CAHIER DU LAMSADE 162

Laboratoire d'Analyse et Modélisation de Systèmes pour l'Aide à la Décision

CNRS UMR 7024

CAHIER DU LAMSADE 162

Juin 1999

Master-slave strategy and polynomial approximation

Laurent Alfandari, Vangelis Th. Paschos

Résumé ii

Abstrat ii

1 Introdution 1

2 Master-slave tratable problems 2

3 Master-slave tratable network-design problems 5

3.1 Minimum spanning treeof depth 2 . . . 5

3.2 Minimum bounded-diameter spanning forest . . . 7

3.3 Edge-overing bytrees . . . 9

4 Disussion: introduing a new kind of redution 11

Referenes 12

A Pseudo-redution and maximization paking problems 14

A.1 Dualmaster-slave approximation . . . 14

A.2 Pseudo-redution and improved approximations for unweighted graph-paking

problems. . . 15

Résumé

De nombreuxproblèmesdeminimisation onsistantàouvrir lessommetsoulesarêtes

d'ungraphe pardessous-graphessontlassiquementmodélisésparunproblèmede seto-

vering.Si lenombredesous-graphesest polynomialen

n

^{,es problèmespeuventalors être}

approhés àrapport logarithmique parl'algorithme glouton standardpourle set overing.

Nous étendons la lasse des problèmes approximables parette approhe à des problèmes

de ouverture et de partitionnement où le nombre de sous-graphes peut être exponentiel

en

n

^{,enrevisitantunetehniqueappelée}

((

maître-eslave

))

^{etenl'étendantàdesproblèmes}

pondérés.Nousappliquonsnalementl'approhemaître-eslaveàdeuxproblèmesdedimen-

sionnementderéseauetunproblèmedeoneptiondeiruitséletroniquesandeproduire

desrésultatspositifsd'approximationpouresproblèmes.

Mots-lé: problèmesombinatoires,omplexité,algorithme polynomiald'approximation,

NP-omplétude,ouvertured'ensembles,dimensionnementdesréseaux,graphe,arbre,fret.

Master-slave strategy and polynomial approximations

Abstrat

A lot ofminimization overingproblems ongraphsonsistin overingvertiesoredges

bysubgraphs verifying aertain property. These problems an beseenas partiular ases

ofset-overing.Ifthenumberofsubgraphsispolynomialintheorder

n

^oftheinput-graph, then these problems an be approximated within logarithmi ratio bythe lassial greedy

set-overing algorithm. We extend the lass of problems approximable by this approah

to overing problems where the number of andidate subgraphs is exponential in

n

^{, by}

revisitinganoldtehniquealledmaster-slaveandextendingittoweightedmasteror/and

slaveproblems. Finally,weusethemaster-slavetoolto provesomepositiveapproximation

resultsfortwonetwork-designandaVLSI-designgraph-problems.

Keywords: ombinatorial problems,omputationalomplexity,polynomial-timeapproxi-

mationalgorithm,NP-ompleteness,set-overing,networkdesign,graph,tree,forest.

Given an NP-hard optimization problem

Π

^{, one is interested in ndinga polynomial time ap-}

proximationalgorithm(PTAA)

A

^{withagoodperformaneratio,thisratiobeingusuallydened}

as the worst, over all instanes of a given size, of the values of the fration measure of the

solution returned byalgorithm

A

^{over the measureof an optimalsolution.}

Let us restrit ourselves to NP-hard minimization overing graph-problems onsisting in

overingvertiesoredgesbysubgraphs verifyingaertainproperty. Someoftheseproblemsan

beapproximated bythe following thought proess: at eah step, one tries to over a maximum

number of elements (verties or edges, depending on the denition of the problem) among the

unoveredvertiesor edges. Themaximization problem solved ateah step isalled the slave

whih servesthemaster, the (original)minimizationproblem. The termsslave and master

aredueto Simon ([13 ℄)whopointsoutthe fatthatiftheslave-problem ispolynomialthen the

masteroneisapproximablewithin

O(log n)

^{. Thenitusesthisfatin ordertoprovethatifsome}

masterproblems areapproximate-equivalent,soarethe orrespondingslaveones(two problems

are saidapproximate-equivalent ifthey are linked byapproximation-preserving redutions, i.e.,

redutionsalong whihapproximationboundsandinaproximabilityresults aretransferredfrom

one to the other).

A lassial example ofmaster-slave approximation isthe one given by Johnson(in [8 ℄,algo-

rithm D3 at the end of setion 7 devoted to graph-oloring). At eah iteration this algorithm

omputes a maximum independent set of the surviving graph, it olors its verties by a new

olor and removes them from the graph. The master problem in this ase is the minimum

graph-oloring, whilethe slave oneisthe maximum independent set (forreasonsofeonomywe

do not dene here well-known NP-omplete problems suh asgraph-oloring, independent set,

set-overing,set-paking,dominatingset, hittingset,et.;theirdenitionsaregivenin[6℄). The

drawbak of D3 is exatly that it is not polynomial, sine it uses a maximum independent set

omputation. In anyase, it ahieves approximationratio

O(ln n)

^{at worst.}

Here,werstextendthemaster-slavestrategytoNP-hardweightedoveringandpartitioning

problems. Butour main purpose is to add a ontribution towards asystemati lassiation of

NP-omplete problems with respet to their approximability and, also, to bring to the fore a

uniedwaytoobtainapproximationresultsforaertainlassofproblemsforwhihahievement

oflogarithmiratiosisnot obvious. Forthis,wedenethe lassM-STofmaster-slavetratable

problems, approximable, in polynomial time, within logarithmi ratio. Informally speaking,

givenapartiular unweighted NP-hardgraph-overingproblem

Π

^{,wetrytoshowthat}

Π

^anbe

modeled asa set-overing problem, nomatterifthe size ofthe obtained set-overing instaneis

exponentialinthesizeofthegeneriinstaneof

Π

^{. The}set-overinginstaneobtainedinludesas groundsetthesetofobjets(vertiesoredges)tobeovered,andasset-systemallthesubgraphs

ableto beinluded into afeasible solution. Via thismodeling,

Π

^{an be solved bythe following}

kind ofgreedy iterative proedure G:at eahiteration tryto over the maximum number of the

unoveredobjets. Ifoneanmodeltheproblemathandintermsofset-overing,andifonean

prove thatovering the maximum number of unovered objetsan beperformedin polynomial

time (even ifthe expliit onstrution of the set-overing instane isexponential), then one has

proved that

Π

^{belongs to the lass M-ST. Sine G is a kind of simulation of the greedy set-}

overing algorithm, one an onlude that

Π

^{, aswell asevery M-ST problem, is}approximable within approximation ratio equal to the one of the greedy set-overing algorithm; this ratio

is

O(log n)

^{([14 ℄). In other words, insteadof developing and analyzing a proper} approximation algorithmforeverygraph-overingproblem,werathertrytoprovethatitsatisestheonditions

ofinlusion in M-ST (letus notethatproof ofthe inlusion ofa problemin lassM-ST isnot

trivialatall). Then,approximabilityof

Π

^{withinlogarithmiratioensues}immediately. Usingthis

wenotethatahievement oflogarithmiapproximationratiosforthemis,toourknowledge,new

and seems non-trivial. At the end of the paper, we try to insribe the master-slave game into

the formalframework ofanewkindofredution,bymeansofwhihwe showthat, insome way,

setovering isompleteforthe lassM-ST.Finally, letus notethatinlusionin lassM-STis

interesting, not only for proving logarithmi ratios, but also for proving lower approximability-

bounds(inapproximability results). Plainly, as it is proved in [1℄, two of the problems studied

arenotapproximable within

(1 − ǫ) ln n

^,

∀ǫ > 0

^{. Inlusionof themin M-STonstitutesaproof}

that master-slave approximation is the best one an do in order to approximately solve these

problems.

In what follows, given an instane

I

^{of an NP-hard problem}

Π

^{and a PTAA A for}

Π

^{, we}

denotebyOPT(

I

^{)theoptimalvalueof}

I

^,byA(

I

^{)thevalueofthesolutionof}

I

^{providedbyA,and}

wesaythatalgorithmA approximates

Π

^withinratio

ρ

^if

A(I)/OP T (I ) ≤ ρ

^{,for everyinstane}

I

of

Π

^.

2 Master-slave tratable problems

We denote by

S P

^{the set of subgraphs of a graph}

G = (V, E)

^{satisfying a property or a pred-}

iate

P

^{, by}

V (G)

^(resp.,

E(G)

^{) the vertex-set (resp., edge-set) of}

G

^{, by}

n

^{its order (}

|V |

^{) and}

dene the following lass.

Denition 1. ConsideranNP-hardminimizationgraph-problem

Π

^andaproperty

P

^{; suppose}

that (i) a feasible solution of

Π

^{is an element of}

2 ^{S P}

^{and (ii) a ost funtion}

¯ c

^{, omputable}

in polynomial time, is assoiated with

S ∈ S P

^{suh that the ost of a solution}

S ^′

^of

Π

^is

c(S ^′ ) = ^P _S∈S ^′ c(S) ¯

^{. Then}

Π

^is master-slave tratable (M-ST) i it satises the following onditions:

[M-ST-1℄ a solution

S ^′

^of

Π

^is

•

^{either a}

P

^{-over (i.e., asubset}

S ^′ ⊂ S P

^suhthat

∪ _S∈S ^′ V (S) = V

^),

•

^{or a}

P

^{-partition (i.e., a subset}

S ^′ ⊂ S P

^{suh that}

∪ S∈S ^′ V (S) = V

^{and for}

S, S ^′ ∈ S ^′

^,

V (S) ∩ V (S ^′ ) = ∅

^{), and every}

P

^{-over of}

G

^{an be} polynomially transformed into a

P

^{-partition of at most the sameost;}

[M-ST-2℄ givenabinaryvetor

~u ∈ {0, 1} ⁿ

^{assoiatedwith}

V

^{,the (slave)problemofndingthe}

subgraph

S ^∗ = argmax _{S∈S P} { ^P _{v∈V (S)} u[v]/¯ c(S)}

^{isin P.}

For the ase ofedge-over or partition, wesimplyreplae

v

^by

e

^and

V

^by

E

^.

Theabovedenitiongeneralizesthelassialmaster-slavemethod([13℄)where,

∀S ∈ S P

^,

c(S) = ¯ 1

^{, i.e., where}

c(S ^′ ) = |S ^′ |

^{. Note thatinthe aseof} partitioning, ifproperty

P

^{ishereditary(i.e.,}

every subgraph of

S

^satises

P

^whenever

S

^satises

P

^{), then ondition[M-ST-1℄of denition1}

is always satised. However, anypartitioning problem doesnot satisfyondition [M-ST-1℄; for

instane, from a set-over one annot systematially obtain a set-partition of the same weight,

or ardinality.

WenowshowthatM-STproblemsareapproximablewithinlogarithmiratiobythefollowing

simulation of the lassial greedy weighted set-overing (WSC) algorithm.

BEGIN /*MSGREEDY*/

(1) FOR

v ∈ V

^DO

u[v] ← 1

^OD

(2)

S ^′ ← ∅

^;

(3) WHILE

∃v ∈ V : u[v] = 1

^DO

(4)

S ^∗ ← argmax _{S∈S P} { ^P _v∈V(S) u[v]/¯ c(S)}

^;

(5)

S ^′ ← S ^′ ∪ {S ^∗ }

^;

(6) FOR

v ∈ V(S ^∗ )

^DO

u[v] ← 0

^OD

(7) OD

(8) OUTPUT

S ^′

^{; (*OUTPUT}

h(S ^′ )

^;*)

END. /*MSGREEDY*/

Inthe abovealgorithm,

u[v]

^{indiates wether vertex}

v

^{is unovered(}

u[v] = 1

^{) or not (}

u[v] = O

⁾

atthe urrentstepof theWHILE loop. Funtion

h

^atline(8) polynomially transformsa

P

^-over

intoa

P

^{-partition(when dealingwitha}partitioning-problem). Foredge-overingorpartitioning problems,simplyreplae

v

^(resp.,

V

^{) by}

e

^(resp.,

E

^).

Theorem 1. If

Π

^{is M-ST, then MSGREEDY} polynomially approximates

Π

^within

min{1 + ln ∆ _Π , ln n − ln ln n + 0.78}

^{iftheostsofeverysubgraphsatisfying}

P

^{are allidential, andwithin}

1 + ln ∆ _Π

^{otherwise, with}

∆ _Π = max _{S∈S P} {|S|}

^.

Proof. We prove the theorem for vertex-overing and partitioning problems, the proof being

quite similarin ase of edge-overing or partitioning. Letus transform an instane

G = (V, E)

of

Π

^{into an instane}

ϕ(G) = (C, S) ¯

^{of WSC in the following way. Let}

C = V

^{be the ground}

elementsetin

ϕ(G)

^{,andforeverysubgraph}

S ∈ S P

^,addaset

S ¯ = V (S)

^withweight

w( ¯ S) = ¯ c(S)

in

S ¯

^{(note that the number of sets}

| S| ¯

^{an be} exponential in

n

^{). Under this} transformation, thereisa1-1orrespondenebetween thesolutionsof

Π

^on

G

^{andthe solutionsofWSCon}

ϕ(G)

onstruted as above, i.e.,

{S ₁ , S ₂ , . . . , S t }

^{is a}

P

^{-over for}

G

ⁱ

{ S ¯ ₁ , S ¯ ₂ , . . . , S ¯ t }

^{is a set-over}

for

ϕ(G)

^{and, moreover the ost of the}

P

^{-over of}

G

^{is the same as the total weight of the}

set-over of

ϕ(G)

^{. Hene,}

OPT(G) = OPT(ϕ(G))

^{. Now, the greedy algorithm for the WSC-}

instane

(C, S) ¯

^{an be re-written in the following way.}

BEGIN /*WSCGREEDY*/

FOR

c ∈ C

^DO

u[c] ← 1

^OD

S ¯ ^′ ← ∅

^;

WHILE

∃c ∈ C : u[c] = 1

^DO

¯ S ^∗ ← argmax ¯ S∈ S ¯ { ^P _c∈ ¯ S u[c]/w( ¯ S)}

^;

S ¯ ^′ ← S ¯ ^′ ∪ { ¯ S ^∗ }

^;

FOR

c ∈ ¯ S ^∗ ∩ C

^DO

u[c] ← 0

^OD

OD

OUTPUT

S ¯ ^′

^;

END. /*WSCGREEDY*/

Sine subgraph

S ∈ S P

^{(resp., vertex-set}

V

^{) in}

G

^{orresponds to set}

S ¯

^{(resp., ground set}

C

⁾

in

ϕ(G)

^{, thensolution}

S ^′ = {S 1 , S ₂ , . . . , S _t }

^{omputedbyalgorithmMSGREEDYfor}

G

^orresponds

to the over

S ¯ ^′ = { S ¯ ₁ , S ¯ ₂ , . . . , S ¯ t }

^for

ϕ(G)

^{, and}

c(S ^′ ) = w( ¯ S ^′ )

^{. Hene,}

c(S ^′ )/OP T (G) = w( ¯ S ^′ )/OP T (ϕ(G))

^.

If a feasible solution for

Π

^{is not a}

P

^{-over but a}

P

-partition, then by ondition [M-ST-1℄

P

^-over

S ^′

^anbetransformedinpolynomialtimeintoa

P

^-partition

S ^′′

^satisfying

c(S ^′′ ) ≤ c(S ^′ )

^.

The approximation ratio of WSCGREEDY is bounded above by

1 + ln ∆ _SC

^{, where}

∆ _SC = max S∈ ¯ S ¯ {| S|} ¯

^{in the weighted ase ([4 ℄), and by}

ln |C| − ln ln |C| + 0.78

^{in the unweighted}

ase([14 ℄). Sine in

ϕ(G)

^,

∆ _SC = ∆ _Π

^and

|C| = n

^{, the proofis ompleted.}

Let us remark here that all problems linked to set-overing by approximation-preserving

redutionsas,for example,thedominating set,the hittingset, aversionoftheoloringproblem

where the input graph has stability number bounded above by a xed positive onstant, et.,

are M-ST ones. Algorithm D3of [8℄ mentioned above is indeeda simulation of WSCGREEDY for

not in P.

Simon,in [13 ℄,mentionsthatifthe(unweighted) slave-problem isnot solvablein polynomial

time, but is approximable within a fator

ρ ≤ 1

^{, then the} (unweighted) master an be polyno- miallyapproximated with aratio

(1/ρ) ln n

^{. Ofourse}

ρ

^{an depend onan} instane-parameter.

Notethatanaïve appliation ofthisresultto graph-oloring,where ateahroundone olorsa

maximal independent set insteadof a maximum one, doesnot improve the bestapproximation

ratio for oloring. Plainly, the best ratio

(log n) ² /n

^{([3℄) for the maximum} independent set impliesa ratio

n/ log n

^{for oloring. Thisratio, obtained byJohnson ([9℄) in 1974,has been re-}

peatedlyimprovedsinethen. WeanextendSimon'sresulttotheweighted asewheretheosts

ofthe subgraphs in

S P

^{arenot onstrained tobeidential. Letus rsttransformdenition1 to}

inlude a larger lassofminimization problems.

Denition 2. Aminimizationgraph-problem

Π

^isextendedmaster-slave tratable (EM-ST)i itsatisesondition[M-ST-1℄ofdenition1and if,given abinaryvetor

~u ∈ {0, 1} ⁿ

^assoiated

with

V

^{,theproblemofndingthe subgraph}

S ^∗ = max _{S∈S P} { ^P _{v∈V (S)} u[v]/¯ c(S)}

^ispolynomially approximable within ratio

ρ ≤ 1

^.

Theorem 2. An EM-ST problem

Π

^ispolynomially approximable within

(1/ρ)(1 + ln ∆ _Π )

^.

Proof. Consider the transformation of an instane

G = (V, E)

^of

Π

^{into a} set-overing in- stane

(C, S) ¯

^{as shown in the proof of theorem 1; now, denote by WSCGREEDY}

^o

^{the version of}

WSCGREEDY where, at eah iteration, insteadof the set maximizing the ratio

P

c∈ S ¯ u[c]/w( ¯ S)

^{, a}

set,theassoiatedratioofwhihisatleast

ρ

^{timesthemaximumone(}

ρ ≤ 1

^{)ishosen,andrevisit}

the analysisof WSCGREEDYpresented in[4℄. Suppose, withoutlossofgenerality, thatthesolution

omputed byWSCGREEDY

o

is,after

r

iterations, the set

{ S ¯ 1 , S ¯ 2 , . . . , S ¯ r }

^{. Nowlet}

u ^r _j = ^P _c∈ S ¯ j u[c]

at step

r

^{of WSCGREEDY}

^o

^,

m = | S| ¯

^and

w j = w( ¯ S j )

^{; let}

x ^∗ = (x ^∗ _j ) j=1,...,n

^{denote the inidene}

vetor of an optimal over and let

s j

^{be the largest supersript}

r

^{suh that}

u ^r _j > 0

^{. The same}

analysisasChvatal's leads tothe following assertion:

WSCGREEDY ^o (C, S) ¯ ≤

m

X

j=1 s j

X

r=1

u ^r _j − u ^r+1 _j w r

u ^r _r !

x ^∗ _j .

Now, set

S ¯ r

^{seleted at step}

r

^satises

u ^r _r /w r ≥ ρ(u ^r _j /w j )

^,

∀j

^{, then}

WSCGREEDY ^o (C, S) ¯ ≤ 1 ρ

m

X

j=1 s j

X

r=1

u ^r _j − u ^r+1 _j u ^r _j

! w j x ^∗ _j .

In[4℄ itis shownthat

P s j

r=1 (u ^r _j − u ^r+1 _j )/u ^r _j ≤ H(| S ¯ _j |)

^{, where}

H(k) = ^{P k} _i=1 (1/i)

^. Consequently,

WSCGREEDY ^o (C, S) ¯ ≤ 1 ρ

m

X

j=1

H(| S ¯ j |)w j x ^∗ _j ≤ 1

ρ H(∆ _SC )OP T (C, S). ¯

As

H(∆ _SC ) ≈ 1 + ln ∆ _SC

^{, we get the ratio of the laim for WSC and transfer this ratio to}

problem

Π

^{thanksto the 1-1} orrespondene between solutions of

Π

^{and solutions of}

(C, S) ¯

^{, as}

shownin the proof oftheorem 1.

In the next setion, we use the result of theorem 1 to study the approximation of some

network-design problems onsistingof overing or partitionning verties or edgesbysubgraphs,

the number of andidate subgraphs being exponential in the size ofthe input-graph.

3.1 Minimum spanning tree of depth 2

Considerthe following ommuniation problem. Givenaset

V

^{ofitiesandan extraity}

r

^,nd

a subset

V ^′ ⊂ V

^{(where one wishes to onstrut relay stations), suh that every ity in}

V ^′

^is

onneted to

r

^{and every ity in}

V \ V ^′

^{is onneted to a ity of}

V ^′

^{, and suh that the total}

length ofonnetions isminimum. Formally, the problemis the following.

Minimum-weight rooted spanningtree of depth 2 (RST2).

Given a omplete graph

G = (V ∪ {r}, E )

^{with positive integer distanes}

d ~

^{on its edges,}

nda tree

T

^spanning

V ∪ {r}

^{, minimizing quantity}

^P _{e∈E(T )} d(e)

^{, and suhthat, for any}

vertex

v ∈ V

^{, the number of edgesin}

T

^{in the path onneting}

v

^to

r

^{isat most 2.}

Reallthata star

S v,X

^{isa treespanning}

v ∪ X

^{suhthatall verties of}

X

^{have degree1 in the}

tree. Let

v

^{be the enter of the star and all the quantity}

|X|

^{the degree of the star. We}

laimthat RST2isequivalent tothe following problem.

Minimum-weight spanning star-forest.

Given a omplete graph

K _{|V |}

^{, a ost-vetor}

d ~

^{on edges anda weight vetor}

w ~

^onverties,

nd a spanning forest of stars, i.e., a olletion of stars

F = {S v 1 ,X 1 , S v 2 ,X 2 , . . . , S v t ,X t }

partitioning

V

^{and minimizing}

c(F ) = ^{P t} _i=1 [w(v i ) + ^P _{x∈X i} d(v i , x)]

^.

In fat, given

G = K _{|V ∪{r}|}

^{, one an onsider the (omplete) subgraph of}

G

^{indued by}

V

^and

set

w(v) = d(vr)

^,

v ∈ V

^{. In the sequel, when speaking for RST2, we will refer to the latter}

problem. The proof of the NP-ompleteness of RST2 aswell as further positive and negative

approximation resultsabout itan be found in [2℄. Finally, Note thatthe total number of stars

in a ompletegraph of order

n

^is

n2 ⁿ⁻¹

^, exponential in

n

^.

Theorem 3. RST2is M-ST;onsequently, it isapproximable within

(1 + ln n)

^.

Proof. We rst prove ondition [M-ST-1℄. Consider a star-over

F = {S v j ,X j : j = 1, . . . , |F|}

and the following proedure.

BEGIN /*POSTPROCESS

(F)

^*/

let

V _c = {v j : j = 1, . . . , |F|}

^;

FOR

j ← 1

^TO

|F|

^DO

onsider

S v _j ,X _j

^;

IF

∃x ∈ X _j , (x ∈ V _c ) OR (x ∈ X _i , i < j)

^THEN

S _{v j ,X j} ← S _{v j ,(X j \{x})}

^FI

IF

∃i < j, v _i = v _j

^THEN

F ← (F \ {S v i ,X i , S _{v j ,X j} }) ∪ {S _{v i ,(X i ∪X j )} }

^FI

OD

OUTPUT

F

^;

END. /*POSTPROCESS

(F)

^*/

Theabove proedureisakindofpost-proessingtransformingastar-overto astar-partitionof

at most the same ost. Star-adjustment operations performed here onsist of edge-removals or

star-groupings whihdonot inreasethe ost ofthe nalsolutionobtained. So,the seond part

of ondition[M-ST-1℄in denition1 issatised.

Letus denoteby

c(S ¯ v,X ) = w(v) + ^P _x∈X d(v, x)

^{the ostfuntionassoiatedwith star}

S v,X

^.

Wenowprovethat,givenabinaryvetor

~u

^on

|V |

^{omponents, theslaveproblemofndingstar}

S v ^∗ ,X ^∗ = argmax

S _v,X



 

 

u[v] + ^P

x∈X

u[x]

¯ c(S v,X )



 

 

= argmin

S _v,X



 

 

w(v) + ^P

x∈X

d(v, x) u[v] + ^P

x∈X

u[x]



 

 

ispolynomial(ondition [M-ST-2℄). Letus denote by

R(S v,X ; ~u)

^{the last ofthe above ratios.}

Observe rst thatif a vertex

x ∈ X ^∗

^satises

u[x] = 0

^{, then removing}

x

^from

X ^∗

^{does not}

hange the value of

R(S v ^∗ ,X ^∗ ; ~u)

^{; so, supposewithoutlossofgenerality, that}

u[x] = 1

^,

∀x ∈ X ^∗

^.

Also suppose that the verties in

V

^{are labelled by}

v _i

^,

i = 1, . . . , n

^{, and onsider the following}

proedurending, given

~u

^{, star}

S v ^∗ ,X ^∗ = argmin _{S v,X} {R(S v,X ; ~u)}

^.

BEGIN /*SLAVE_RST2(

~ u

^)*/

min

^_

ratio ← ∞

^;

FOR

i ← 1

^TO

n

^DO

V i ← {v ∈ V \ {v i } : u[v] = 1}

^;

FOR

j ← |V i |

^{DOWNTO 1 DO}

let

X _j = {v ^′ ₁ , v ^′ ₂ , . . . , v ^′ _j }

^{be the list of verties of}

V _i

sorted in suh a way that

d(v i , v ^′ ₁ ) ≤ d(v i , v ^′ ₂ ) ≤ . . . ≤ d(v i , v ^′ _j )

^;

OD

j ← 0

^;

X 0 ← ∅

^;

WHILE

R(S v _i ,X _j + 1 ;~ u) < R(S v _i ,X _j ; ~ u)

^AND

j < |V i |

^DO

j ← j + 1

^OD

IF

R(S _{v i ,X j} ; ~ u) < min

^_

ratio

^THEN

min

^_

ratio ← R(S v i ,X j ;~ u)

^;

SOL ← S v _i ,X _j

^;

FI

OD

OUTPUT SOL;

END; /*SLAVE_RST2(

~ u

^)*/

InordertoprovethatproedureSLAVE_RST2orretlyndsinpolynomialtimethestarminimiz-

ingratio

R

^{over allstarsof}

G

^{, we rst provethat, for axed}

v i

^{, itorretlynds in polynomial}

time the star minimizing

R

^{over all stars with enter}

v _i

^{. Then the} orretness-result follows immediatelysineallvertiesareexaminedaspossiblestar-enters. Forreasonsofsimpliity, we

restrit ourselves to the ase where

u[v i ] = 1

^{, the proof for the ase}

u[v i ] = 0

^{being ompletely}

analogous. Consider rst the problemof determining, for a xed

|X| = j

^,

j = 0, 1, . . . , |V i |

^{, the}

quantity

min X {(w(v i ) + ^P _x∈X d(v i , x))/(|X| + 1)}

^{. Sine the list}

(v ₁ ^′ , v ^′ ₂ , . . . , v _|V ^′

i | )

^{is sorted in}

inreasing orderwith respetto the distanes of itselements from

v i

^{, we get(setting}

v ^′ ₀ = v i

^):

|X|=j min {R(S v i ,X ; ~u)} =

w(v i ) +

j

P

k=0

d(v i , v _k ^′ ) j + 1

min X



 

 

w(v i ) + ^P

x∈X

d(v i , x) (|X| + 1)



 

 

= min

j=0,...,|V i |



 

 

 

 

w(v _i ) + ^{P j}

k=0

d(v _i , v _k ^′ ) j + 1



 

 

 

  .

Let

U j = R(S v _i ,X _j ; ~u) = (w(v i ) + S j )/(j + 1)

^with

S j = ^{P j} _k=1 d(v i , v ^′ _k )

^{. Proedure SLAVE_RST2}

inrements

j

^while

U j

^{dereases in}

j

^{and stops as soon as}

U j

^{inreases. In order to omplete}

the proof, we now prove that when

U j

^{starts inreasing, it will never derease again, i.e., if}

U _j+1 − U j ≥ 0

^{, then}

U _j+2 − U _j+1 ≥ 0

^{. Let usstudy the sign of}

U _j+1 − U j

^.

U _j+1 − U j = w(v i ) + S j + d(v i , v ^′ _j+1 )

j + 2 − w(v i ) + S j

j + 1

= 1

(j + 1)(j + 2)

h (j + 1) ^h w(v i ) + S j + d(v i , v _j+1 ^′ ) ⁱ − (j + 2)(w(v i ) + S j ) ⁱ

= 1 (j + 1)(j + 2)

h (j + 1)d(v _i , v ^′ _j+1 ) − w(v _i ) − S _j ⁱ = 1 j + 2

h d(v _i , v ^′ _j+1 ) − U _j ⁱ .

Supposenow that

U _j+1 − U j ≥ 0

^{; we shallshowthat}

U _j+2 − U _j+1 ≥ 0

^.

U j+2 − U j+1 = 1

j + 3

d(v i , v _j+2 ^′ ) − U j+1

= 1

j + 3 d(v i , v _j+2 ^′ ) − (j + 1)U j + d(v i , v _j+1 ^′ ) j + 2

!

≥ 1

j + 3 d(v _i , v ^′ _j+1 ) − (j + 1)U j + d(v i , v _j+1 ^′ ) j + 2

!

= 1

j + 3

j + 1 j + 2

d(v i , v _j+1 ^′ ) − U j

=

j + 1 j + 3

(U _j+1 − U j ) ≥ 0

andtheorretnessof SLAVE_RST2follows;sineitrunsin

O(n ² )

^{, ondition[M-ST-2℄issatised.}

Consider now the following algorithm for RST2.

BEGIN /*MASTER_RST2*/

F ← ∅

^;

FOR

v ∈ V

^DO

u[v] ← 1

^;

WHILE

∃v ∈ V

^{suh that u[v℄ = 1 DO}

S v ^∗ ,X ^∗ ← SLAVE

^_

RST2(~ u)

^;

F ← F ∪ {S v ^∗ ,X ^∗ }

^;

FOR

v ∈ S v ^∗ ,X ^∗

^DO

u[v] ← 0

^OD;

OD

OUTPUT POSTPROCESS(F);

END. /*MASTER_RST2*/

Obviously, the omplexity of algorithm MASTER_RST2 is

O(n ³ )

^{. In all, we have proved that}

denition1 appliesfor RST2; furthermore,

∆ _RST2 ≤ n

^{andthe result ofthe theorem follows.}

Theresultoftheorem3anbeslightlymodiedtoapplyeveninthe restritedaseofRST2

where the degree of a star in anysolution hasto be bounded by a positive integer onstant

k

^.

It suesto add, in the WHILE loop of proedure SLAVE_RST2, an exit-riterion ifthe number

j

of unovered(i.e., having

~u

^{-values equal to 1) neighbours of}

v i

^{beomesgreater than}

k

^{and the}

following orollaryholds.

Corollary 1 . There exists a PTAA with ratio

1 + ln (k + 1)

^{for the restrition of RST2 where}

the degree of a starin any solutionhas tobe at most

k

^.

Sineatree ofdepth 2isofdiameter4,appliation ofalgorithm MASTER_RST2onsidering every

vertexof

V

^{asa potential rootleads to the following orollary.}

Corollary 2. The 4-diameter minimum spanning tree problem is approximable within ratio

1 + ln n

^.

3.2 Minimum bounded-diameter spanning forest

In this setion we prove that the minimum

k

^{-diameter spanning forest problem, alled}

k

^-DSF

in what follows, is M-ST and, sine itis an unweighted problem, itadmits a PTAA ahieving

ratio

ln n − ln ln n + 0.78

^.

Minimum k-diameter spanning forest.

Givenagraph

G = (V, E)

^{, nda}minimum-ardinalitypartitionof

V

^{intotreesofdiameter}

at most

k

^{, (the diameter of a tree}

T

^{is the maximum number of edges in a path between}

any pairof verties in

T

^).

Thisproblem isNP-hard (redution from minimum dominating set for

k = 2

^{, [6℄). Therest of}

the setionis devotedto the proof ofthe following theorem.

Theorem 4.

k

^{-DSFis M-STand,} onsequently, approximable within

ln n − ln ln n + 0.78

^.

Proof. Let us rst prove satisfation of ondition [M-ST-1℄; onsider the following proedure,

reeivingasinputs a set

V ₀ ⊂ V

^{and apositiveinteger}

r

^.

BEGIN /*SPANF(

V ₀ , r

^)*/

F ← ∅

^;

V ^c ← V 0

^;

FOR

i ← 1

^TO

r

^DO

V _i ← Γ(V _i−1 ) \ V ^c

^;

FOR

v ∈ V i

^DO

let

v ^′

^{be a vertex of}

Γ(v) ∩ V _i−1

^;

add

vv ^′

ⁱⁿ

F

^;

OD

V ^c ← V ^c ∪ V i

^;

OD

OUTPUT

F

^;

END; /*SPANF(

V 0 , r

^)*/

In the output

F

^{of proedure SPANF,every vertex of}

V i

^{is linked to a unique vertex of}

V ₀

^{by a}

unique path of length

i

^. Consequently,

F

^{is a forest of size}

|V 0 |

^{and ofdepth}

r

^{. Moreover, this}

forest is maximal, in the sense that it ontains all verties of

V

^{linked to some vertexof}

V ₀

^by

a path of at most

r

^{edges. Now, let}

F = {T 1 , . . . , T t }

^{be a over of}

V

^{by trees of diameter at}

most

k

^{(over in the sense thatavertex of}

V

^{an belongto more than one treeof}

F

^).

•

^If

k = 2r

^{, let}

r i

^{be a root of}

T i

^{(i.e., a vertex of}

V (T i )

^{suh that anyother vertex in}

T i

^is

linked to

r i

^{by a path of at most}

r

^{edges) and onsider}

V ₀ = {r i : i = 1, . . . , t}

^{. For this}

ase, proedure SPANF

(V ₀ , r)

polynomially transforms tree-over

F

^{into a} tree-partition of at most the sameardinality, satisfying ondition[M-ST-1℄.

•

^{For the ase where}

k = 2r + 1

^{, a tree of diameter at most}

k

^{an be seen as omposed}

of two rootedtrees of depth at most

r

^{having their roots onneted byan edge whih we}

will all edge-root in the sequel; let

r i r _i ^′

^{be an edge-root of}

T i

^{(i.e., an edge of}

E(T i )

suh that any othervertex in

T i

^{is onneted either to}

r i

^{, or to}

r _i ^′

^{by a path of at most}

r

edges) and onsider

E ₀ = {r i r _i ^′ : i = 1, . . . , t}

^{; moreover, let}

V ₀

^{be the set of verties}

endpoints of an edge in

E ₀

^{. A modied version of SPANF an be used here. It onsists}

of rst alling SPANF

(V 0 , r)

^{to normally produe a set of trees of depth}

r

^{, next of nding}

in

E ₀

^{a maximal mathing}

M ⊂ E ₀

^{and, nally, of unifying trees having edges of}

M

^as

edge roots; let us denote by MSPANF

(E ₀ , r)

^{this modied version of SPANF and by}

F ^′

^the

forestomputed. Sineeveryvertexof

V

^notin

V ₀

^{iswithindistaneatmost}

r

^from

V ₀

^,

F ^′

is indeeda partition of

V

^{into trees of diameter at most}

k = 2r + 1

^{, ontaining}

|M|

^trees

having edgesof

M

^{asedgeroots, and}

|V ^′ |

^{other trees(}

V ^′ ⊂ V ₀

^{denoting the setofverties}

of

V ₀

unsaturatedby

M

^{). Now,sine}

M

^ismaximalfor

E ₀

^{,everyvertexof}

V ^′

^isonneted

by an edge (in

E ₀

^{) to a vertex of}

V ₀ \ V ^′

^{. Hene,}

|E 0 | ≥ |M | + |V ^′ |

^{, i.e.,}

|F | ≥ |F ^′ |

^and

ondition[M-ST-1℄is satised.

Consider now the following algorithm for

k

^-DSF. Instrutions between parentheses refer to the ase

k = 2r + 1

^{; in this ase, remark that, when alled from a single edge}

e

^{, proedure}

MSPANF(e,r) onstruts a maximal tree of depth

r

^{and edge-root}

e

^{, i.e., a tree ontaining all}

verties of

V

^{lyingwithin distaneat most}

r

^{from anendpoint of}

e

^.

(1)

r ← ⌈k/2⌉

^;

(2)

F ← ∅

^;

(3)

V ₀ ← ∅

^;

(4) FOR

v ∈ V

^DO

u[v] ← 1

^OD

(5) WHILE

∃v ∈ V, u[v] = 1

^DO

(6)

max ← 0

^;

(7) mark all verties of V (* all edges of E *) as unvisited;

(8) WHILE

∃v 0 ∈ V

^(*

∃v 0 v ^′ ₀ ∈ E

^{*) unvisited DO}

(9)

T ← SPANF({v 0 }, r)

^(*

MSPANF({v 0 v ^′ ₀ }, r)

^*);

(10) IF

P

v∈T u[v] > max

^THEN

(11)

T ^∗ ← T

^;

(12)

max ← ^P _v∈T u[v]

^;

(13) FI

(14) mark

v 0

^(*

v 0 v ^′ ₀

^{*) as visited;}

(15) OD

(16)

F ← F ∪ {T ^∗ }

^;

(17)

V 0 ← V 0 ∪ {v 0 }

^{; (*}

E 0 ∪ {v 0 v ^′ ₀ }

^*)

(18) FOR

v ∈ V(T ^∗ )

^DO

u[v] ← 0

^OD

(19) OD

(20) OUTPUT

SPANF(V 0 , r)

^{; (*}

MSPANF(E 0 , r)

^*)

END.

We shall nally prove that lines (6) to (15) of algorithm MASTER_k-DSF orretly solve the

slave problem, hene satisfying ondition [M-ST-2℄, i.e., that tree

T ^∗

^{at line (16) maximizes}

quantity

P

v∈V (T ) u[v]

^{, where}

T

^{ranges over all trees of diameter at most}

k

^{. We shallall}

T opt

an optimal tree. Sine

T opt

^{has diameter at most}

k

^{, then if}

k

^{is even (resp., odd) there existsa}

vertex

v opt

^{(resp.,anedge}

e opt

^{)suhthateveryvertexof}

T opt

^islinkedto

v opt

^{(resp.,tooneofthe}

endpoints of

e opt

^{) by a path of length at most}

r = ⌈k/2⌉

^{. On the other hand,proedure SPANF}

(resp., MSPANF),alled byalgorithm MASTER_k-DSFfromeveryvertex

v ₀

^{(resp., edge}

v ₀ v ^′ ₀

^{) of}

G

produes a maximal tree of root

v ₀

^{(resp., edge-root}

v ₀ v ₀ ^′

^{) and depth}

r

^{. Let}

T ˆ

^{be the tree}

omputed by SPANF (resp., MSPANF), when alled from

v _opt

^(resp.,

e _opt

^{). As}

T ˆ

^{is maximal, it}

ontains all the verties linked to

v opt

^(resp.,

e opt

^{) by paths of length at most}

r

^; onsequently,

V (T opt ) ⊂ V ( ˆ T )

^and

^P _{v∈V (T opt )} u[v] ≤ ^P _{v∈V ( ˆ T )} u[v] ≤ ^P _{v∈V (T} ^∗ ₎ u[v]

^{; hene, tree}

T ^∗

^{is also}

optimal.

Obviously, the above algorithm is polynomial; moreover,

∆ _kDSF ≤ n

^{andthis ompletes the}

proofof theorem4.

3.3 Edge-overing by trees

Thelastproblemstudiedin thispaper, denotedbyECT,isanedge-overingproblemfrequently

met inVLSI-design whenever one tries tominimize the ost ofthe iruit onnexions ([11 ℄).

Edge-overing by weighted trees.

Given a graph

G = (V, E)

^{and positive integer valuations}

d ~

^{on its edges, nd a olle-}

tion of trees

F = {T 1 , . . . , T _t }

^satisfying

∪ i=1,...,t {E(T i )} = E

^{and minimizing}

c(F ) = P

1≤i≤t c(T ¯ i )

^,where

¯ c(T ) = max _{e∈E(T )} {d(e)}

^.

To our knowledge, the omplexity of ECT is still open. Therefore, sine no exat polynomial

algorithm is known, wedevise a PTAA ahievingapproximation ratio

O(ln n)

^.

mation ratio bounded above by

1 + ln n

^.

Proof. Condition [M-ST-1℄ of denition 1 issatised sine every feasible solution of ECT is a

over of

E

^{by trees (theostof whih isthe sumof the ostsofthe} solution-trees).

Weshallnowshowthatondition[M-ST-2℄isalso satised. Given a0-1vetor

~u

^assoiated

with

E

^{, we showthatsolution of the} slave-problem, i.e., omputation of

T ^∗ = argmax

T



 

 

R(T ; ~u) = P

e∈E(T )

u[e]

¯ c(T )



 

 

an be performedin polynomial time.

Let

E _δ = {e ∈ E : d(e) ≤ δ}

^{, and}

G _δ = (V, E _δ )

^{. Moreover, we denote by}

d(G)

^the

quantity

max e∈E {d(e)}

^{, and by}

d(G)

^{, the quantity}

min e∈E {d(e)}

^{. Consider now the following}

proedureoptimally solving the slave problem.

BEGIN /*SLAVE_ECT(

~ u

^)*/

(1)

max ← 0

^;

(2)

δ ← d(G)

^;

(3) WHILE

δ ≥ d(G)

^DO

(4)

argmax T { ^P _e∈E(T) u[e]} = T ^∗ ← KRUSKAL(G δ )

^;

(5)

δ ← ¯ c(T ^∗ ) − 1

^;

(6) IF

R(T ^∗ ;~ u) > max

^THEN

(7)

max ← R(T ^∗ ;~ u)

^;

(8) retain

T ^∗

^;

(9) FI

(10) OD

(11) output

T ^∗

^;

END; /*SLAVE_ECT(

~ u

^)*/

Algorithm KRUSKAL in line (4) is a modied version of the lassial minimum weight spanning

tree algorithm ([10℄) where, instead of a minimum-distane edge, a maximum-distane one is

seleted at eah step in graph

G δ

^{. Moreover, let us note that}

KRUSKAL(G δ )

^treats

u[e]

^{'s (and}

not

d[e]

^'s)asedge-distanes (

e ∈ E δ

^{). SineKRUSKAL runsin time}

O(|E| log |E|)

^{, andthe WHILE}

loop willbe exeutedat worst

|E|

^{times, then the whole proedure is}polynomial.

We now prove that proedure SLAVE_ECT orretly solves the slave problem for ECT, i.e.,

the output-tree

T ^∗

^satises

T ^∗ = argmax _T R(T ; ~u)

^{. Suppose that the WHILE loop of the above}

proedure is exeuted

p

^{times and denote, for}

i = 1, . . . , p

^{, by}

δ i

^{the value of}

δ

^{onsidered in}

line (3) and by

T _i ^∗

^{, the tree}

T ^∗

^{omputed in line (4), at the}

i

^{th exeution of the WHILE loop.}

We have, for

i = 1, . . . , p

^,

c(T ¯ _i ^∗ ) = δ _i+1 + 1

^{. Moreover, set}

δ _p+1 = δ p − 1

^{, and observe that}

δ p = d(G)

^and

δ ₁ = d(G)

^{. Consider now, at some exeution}

i

^{ofthe WHILEloop,}

1 ≤ i ≤ p − 1

^,

a tree

T

^{suh that}

δ _i+1 + 1 ≤ ¯ c(T ) ≤ δ _i

^{. We have}

^P _{e∈E(T )} u[e] ≤ ^P _e∈E(T ^∗

i ) u[e]

^{, otherwise}

T

would have been seleted instead of

T _i ^∗

^{by KRUSKAL. Sine}

¯ c(T ) ≥ ¯ c(T _i ^∗ )

^{, one} immediately gets

R(T _i ^∗ ; ~u) ≥ R(T; ~u)

^{. So,}

T _i ^∗ = argmax _{δ i+1 +1≤¯ c(T )≤δ i} {R(T ; ~u)}

^{andfor thenaltree}

T ^∗

^obtained

at line (11),the following holds:

T ^∗ = argmax

i=1,...,p {R(T _i ^∗ ; ~u)} = argmax

i=1,...,p

(

δ i+1 +1≤¯ max c(T )≤δ i

R(T ; ~u) )

= argmax

δ p ≤¯ c(T )≤δ 1

{R(T; ~u)} .

So,sine

δ p = d(G)

^and

δ ₁ = d(G)

^{, the optimalityofproedureSLAVE_ETCforthe slaveproblem}

isonluded.

for ECT.

BEGIN /*MASTER_ECT*/

C ← ∅

^;

FOR

e ∈ E

^DO

u[e] ← 1

^OD

WHILE

∃e ∈ E

^with

u[e] = 1

^DO

T ^∗ ← SLAVE

^_

ECT(~ u)

^;

C ← C ∪ {T ^∗ }

^;

FOR

e ∈ E(T ^∗ )

^DO

u[e] ← 0

^OD

OD

OUTPUT

C

^;

END. /*MASTER_ECT*/

Let us note that, if all edge-distanes are idential, ECT onsists in nding the minimum-

ardinality edge-over by trees. In this ase, proedure SLAVE_ECT is nothing else but a single

all of KRUSKAL(G)(line (4)).

Before losing this setion, letus reall our introdutoryremark thatthe rsttwo problems

are provably inapproximable within

(1 − ǫ) ln n

^{([1 ℄), and thus lie among the hardest problems}

of the lassAPX

(log n)

^. Consequently, the ratio attained hereis asymptotiallyoptimal.

4 Disussion: introduing a new kind of redution

A usual way for transferring (positive or negative) approximation results from a problem to

another is by approximation-preserving redutions (

A

^{-redutions [12℄,}

P

^{-redutions [5℄, ontin-}

uous redutions [13℄, et.). In these redutions the general underlying idea is that, given two

problems

Π

^and

Π ^′

^{(let us suppose wlog that these problems are} minimization ones), one an polynomially transform a generi instane

I

^of

Π

^{into an instane}

I ^′

^of

Π ^′

^{; next, on the hy-}

pothesis that a

ρ

-approximation PTAA A exists for

Π ^′

^{, one shows how a}

Π ^′

^-solution

S ^′

^for

I ^′

an be polynomially transformed to a

Π

^-solution

S

^for

I

^{suh that if}

|S ^′ |/OPT(I ^′ ) ≤ ρ

^then

|S|/OPT(I ) ≤ c(ρ)

^{, for some}

c : [1; ∞[→ [1; ∞[

^{, where}

|S ^′ | = A(I ^′ )

^and

|S|

^{denotes the size}

(ardinalityorvalue)ofsolution

S

^{. Usuallypeopleall}approximation-preserving theredutions where

c(ρ) = ρ

^.

Intuitively, theorem1 (aswell asthe results of setion3) desribes, in some sense,a kindof

redution fromanM-ST problemto WSC.Ofourse,sine thisredution isnot polynomial

(thenumber of setsisin all theases studied exponential in

n

^{,the order ofthe graph), itisnot}

approximation-preserving in the ommon sense. However, we feel that there still exist strong

approximationlinksbetweentheseproblems andthegreedy algorithmfor setovering,and that

these linksan bewritten in a formalway. We thus introdue the following kindof redution.

Denition 3. Let

Π

^and

Π ^′

^{be two} minimization problems, and

A _Π ^′

^{be a PTAA for}

Π ^′

^{. A}

pseudo-redution from

Π

^to

(Π ^′ , A _Π ^′ )

^{isa triple}

(ϕ, ψ, A Π )

^{suh that}

[1℄

ϕ

^{transformsaninstane}

I

^of

Π

^{into aninstane}

I ^′ = ϕ(I )

^of

Π ^′

^;

[2℄

ψ

^transformsa

Π ^′

^-solution

S ^′

^for

I ^′

^intoa

Π

^-solution

S = ψ(S ^′ )

^for

I

^suhthat

∃c : [1; ∞[→

[1; ∞[

^,

|S ^′ |/OPT(I ^′ ) ≤ ρ = ⇒ |S|/OPT(I) ≤ c(ρ)

^;

[3℄

A _Π

^{is a PTAA for}

Π

^{and, if}

S

^and

S ^′

^{denote the solutions returned by}

A _Π

^and

A _Π ^′

^on

I

and

I ^′

^, respetively, then

S = ψ(S ^′ )

^.

We willdenoteby

Π ֒ → (Π ^′ , A _Π ^′ )

^thefatthat

Π

^ispseudo-reduibleto

(Π ^′ , A _Π ^′ )

^with

c(ρ) = ρ

^.

Pseudo-redution diers from lassial redutions by the fat that funtions

ϕ

^and

ψ

^{are not}

onstrained to be omputable in time polynomial in

|I |

^{, and that, givena PTAA}

A _Π ^′

^for

Π ^′

^{, we}

areonlyinterested,in solutionsomputed byalgorithm

A _Π ^′

ⁱⁿ

I ^′

^{. Anothermeaningfuldierene}

isalsothatsolution

S = ϕ(S ^′ )

^for

I

^{isnotontruted} polynomiallyfrom

S ^′

⁽

I ^′

^{isnot,ingeneral,}

onstrutible in polynomial time), but it is onstruted diretly on

I

^{by a kind of simulation}

of algorithm

A _Π ^′

^{. Sine}

A _Π ^′ (I ^′ )/OPT(I ^′ ) ≤ ρ = ⇒ A _Π (I )/OPT(I ) ≤ c(ρ)

^{, if}

c(ρ) = ρ

^{, then the}

pseudo-redution an be onsidered as approximation-preserving. We thus have the following

proposition.

Proposition 1. If

Π ֒ → (Π ^′ , A _Π ^′ )

^and

A _Π ^′

approximately solves (inpolynomialtime)

Π ^′

^within

ratio

ρ

^{, then}

Π

^is polynomially approximable within

ρ

^.

We haveseen in the proofoftheorem 1that, for overing (resp., partitioning) M-STproblems,

triple

(ϕ, ψ, MSGREEDY)

^(resp.,

(ϕ, ψ ◦ h, MSGREEDY)

^{) satisesdenition3. We an thus reover a}

onept ofompleteness to our pseudo-redution.

Proposition 2. WSCisM-ST-hard, inthesensethat

∀Π ∈

^M-ST,

Π ֒ → (WSC, WSCGREEDY)

^.

From propositions 1 and 2 we nd again our former result, namely every M-ST problem is

approximable within logarithmi ratio.

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