFL(S) ≤ (cid:88) 2f∗+ (cid:88) (cid:16)t+1 ·c∗− t−1 ·c (cid:17)+ (cid:88) 2c∗. (2)
i t j t j j
P∈Pc(s) j∈D∗(o) j∈D({s}∪T(s))
i∈P j∈/D∗(o)
6
Proof : We can view this multi-location swap as doing swap(start(P),end(P)) for each P ∈ P (s) and
c
(cid:0) (cid:1)
swap(s,o)simultaneously. (NoticethatnopathP ∈ P (s)containss,sinces = σ end(P) ∈/ P.) Foreach
c
swap(start(P),end(P))themovement-costincreaseisboundedbyshift(P) ≤ (cid:80) 2f∗. Forswap(s,o)
i∈P i
wemovethefacilityi,wheres = s ,too,sotheincreaseinmovementcostisatmostc(s,o) = c(σ(o),o) ≤
i
c(σ(j),o) ≤ c +c∗ foreveryj ∈ D∗(o). Sosince|D∗(o)| > t,wehavec(s,o) ≤ (cid:80) cj+c∗j. Thus,
j j j∈D∗(o) t
theincreaseintotalmovementcostisatmost
WeupperboundthechangeinassignmentcostbyreassigningtheclientsinD∗(o)∪D(X)asfollows.
We reassign each j ∈ D∗(o) to o. Each j ∈ D(X) \ D∗(o) is assigned to σ∗(j), if σ∗(j) ∈ Y, and
otherwise to s(cid:48) = σ(σ∗(j)). Note that s(cid:48) ∈/ X: s(cid:48) (cid:54)= s since σ∗(j) ∈/ cap(s), and s(cid:48) ∈/ T(s) since
(cid:83) cap(s(cid:48)(cid:48)) = ∅. Thechangeinassignmentcostforeachsuchclientj isatmost2c∗ byLemma3.2.
s(cid:48)(cid:48)∈T(s) j
Thus the change in total assignment cost is at most (cid:80) (c∗ −c )+(cid:80) 2c∗. Combining
j∈D∗(o) j j j∈D(X)\D∗(o) j
thiswiththeboundonthemovement-costchangeprovesthelemma.
Wenowdefineakeyingredientofouranalysis,calledaninterval-swapoperation,thatisusedtobound
themovementcostoftheS -andS -locationsandtheassignmentcostoftheclientstheyserve. (Webuild
1 0
upon this in Section 4 to give a tighter analysis proving a 3-approximation.) Let S(cid:48) = {s(cid:48),...,s(cid:48)} ⊆
1 r
S ∪S , r ≤ t2 be a subset of at most t2 locations on a path or cycle Z in P ∪C, where s(cid:48) is the next
0 1 q+1
locationin(S ∪S )∩Z afters(cid:48). LetO(cid:48) = {o(cid:48),...,o(cid:48)} ⊆ O whereo(cid:48) = center(s(cid:48))forq = 2,...,r
0 1 q 1 r q−1 q
and o(cid:48) is an arbitrary location that appears after s(cid:48) (and before s(cid:48)) on the corresponding path or cycle.
r r 1
Consider each s(cid:48). If |cap(s(cid:48))| > t, choose a random path P ∈ P (s(cid:48)) with probability 1 , and set
q q c q |Pc(s(cid:48)q)|
X = {start(P)} and Y = {o(cid:48)}. If |cap(s(cid:48))| ≤ t, set X = {s(cid:48)}∪T(s(cid:48)), and Y = {o(cid:48)}∪H(s(cid:48)). Set
q q q q q q q q q q
X = (cid:83)r X andY = (cid:83)r Y . Notethat|X| = |Y| ≤ t3 since|X | = |Y | ≤ tforeveryq = 1,...,r.
q=1 q q=1 q q q
NoticethatX isarandomset,butY = O(cid:48) ∪H(S(cid:48) ∩S )isdeterministic. Toavoidcumbersomenotation,
3
weuseswap(X,Y)torefertothedistributionofswap-movesthatresultsbytherandomchoicesabove,and
call this the interval swap corresponding to S(cid:48) and O(cid:48). We bound the expected change in cost due to this
movebelow. Let1(s)betheindicatorfunctionthatis1ifs ∈ S and0otherwise.
3
Lemma3.5 LetS(cid:48) = {s(cid:48),··· ,s(cid:48)} ⊆ S ∪S , r ≤ t2andO(cid:48)beasgivenabove. Leto(cid:48) := center(s(cid:48)) = o ,
1 r 0 1 0 1 ˆi
where o(cid:48) = nil and D∗(o(cid:48)) = ∅ if s(cid:48) ∈ S . Consider the interval swap swap(cid:0)X = (cid:83)r X ,Y =
0 0 1 0 q=1 q
(cid:83)r Y (cid:1)correspondingtoS(cid:48) andO(cid:48),asdefinedabove. Wehave
q=1 q
r
0 ≤ E(cid:2)MFL(cid:0)(S \X)∪Y(cid:1)−MFL(S)(cid:3) ≤ (cid:88)shift(s(cid:48),o(cid:48))+ (cid:88) 2f∗+ (cid:88) (c∗−c )
q q i j j
q=1 P∈Pc(S(cid:48)),i∈P j∈D∗(O(cid:48)) (3)
+ (cid:88) 2c∗+ (cid:88) 2c∗j +1(s(cid:48)) (cid:88) (f∗+f +c∗).
j t 1 ˆi ˆi j
j∈D(T(S(cid:48)∩S3)∪(S(cid:48)∩S3)) j∈D(T(S(cid:48)\S3)) j∈D∗(o(cid:48)0)
Proof: LetZ bethepathinP orcycleinC suchthatS(cid:48)∪O(cid:48) ⊆ Z.
We first bound the increase in movement cost. The interval swap can be viewed as a collection of
simultaneous swap(X ,Y ), q = 1,...,r moves. If X = {start(P)} for a random path P ∈ P (s(cid:48)),
q q q c q
the movement-cost increase can be broken into two parts. We do a shift along P, but move the last initial
locationonP tos(cid:48),andthendoshiftonZ froms(cid:48) too(cid:48). Sotheexpectedmovement-costchangeisatmost
q q q
1 (cid:88) (cid:0)shift(P)+c(end(P),s(cid:48))(cid:1)+shift(s(cid:48),o(cid:48)) ≤ 1 (cid:88) 2f∗+shift(s(cid:48),o(cid:48))
|P (s(cid:48))| q q q |P (s(cid:48))| i q q
c q P∈Pc(s(cid:48)q) c q P∈Pc(s(cid:48)q),i∈P
whichisatmost(cid:80) 2f∗+shift(s(cid:48),o(cid:48)). Similarly,if|cap(s(cid:48))| ≤ t,wecanbreakthemovement-
cost increase into sPhi∈fPt(cP(s(cid:48)q)),≤i∈P(cid:80) i 2f∗ forqallqP ∈ P (s(cid:48)) and shiftq(s(cid:48),o(cid:48)). Thus, the total increase in
i∈P i c q q q
7
movementcostisatmost
r
(cid:88) (cid:88)
shift(s(cid:48),o(cid:48))+ 2f∗. (4)
q q i
q=1 P∈Pc(S(cid:48)),i∈P
Next, we bound the change in assignment cost by reassigning clients in D(cid:98) = D∗(O(cid:48)) ∪ D(X) as
follows. We assign each client j ∈ D∗(O(cid:48)) to σ∗(j). If |cap(s(cid:48))| > t, then s(cid:48) ∈/ X. For every client
1 1
j ∈ D(cid:98) \ (D∗(O(cid:48))), observe that either σ∗(j) ∈ Y or σ(σ∗(j)) ∈/ X. To see this, let o = σ∗(j) and
s = σ(o). Ifo ∈/ Y thens ∈/ S(cid:48)∩S ;alsos ∈/ T(S(cid:48)),andsos ∈/ X. Soweassignj toσ∗(j)ifσ∗(j) ∈ Y
3
andtoσ(σ∗(j))otherwise;thechangeinassignmentcostofj isatmost2c∗ (Lemma3.2).
j
Now suppose |cap(s(cid:48))| ≤ t, so s(cid:48) ∈ X. For each j ∈ D(cid:98) \ (D∗(O(cid:48)) ∪ D∗(o(cid:48))), we again have
1 1 o
σ∗(j) ∈ Y orσ(σ∗(j)) ∈/ X,andweassignj toσ∗(j)ifσ∗(j) ∈ Y andtoσ(σ∗(j))otherwise. Weassign
everyj ∈ D(cid:98) ∩D∗(o(cid:48)0)tosˆi (recallthato(cid:48)0 = oˆi),andoverestimatetheresultingchangeinassignmentcost
by (cid:80) (c∗ + f∗ + f ). Finally, note that we reassign a client j ∈ D(T(S(cid:48) \ S )) \ D∗(O(cid:48)) with
j∈D∗(o(cid:48)o) j ˆi ˆi 3
probabilityatmost 1 (sinceσ(j) ∈ X withprobabilityatmost 1). Sotakingintoaccountallcases,wecan
t t
boundthechangeintotalassignmentcostby
(cid:88) (c∗−c )+ (cid:88) 2c∗+ (cid:88) 2c∗j +1(s(cid:48)) (cid:88) (f∗+f +c∗). (5)
j j j t 1 ˆi ˆi j
j∈D∗(O(cid:48)) j∈D(T(S(cid:48)∩S3)∪(S(cid:48)∩S3)) j∈D(T(S(cid:48)\S3)) j∈D∗(o(cid:48)0)
In(5),wearedouble-countingclientsinD(cid:0)T(S(cid:48))∪(S(cid:48) ∪S )(cid:1)∩D∗(O(cid:48)). Wearealsooverestimatingthe
3
change in assignment cost of a client j ∈ D(X)∩D∗(o(cid:48)) since we include both the 1(s(cid:48))(c∗ +f∗ +f )
0 1 j ˆi ˆi
2c∗
term,andthe2c∗ or j terms. Adding(4)and(5)yieldsthelemma.
j t
Notice that Lemma 3.4 immediately translates to a bound on the assignment cost of the clients in
D∗(center(s))fors ∈ S . In contrast, itis quite unclear howLemma 3.5 may beuseful, since the expres-
2
sion(cid:80) (f∗ +f )intheRHSof(3)maybeaslargeast(f∗ +f )(butnomoresince|D∗(o(cid:48))| ≤ t
j∈D∗(o(cid:48)0) ˆi ˆi ˆi ˆi 0
if 1(s(cid:48)) = 1) and it is unclear how to cancel the contribution of f on the RHS. One of the novelties of
1 ˆi
our analysis is that we show how to amortize such expensive terms and make their contribution negligible
by considering multiple interval swaps. We cover each path or cycle Z in t2 different ways using intervals
comprisingconsecutivelocationsfromS ∪S . Wethenarguethataveraging,overtheset2 coveringways,
0 1
the inequalities obtained from the corresponding interval swaps yields (among other things) a good bound
onthemovement-costofthe(S ∪S )-locationsonZ andtheassignmentcostoftheclientstheyserve.
0 1
Lemma3.6 LetZ ∈ P∪C,S(cid:48) = {s(cid:48),...,s(cid:48)} = S ∩Z,wheres(cid:48) isthenextS -locationonZ afters(cid:48),
1 r 1 q+1 1 q
andO(cid:48) = {center(s(cid:48)),...,center(s(cid:48))}. Leto(cid:48) = end(Z)ifZ ∈ P andcenter(s(cid:48))otherwise. Forr ≥ t2,
1 r r 1
(cid:88)(cid:16) (cid:17) (cid:88) (cid:88) (cid:16) (cid:17)
0 ≤ t+1 ·f∗− t−1f + 2f∗+ 1 ·c + t+1 ·c∗
t i t i i t j t2 j
i∈Z P∈Pc(S(cid:48)),i∈P j∈D∗(Z∩O)
(6)
+ (cid:88) (c∗−c )+ (cid:88) 2c∗+ (cid:88) 2c∗j.
j j j t
j∈D∗(O(cid:48)∪{o(cid:48)r}) j∈D(T(Z∩S3)∪(Z∩S3)) j∈D(T(S(cid:48)\S3))
Proof : We first define formally an interval of (at most) t2 consecutive (S ∪S ) locations along Z. As
0 1
before, leto(cid:48) = center(s(cid:48))forq = 1,...,r. Fora pathZ, defines(cid:48) = start(Z)forq ≤ 0ands(cid:48) = nil
q−1 q q q
for q > r. Also define o(cid:48) = o(cid:48) for q ≤ 0 and o(cid:48) = end(Z) for q ≥ r. If Z is a cycle, we let our indices
q 0 q
wraparoundandbe mod r,i.e.,s(cid:48) = s(cid:48) , o(cid:48) = o(cid:48) forallq (soo(cid:48) = o(cid:48) = center(s(cid:48))).
q qmodr q qmodr r 0 1
For 1 − t2 ≤ h ≤ r, define S(cid:48) = {s(cid:48) ,s(cid:48) ,...,s(cid:48) } to be an interval of length at most t2 on
h h h+1 h+t2−1
Z. Define O(cid:48) = {o(cid:48) ,o(cid:48) ,...,o(cid:48) }. Note that we have 1 ≤ |S(cid:48)| = |O(cid:48)| ≤ t2 if Z is a path, and
h h h+1 h+t2−1 h h
8
|S(cid:48)| = |O(cid:48)| = t2 ifZ isacycle. Considerthecollectionofintervals,{S(cid:48) ,S(cid:48) ,··· ,S(cid:48)}. Foreach
h h −t2+1 −t2+2 r
S(cid:48),O(cid:48),where−t2+1 ≤ h ≤ r,weconsidertheintervalswap(X ,Y )correspondingtoS(cid:48),O(cid:48). Weadd
h h h h h h
theinequalities 1×(3)forallsuchh. Sinceeachs(cid:48) ∈ S(cid:48)∪{s(cid:48)}participatesinexactlyt2 suchinequalities,
t2 0
andeachs(cid:48) ∈ S(cid:48) isthestartofonlytheintervalS(cid:48),weobtainthefollowing.
h h
r
(cid:88) 1 (cid:88) 1
0 ≤ ·t2·shift(s(cid:48),o(cid:48))+ ·t2·2f∗
t2 q q t2 i
q=0 P∈Pc(S(cid:48)),i∈P
+ (cid:88) t12 ·t2·(c∗j −cj)+ (cid:88) t12 ·t2·2c∗j + (cid:88) t12 ·t2· 2ct∗j (7)
j∈D∗(O(cid:48)∪{o(cid:48)r})) j∈D(T(Z∩S3)∪(Z∩S3)) j∈D(T(S(cid:48)\S3))
(cid:88) 1 (cid:88)
+ 1(σ(o ))· · (f∗+f +c∗).
i t2 i i j
i:σ(oi)∈Z j∈D∗(oi)
Notice that the S-locations other than s(cid:48) on the s(cid:48) (cid:32) o(cid:48) sub-paths of Z lie in S , and for each i such
q q q 2
thatσ(o ) ∈ Z ∩S ,wehavec(o ,σ(o )) ≤ (cid:80) cj+c∗j. Thus,using(1),wehave
i 2 i i j∈D∗(oi) t
(cid:88)r shift(s(cid:48),o(cid:48)) = (cid:88)(f∗−f )+ (cid:88) c(o ,σ(o )) ≤ (cid:88)(f∗−f )+ (cid:88) cj +c∗j. (8)
q q i i i i i i t
q=0 i∈Z i:σ(oi)∈Z∩S2 i∈Z j∈D∗(Z∩O)
Since1(σ(o)) = 1meansthatσ(o) ∈ S ,andso|D∗(o)| ≤ t,wehave
3
(cid:88) (cid:88) fi∗+fi+c∗j ≤ (cid:88)(cid:16)fi∗+fi +(cid:80)j∈D∗(oi)c∗j(cid:17) ≤ (cid:88) fi∗+fi + (cid:88) c∗j. (9)
t2 t t2 t t2
i:σ(oi)∈Z∩S3j∈D∗(oi) i∈Z i∈Z j∈D∗(Z∩O)
Incorporating(8)and(9)in(7),andsimplifyingyieldsthedesiredinequality.
For a path or cycle Z where |S ∩Z| < t2, we obtain an inequality similar to (6). Since we can now
1
coverZ withasingleinterval,weneverhaveaclientj suchthatnoneofσ(j), σ∗(j), σ(σ∗(j))areinour
newsetoffinallocations. Sotheresultinginequalitydoesnothaveany fi∗+fi + c∗j terms.
t t2
Lemma3.7 LetZ ∈ P∪C,S(cid:48) = {s(cid:48),...,s(cid:48)} = S ∩Z,wheres(cid:48) isthenextS -locationonZ afters(cid:48),
1 r 1 q+1 1 q
andO(cid:48) = {center(s(cid:48)),...,center(s(cid:48))}. Leto(cid:48) = end(Z)ifZ ∈ P andcenter(s(cid:48))otherwise. Forr(cid:48) < t2,
1 r r 1
0 ≤ (cid:88)(f∗−f )+ (cid:88) 2f∗+ (cid:88) cj +c∗j
i i i t
i∈Z P∈Pc(S(cid:48)),i∈P j∈D∗(Z∩O) (10)
+ (cid:88) (c∗−c )+ (cid:88) 2c∗+ (cid:88) 2c∗j.
j j j t
j∈D∗(O(cid:48)∪{o(cid:48)r}) j∈D(T(Z∩S3)∪(Z∩S3)) j∈D(T(S(cid:48)\S3))
Proof: TheproofissimilartothatofLemma3.6,exceptthatsincewecancoverZ withasingleinterval,
weonlyneedtoconsiderasingle(multi-location)swap. Weconsidertwocasesforclarity.
1. Z is a path, or r > 0. As before, let o(cid:48) = center(s(cid:48)) for q = 1,...,r. If Z is a path, define
q−1 q
s(cid:48) = start(Z). IfZ isacycle,weagainsets(cid:48) = s(cid:48) , o(cid:48) = o(cid:48) forallq. Considertheinterval
0 q qmodr q qmodr
swap (X,Y) corresponding to S(cid:48) ∪{s(cid:48)},O(cid:48) ∪{o(cid:48)}. The inequality generated by this is similar to (3)
0 r
exceptthatwedonothaveany(f∗+f +c∗)termssinceforclientj ∈ D(X)∪D∗(Y),wealwayshave
ˆi ˆi j
thateitherσ∗(j) ∈ Y orσ(σ∗(j)) ∈/ X. Thus,(3)translatestothefollowing.
r
0 ≤ (cid:88)shift(s(cid:48),o(cid:48))+ (cid:88) 2f∗+ (cid:88) (c∗−c )+ (cid:88) 2c∗+ (cid:88) 2c∗j.
q q i j j j t
q=0 P∈Pc(S(cid:48)),i∈P j∈D∗(O(cid:48)∪{o(cid:48)r}) j∈D(T(Z∩S3)∪(Z∩S3)) j∈D(T(S(cid:48)\S3))
9
Substituting (cid:80)r shift(s(cid:48),o(cid:48)) ≤ (cid:80) (f∗ −f )+(cid:80) cj+c∗j as in (8) yields the stated in-
q=0 q q i∈Z i i j∈D∗(Z∩O) t
equality.
2. Z isacyclewithr = 0. Here,Lemma3.3yields0 ≤ (cid:80) (f∗ −f )+(cid:80) cj+c∗j (whichis
i∈Z i i j∈D∗(Z∩O) t
thespecialcaseoftheearlierinequalitywiths(cid:48) = nil = o(cid:48), S(cid:48) = O(cid:48) = Z ∩S = ∅).
0 r 3
ProofofTheorem3.1: Weconsiderthefollowingsetofswaps.
(cid:0) (cid:1)
A1. Foreverys ∈ S ,themoveswap {s}∪T(s),cap(s) .
2
A2. ForeverypathorcycleZ with|Z∩S | ≥ t2,the 1-weightedintervalswapsasdefinedinLemma3.6.
1 t2
A3. ForeverypathorcycleZ with|Z ∩S | < t2,theintervalswapdefinedinLemma3.7.
1
Notice that every location o ∈ O is swapped in to an extent of at least 1 and at most 2. (By “extent”
we mean the total weight of the inequalities involving o.) To see this, suppose first o = end(Z) for some
path Z, then o is involved to an extent of 1 in the interval swaps for Z in A2 or A3. In this case, we say
that the interval swap for Z is responsible for o. Additionally, if s = σ(o) ∈ S , then o belongs to the
2
multi-swap for s in A1, else if s ∈ S then o is part of the interval swap for the path/cycle containing s in
3
A2orA3. Nowsupposeo = center(s). Ifs ∈ S ,thenoisincludedinthemulti-swapforsinA1. Wesay
2
thatthismulti-swapisresponsibleforo. Ifs ∈ S ,thenoisincludedintheintervalswapforthepath/cycle
1
containingsinA2orA3,andwesaythatthisintervalswapisresponsibleforo.
Consider the compound inequality obtained by summing (2), (6), and (10) corresponding to the moves
considered in A1, A2, and A3 respectively. The LHS of this inequality is 0. We now need to do some
bookkeeping to bound the coefficients of the f∗,f ,c∗,c terms on the RHS. We ignore o(1) coefficients
i i j j
like 1, 1 in this bookkeeping, since for a given {f∗,f ,c∗,c } term, such coefficients appear in only a
t t2 i i j j
constant number of inequalities, so they have an o(1) effect overall. Let F and C denote respectively the
movement-andassignment-costofthelocaloptimum.
Contribution from the c∗ and c terms. First, observe that for each o ∈ O, we have labeled exactly one
j j
move involving o as being responsible for it. Consider a client j ∈ D(s)∩D∗(o). Observe that c∗ or c
j j
termsappear(withaΘ(1)-coefficient)inaninequalitygeneratedbyamoveif(i)j isreassignedbecausethe
moveisresponsibleforo;or(ii)sisswappedout(toanextentof1)bythemove(sothisexcludesthecase
wheres ∈ T(s(cid:48)), s(cid:48) ∈ S \S andthemoveistheintervalswapforthepathcontainings(cid:48)). If(i)applies,
1 3
then the inequality generates the term (c∗ −c ). If (ii) applies then the term 2c∗ appears in the inequality.
j j j
Finally,notethatthereareatmosttwoinequalitiesforwhich(ii)applies:
– Ifs = start(Z) ∈ S ,then(ii)appliesfortheinterval-swapmoveforZ. Ifs(cid:48) = σ(end(Z)) ∈ S ∪S ,
0 2 3
then (ii) again applies, for the multi-swap move for s(cid:48) if s(cid:48) ∈ S , or for the interval swap for the path
2
containings(cid:48) ifs(cid:48) ∈ S .
3
– Ifs ∈ S ∩S ,then(ii)appliesfortheintervalswapforthepathcontainings.
1 3
– Ifs ∈ S ,then(ii)appliesforthemulti-swapmovefors.
2
So overall, we get a 5c∗ −c contribution to the RHS, the bottleneck being the two inequalities for which
j j
(ii)applieswhens ∈ start(Z)andσ(end(Z)) ∈ S ∪S .
2 3
Contribution from the f∗ and f terms. For every i ∈ F, the expression (f∗ − f ) is counted once in
i i i i
the RHS of the inequality (6) or (10) for the unique path or cycle Z containing i. The total contribution
of all these terms is therefore, F∗ − F. The remaining contribution comes from expressions of the form
(cid:80) 2f∗ on the RHS of (2), (6), and (10). The paths P involved in these expressions come from
P P(S∈P)c(∪s)(cid:0),i(cid:83)∈P i P (Z ∩S )(cid:1) ⊆ P. Therefore,thetotalcontributionofthesetermsisatmost2F∗.
c 2 Z∈P∪C c 3
Thus,weobtainthecompoundinequality
0 ≤ (cid:0)5+o(1)(cid:1)C∗+(cid:0)3+o(1)(cid:1)F∗−(cid:0)1−o(1)(cid:1)(F +C) (11)
10