Degree-Constrained Subgraph Problems: Hardness and Approximation Results

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Approximation Results

Omid Amini, David Peleg, Stéphane Pérennes, Saket Saurabh

To cite this version:

Omid Amini, David Peleg, Stéphane Pérennes, Saket Saurabh. Degree-Constrained Subgraph

Prob-lems: Hardness and Approximation Results.

[Research Report] RR-6690, INRIA. 2008.

�inria-00331747�

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Sous-graphes avec contraintes sur le degré :

difficulté et approximation

Omid Amini — David Peleg — Stéphane Pérennes — Ignasi Sau — Saket Saurabh

N° 6690

Omid Amini

†

, DavidPeleg

‡

, Stéphane Pérennes

§

,Ignasi Sau

¶k

, Saket Saurabh

∗∗

ThèmeCOMSystèmes ommuni ants ProjetMASCOTTE

Rapportdere her he n°6690September200829pages

Résumé:Ageneralinstan eofaDegree-Constrained Subgraphproblem onsistsof anedge-weightedorvertex-weightedgraph

G

andtheobje tiveistondanoptimalweighted subgraph,subje tto ertaindegree onstraintsontheverti esof thesubgraph.This lass of ombinatorialproblemshasbeenextensivelystudiedduetoitsnumerousappli ationsin network design. If the input graph is bipartite, these problems are equivalent to lassi al transportationandassignmentproblemsin operationsresear h. Thispaper onsidersthree naturalDegree-ConstrainedSubgraphproblemsandstudiestheirbehaviorintermsof approximationalgorithms. These problemstakeas inputanundire ted graph

G = (V, E)

, with

|V | = n

and

|E| = m

. Ourresults,togetherwith thedenitionofthethree problems, arelistedbelow.

∗

This workhas been partially supported byEuropean proje t IST FETAEOLUS, PACAregion of Fran e,MinisteriodeEdu a iónyCien iaofSpain,EuropeanRegionalDevelopmentFundunder proje t TEC2005-03575,CatalanResear h Coun ilunder proje t2005SGR00256 andCOSTa tion293GRAAL, andhasbeendoneinthe ontextofthe r CorsowithFran eTele om.

†

Max-Plan k-InstitutfürInformatik,Saarbrü ken,GERMANY.aminimpi-inf.mpg.de

‡

WeizmannInstituteofS ien e,Rehovot,ISRAEL.david.pelegweizmann.a .i l

§

Mas otte joint proje t - INRIA/CNRS-I3S/UNSA, Sophia-Antipolis, FRANCE. Stephane.Perennessophia. inri a.f r

¶

Mas otte joint proje t - INRIA/CNRS-I3S/UNSA, Sophia-Antipolis, FRANCE. Ignasi.Sausophia.inria.f r

k

GraphTheoryandCombinatori sgroupatAppliedMathemati sIVDepartmentofUPC,Bar elona, SPAIN.

∗∗

 The Maximum Degree-Bounded Conne ted Subgraph (MDBCS

d

) problem takes as input a weight fun tion

ω : E

→ R

+

and an integer

d

≥ 2

, and asks for a subset

E

′

_{⊆ E}

su h that the subgraph

G

′

_{= (V, E}

′

₎

is onne ted, has maximum degreeat most

d

,and

P

e∈E

′

ω(e)

ismaximized. This problemis one ofthe lassi al NP-hard problems listed by Garey and Johnson in (Computers and Intra tability, W.H.Freeman,1979), buttherewerenoresultsintheliteratureex eptfor

d = 2

.We provethatMDBCS

d

isnotinApxforany

d

≥ 2

(thiswasknownonlyfor

d = 2

)and weprovide a

(min

{m/ log n, nd/(2 log n)})

-approximationalgorithm for unweighted graphs,anda

(min

{n/2, m/d})

-approximationalgorithmforweightedgraphs.Wealso provethat when

G

a eptsalow-degreespanningtree,in termsof

d

,then MDBCS

d

anbeapproximatedwithin asmall onstantfa torinunweightedgraphs.

 The Minimum Subgraph of Minimum Degree

≥d

(MSMD

d

) problem onsists in ndingasmallestsubgraphof

G

(intermsofnumberofverti es)withminimumdegree at least

d

. We prove that MSMD

d

is not in Apx for any

d

≥ 3

and we providean

O(n/ log n)

-approximationalgorithmforthe lassesofgraphsex ludingaxedgraph asaminor,usingdynami programmingte hniques andaknownstru turalresulton graph minors. In parti ular, this approximation algorithm applies to planar graphs andgraphsofboundedgenus.

 The Dual Degree-Dense

k

-Subgraph (DDD

k

S) problem onsists in nding a subgraph

H

of

G

su h that

|V (H)| ≤ k

and

δ

H

is maximized, where

δ

H

is the minimum degree in

H

. We present a deterministi

O(n

δ

₎

-approximation algorithm in generalgraphs,forsomeuniversal onstant

δ < 1/3

.

Mots- lés: ApproximationAlgorithms,Degree-ConstrainedSubgraphs,HardnessofApproximation, Apx,PTAS,DenseSubgraphs,GraphMinors,Ex ludedMinor.

Abstra t: Ageneralinstan eofaDegree-Constrained Subgraphproblem onsistsof anedge-weightedorvertex-weightedgraph

G

andtheobje tiveistondanoptimalweighted subgraph,subje tto ertaindegree onstraintsontheverti esof thesubgraph.This lass of ombinatorialproblemshasbeenextensivelystudiedduetoitsnumerousappli ationsin network design. If the input graph is bipartite, these problems are equivalent to lassi al transportationandassignmentproblemsin operationsresear h. Thispaper onsidersthree naturalDegree-ConstrainedSubgraphproblemsandstudiestheirbehaviorintermsof approximationalgorithms. These problemstakeas inputanundire ted graph

G = (V, E)

, with

|V | = n

and

|E| = m

. Ourresults,togetherwith thedenitionofthethree problems, arelistedbelow.

 The Maximum Degree-Bounded Conne ted Subgraph (MDBCS

d

) problem takes as input a weight fun tion

ω : E

→ R

+

and an integer

d

≥ 2

, and asks for a subset

E

′

_{⊆ E}

su h that the subgraph

G

′

_{= (V, E}

′

₎

is onne ted, has maximum degree at most

d

, and

P

e∈E

′

ω(e)

is maximized. This problem is one of the lassi- alNP-hardproblemslistedbyGareyandJohnsonin(Computers andIntra tability, W.H.Freeman,1979), buttherewerenoresultsintheliteratureex eptfor

d = 2

.We provethatMDBCS

d

isnotinApxforany

d

≥ 2

(thiswasknownonlyfor

d = 2

)and weprovide a

(min

{m/ log n, nd/(2 log n)})

-approximationalgorithm for unweighted graphs,anda

(min

{n/2, m/d})

-approximationalgorithmforweightedgraphs.Wealso provethat when

G

a eptsalow-degreespanningtree,in termsof

d

,then MDBCS

d

anbeapproximatedwithin asmall onstantfa torinunweightedgraphs.

 The Minimum Subgraph of Minimum Degree

≥d

(MSMD

d

) problem onsists in ndingasmallestsubgraphof

G

(intermsofnumberofverti es)withminimumdegree at least

d

. We prove that MSMD

d

is not in Apx for any

d

≥ 3

and we providean

O(n/ log n)

-approximationalgorithmforthe lassesofgraphsex ludingaxedgraph asaminor,usingdynami programmingte hniques andaknownstru turalresulton graph minors. In parti ular, this approximation algorithm applies to planar graphs andgraphsofboundedgenus.

 TheDualDegree-Dense

k

-Subgraph(DDD

k

S)problem onsistsinndinga sub-graph

H

of

G

su h that

|V (H)| ≤ k

and

δ

H

ismaximized, where

δ

H

istheminimum degree in

H

. We present a deterministi

O(n

δ

₎

-approximation algorithm in general graphs,forsomeuniversal onstant

δ < 1/3

.

Key-words: ApproximationAlgorithms,Degree-ConstrainedSubgraphs,Hardnessof Ap-proximation,Apx,PTAS,DenseSubgraphs,GraphMinors,Ex ludedMinor.

1 Introdu tion

In this paper we onsider three natural Degree-Constrained Subgraph problems and study them in termsof approximation algorithms. A general instan e of a Degree-ConstrainedSubgraphproblem[1,5,24℄ onsistsofanedge-weightedorvertex-weighted graph

G

andtheobje tiveistondanoptimalweightedsubgraph,subje tto ertaindegree onstraintsontheverti esofthesubgraph.Theseproblemshaveattra tedalotofattention inthelastde adesandhaveresultedinalargebodyofliterature[1,5,10,1214,16,19,22,24℄. Themostwell-studied onesareprobablytheMinimum-Degree SpanningTree [12℄and theMinimum-DegreeSteiner Tree [13℄problems.

Beyondtheestheti andtheoreti alappealofDegree-Constrained Subgraph prob-lems,thereasonsforsu hintensivestudyarerootedintheirwideappli abilityintheareas of inter onne tion networks and routingalgorithms, among others. For instan e, givenan inter onne tionnetworkmodeledbyanundire tedgraph,onemaybeinterestedinnding asmallsubsetofnodeshavinghighdegreeof onne tivityforea hnode.Thistranslatesto ndingasmallsubgraphwithalowerboundonthedegreeofitsverti es,i.e.totheMSMD

d

problem.Notethatiftheinputgraphisbipartite,theseproblemsareequivalentto lassi al transportationandassignmentproblemsin operationresear h.

The rst problem studied in the paper is a lassi al NP-hard problem listed in [15℄ ( f. Problem[GT26℄fortheunweightedversion):

Maximum Degree-Bounded Conne tedSubgraph (MDBCS

d

) Input : A graph

G = (V, E)

, a weight fun tion

ω : E

→ R

+

and an integer

d

_{≥ 2}

.

Output : Asubset

E

′

_{⊆ E}

su h that thesubgraph

G

′

_{= (V, E}

′

₎

is onne ted, hasmaximumdegreeatmost

d

,and

P

e∈E

′

ω(E)

ismaximized.

For

d = 2

,theunweightedMDBCS

d

problem orrespondstotheLongest Pathproblem. Indeed, given the input graph

G

(whi h an be assumed to be onne ted), let

P

and

G

′

be optimal solutions of Longest Path and MDBCS

2

in

G

, respe tively. Then observe that

|E(G

′

₎

_{| = |E(P )|}

unless

G

is Hamiltonian, in whi h ase

|E(G

′

₎

_{| = |E(P )| + 1}

.One ould also ask the question : what happens when

G

′

is not required to be onne ted in the denition of MDBCS

d

? It turns out that without the onne tivity onstraint, both the edge version and the vertex version(where the goal is to maximize the total weight of theverti esof a subgraphrespe ting the degree onstraints) of theMDBCS

d

problem areknownto besolvablein polynomialtime using mat hingte hniques [7,15,18℄.In fa t, without onne tivity onstraints,evenamoregeneralversionwhere theinput ontainsan intervalofalloweddegreesforea hnodeisknownto besolvableinpolynomialtime.

The most generalversionof Degree-Constrained Subgraph problems is to nd a subgraphunder onstraintsgivenbylowerandupperboundsonthedegreeofea hvertex,the obje tivebeingtominimizeormaximizesomeparameter(usuallythesizeofthesubgraph). A ommon variantignoresthelowerboundon thedegreeandjust requiresthe verti esof thesubgraphstohaveagivenmaximumdegree[22℄,in whi h asethetypi aloptimization riterionistomaximizethesizeofasubgraphsatisfyingthedegree onstraints.Theresulting

problem is also alled an Upper Degree-Constrained Subgraph problem in [14℄.In ontrast,weare unawareof existing results onsideringjust alowerbound onthe degrees oftheverti esofthesubgraph,ex eptfor ombinatorial onditionsontheexisten eofsu h a subgraph [10℄. In an attempt to ll this void in the literature, the last two problems onsideredinthispaperaimatminimizingthesizeofasubgraphandmaximizingthelower bound on the minimum degree, respe tively. For a graph

H

, let

δ

H

denote the minimum degreeoftheverti esin

H

.

Minimum Subgraph of Minimum Degree

≥d

(MSMD

d

) Input:An undire tedgraph

G = (V, E)

andaninteger

d

≥ 2

.

Output:Asubset

S

⊆ V

su hthatfor

H = G[S]

,

δ

H

≥ d

and

|S|

isminimized. DualDegree-Dense

k

-Subgraph(DDD

k

S)

Input:An undire tedgraph

G = (V, E)

andapositiveinteger

k

.

Output:Anindu edsubgraph

H

ofsize

|V (H)| ≤ k

,su hthat

δ

H

ismaximized. MSMD

d

is losely related to MDBCS

d

. Indeed, MSMD

d

orresponds exa tly to the dual(unweighted)node-minimizationversionofMDBCS

d

.MSMS

d

isalsoageneralization oftheGirthproblem(ndingashortest y le),whi h orrespondsexa tlytothe ase

d = 2

. InAmini et al.[4℄, the MSMD

d

problem was introdu ed and studied in the realmof the parameterized omplexity. It was shown that MSMD

d

is W[1℄-hardfor

d

≥ 3

andexpli it FPT algorithmswere givenforthe lassof graphsex ludingaxedgraphas aminorand graphsof bounded lo al-treewidth. Besidesthe abovedis ussion, our main motivation for studyingMSMD

d

isits loserelationtothewellstudiedDense

k

-Subgraph(D

k

S)[11,17℄ andTraffi Grooming[3℄problems.Indeed,ifgoodapproximatesolutions ouldbefound for the MSMS

d

problem, then one ould also nd good approximate solutions (up to a onstantfa tor)fortheD

k

SandTraffi Groomingproblems.Roughly,theideaisthat asmall subgraphwithminimumdegreeatleast

d

hasdensityatleast

d

2

,and thisprovides anapproximationforthedensestsubgraph (infa t,Traffi Grooming an beredu ed, essentially,to ndingdensesubgraphs).See[3,4℄forfurtherdetails.

Theabovedis ussionillustratesthat thestudyoftheabovementionedproblemsisvery naturalandthat theresultsobtainedfor them an reverberatein several otherimportant optimizationproblems, omingfromboththeoreti alandpra ti aldomains.

Our Results : In this paper we obtain both approximation algorithms and results on hardnessof approximation.All thehardnessresultsare basedonthe hypothesis P

6=

NP. Morepre isely,ourresultsarethefollowing:

 We prove that the MDBCS

d

problem is not in Apx for any

d

≥ 2

. On the other hand, wegivean approximationalgorithm for generalunweightedgraphswith ratio

min

_{{m/ log n, nd/(2 log n)}}

, and an approximation algorithm for general weighted graphswith ratio

min

{n/2, m/d}

.The rstalgorithm usesan algorithmintrodu ed in [2℄, that is based on the olor- oding method. We also present a onstant-fa tor approximation in Appendix D when the input grapha epts alow-degreespanning tree,in termsoftheinteger

d

.

 WeprovethattheMSMD

d

problemisnotinApxforall

d

≥ 3

.Theproofisobtained bythefollowingtwosteps.First,byaredu tionfromVertex Cover,weprovethat MSMD

d

doesnotadmitaPTAS.Inparti ular,thisimpliesthatMSMD

d

isNP-hard forany

d

≥ 3

.Then,weusetheerrorampli ation te hniquetoprovethatMSMD

d

is notinApxforany

d

≥ 3

.Onthepositiveside,wegivean

O(n/ log n)

-approximation algorithmforthe lassofgraphsex ludingaxedgraph

H

asaminor,usingaknown stru turalresultongraphminorsanddynami programmingovergraphsofbounded treewidth.Inparti ular,thisgivesan

O(n/ log n)

-approximationalgorithmforplanar graphsand graphsofboundedgenus.

 We givea deterministi

O(n

δ

₎

-approximationalgorithm for theDDD

k

S problem in general graphs,for someuniversal onstant

δ < 1/3

. We also provide a randomized

O(

√

n log n)

-approximationalgorithmin AppendixI,whi h is ompletelydierentin nature.Althoughtheapproximationratiois signi antlyworse,theideaoftheproof isquitesimpleand ni e.

Organization of the paper : In Se tion 2 we establishthat MDBCS

d

is not in Apx for any

d

≥ 2

, and in Se tion 3 we present twoapproximation algorithms for unweighted andweightedgeneralgraphs,respe tively.The onstant-fa torapproximationforMDBCS

d

when the input grapha epts a low-degreespanning tree is provided in Appendix D for unweightedgraphs.InSe tion4weprovethatMSMD

d

isnotinApxforany

d

≥ 3

,andin Se tion5wegivean

O(n/ log n)

-approximationalgorithmforthe lassofgraphsex luding a xed graph

H

as a minor. In Se tion 6 we give two approximation algorithms for the DDD

k

Sproblem.Finally,we on ludewithsomeremarksandopenproblemsinSe tion7. Theomittedproofsandsomebasi denitions an befoundintheappendi es.

2 Hardness of Approximating MDBCS

d

As mentionedinSe tion 1,MDBCS

2

isexa tlytheLongest Pathproblem, whi his knowntonota eptany onstant-fa torapproximation[16℄,unlessP

=

NP.Inthisse tion we extend this result and prove that, under the assumption P

6=

NP, MDBCS

d

is not in Apxforany

d

≥ 2

,provingrstthatMDBCS

d

isnotinPTASforany

d

≥ 2

.Wereferto Appendix A.1forthedenitions of the omplexity lassesApx,PTAS andforthe notion ofgap-preserving redu tion, whi hwill beusedfreelythroughoutthepaper.

Theorem 2.1 MDBCS

d

does notadmit a PTAS for any

d

≥ 2

,unless P

=

NP.

Proof: We prove the result for the ase when

d

≥ 3

. The result for the ase

d = 2

followsfrom [16℄.Wegiveourredu tionfromTSP

(1, 2)

,whi hdoesnothavePTASunless P

=

NP[21℄.Aninstan eofTSP

(1, 2)

onsistsofa ompletegraph

G = (V, E)

on

n

verti es and aweight fun tion

f : E

→ {1, 2}

on itsedges, and theobje tiveis to ndatraveling salesmantourof minimumedgeweightin

G

.

WeshowthatifwehaveaPTASforMDBCS

d

,

d

≥ 3

,thenwe an onstru taPTAS forTSP

(1, 2)

. Towardsthis, we transformthe graph

G

into anewgraph

G

′

with a mod-ied weight fun tion

g

on its edges. For every vertex

v

∈ V

we add

d

− 2

new verti es

{v

1

,

· · · , v

d−2

}

andweaddanedgefrom

v

toeveryvertex

v

i

,

1

≤ i ≤ d − 2

.This on ludes thedes riptionof

G

′

.Let

V

′

₌

_{{v

1

,

· · · , v

d−2

} | v ∈ V }

bethesetofnewverti es,and let

E

′

₌

_{(v

i

, v)

| 1 ≤ i ≤ d − 2, v ∈ V }

bethesetofnewedges.Wedene theweightfun tion

g

of

G

′

as follows :

g(e) = 3

− f(e)

if

e

∈ E

(weightsof original edges get ipped), and

g(e) = 3

if

e

∈ E

′

.

Next weprovea laim showingthestru ture ofthemaximal solutionsof MDBCS

d

in

G

′

. Essentially,weshow that givenanysolution

G

1

of MDBCS

d

in

G

′

with value

W

, we an transformit into anothersolution

G

2

of MDBCS

d

in

G

′

withvalue at least

W

,su h that

G

2

ontainsallthenewlyaddededgesandindu esahamiltonian y lein

G

.Theproof hasbeenmovedtoAppendixBduetola kofspa e.

Claim1 Given a solution

G

1

= (V

∪ V

′

_{, E}

1

)

to MDBCS

d

in

G

′

, we an transform it in polynomial time into a solution

G

2

= (V

∪ V

′

_{, E}

2

)

of MDBCS

d

in

G

′

su h that (a)

G

3

= (V, E

∩ E

2

)

isahamiltonian y lein

G

and; (b)

P

e∈E

2

g(e)

≥

P

e

′

_∈E

1

g(e

′

₎

.

Supposethat there exists aPTAS forMDBCS

d

realizedby anapproximations heme

A

δ

.Thisfamilyofalgorithmstakesasinputagraph

G

′′

andaparameter

δ > 0

,andreturns asolutionof MDBCS

d

ofweightat least

(1

− δ)OP T

G

′′

,where

OP T

G

′′

isthevalueofan optimumsolutionofMDBCS

d

in

G

′′

.Nowwepro eedto onstru taPTASforTSP

(1, 2)

. Givenagraph

G

,aninstan eof TSP

(1, 2)

,and

ε > 0

,wedoas follows:

1. Fix

δ = h(ε, d)

(to bespe iedlater)and run

A

δ

on

G

′

(the graphobtainedfrom

G

withthetransformationdes ribedabove).

2. Apply thepolynomial time transformationdes ribed in Claim 1 onthe solution ob-tainedby

A

δ

on

G

′

.Letthenewsolutionbe

G

∗

_{= (V}

_{∪ V}

₁

_{, E}

∗

₎

. 3. Return

E

∗

_{∩ E}

as thesolutionof TSP

(1, 2)

.

Now we prove that the solution returned by our algorithm is of desired kind, that is

P

e∈E

∗

_∩E

f (e)

≤ (1 + ε)O

T

,where

O

T

istheweightofanoptimumtourin

G

. Letsu han optimumtour ontain

a

edgesofweight

1

and

b

edgesofweight

2

.Then

O

T

= a + 2b

and

a + b = n

.Equivalently

a = 2n

− O

T

and

b = O

T

− n

.Let

O

D

bethevalueofanoptimum solutionof MDBCS

d

in

G

′

.

ThenbyClaim 1andtheippingnature ofthefun tion

g

, wehavethat

O

D

= (d

− 2)3n + 2a + b.

(1)

Let

3(d

− 2)n + O

∗

D

bethevalueofthesolutionreturnedby

A

δ

,where

O

∗

D

isthesumofthe weightsoftheedgesofthehamiltonian y lein

G

,thatis

O

∗

D

=

P

e∈E

∗

_∩E

g(e)

.Sin e

A

δ

is aPTAS,

3(d

− 2)n + O

∗

D

≥ (1 − δ)O

D

.

(2)

CombiningEquation(1)andInequality(2)gives

Ontheotherhand,thevalueofthesolutionreturnedbyouralgorithmforTSP

(1, 2)

is

O

∗

T

= 3n

− O

D

∗

(sin eif

O

∗

D

= 2x + y

,

x

beingthenumberofedgesofweight

2

and

y

being thenumberofedgesofweight

1

,with

x + y = n

,thenthevalueofthesolutionforTSP

(1, 2)

is

x + 2y

).Substituting

O

∗

D

= 3n

− O

∗

T

in Inequality(3)andusingthat

O

T

≥ n

yields

O

∗

T

≤ O

T

− δO

T

+ n(3d

− 3)δ ≤ O

T

− δn + n(3d − 3)δ.

(4) Toshowthat

O

∗

T

≤ (1 + ε)O

T

,by(4)itisenoughtobound

−δn + n(3d − 3)δ ≤ ε · O

T

. Ratherwewillshowthat

−δn+n(3d−3)δ ≤ εn

,whi hwillautomati allyimplytherequired bound. This an bedone by setting

δ = h(ε, d) =

ε

3d−4

, yielding a PTAS for TSP

(1, 2)

. Sin eTSP

(1, 2)

doesnotadmitaPTAS[21℄,thelastassertionalsorulesouttheexisten e ofaPTASforMDBCS

d

forany

d

≥ 3

,unlessP

=

NP.

2

We are now ready to statethe main result ofthis se tion. The proof onsists in using theinnaproximability onstant givenbyTheorem 2.1and applyingtheerror ampli ation te hniqueto ruleouttheexisten eofa onstant-fa torapproximation.Thewhole proofof Theorem2.2hasbeenmovedtoAppendix Cdue tola kofspa e.

Theorem 2.2 MDBCS

d

,

d

≥ 2

, doesnotadmitany onstant-fa torapproximation,unless P

=

NP.

3 Approximating MDBCS

d

Inthisse tionwefo usonapproximatingMDBCS

d

.AsseeninSe tion2,MDBCS

d

does notadmitany onstant-fa torapproximationingeneralgraphs.InAppendixDweshowthat whentheinputgraphhasalow-degreespanningtree(in termsof

d

),theproblem be omes easytoapproximate.Spe i ally,PropositionD.1providesa onstant-fa torapproximation forsu hgraphs.

In this se tion we deal with general graphs. Con erning the Longest Path problem (whi h orresponds to the ase

d = 2

of MDBCS

d

as dis ussed in the introdu tion) the bestapproximationalgorithm[6℄hasapproximationratio

O(n(log log n/ log n)

2

₎

,whi h im-provedthe ratio

O(n/ log n)

of[2℄. Using theresultsof [2℄, we provide in Theorem3.2 an approximationalgorithmforMDBCS

d

ingeneralunweightedgraphsforany

d

≥ 2

.Thenwe turn to weighted graphs,providing a ompletely newapproximationalgorithm forgeneral weightedgraphsinTheorem3.3.Finallywe omparebothalgorithmsforunweightedgraphs. Tothebest ofourknowledge,thesearetherstapproximationalgorithmsforMDBCS

d

in generalgraphs.

Weneedapreliminary lemma,thatusesthefollowingresult:

Proposition 3.1( [20℄) Anyunorderedtreeon

n

nodes anberepresentedusing

2n + o(n)

bits withadja en y beingsupportedin

O(n)

time.

Let

T

n,d

betheset ofnon-isomorphi unlabeledtreeson

n

nodeswithmaximumdegreeat most

d

.

Lemma3.1 The set

T

log n,d

anbe generatedinpolynomial timeon

n

. Proof: It is well known [23℄ that

|T

n,n−1

| ∼ Cα

n

_n

−5/2

as

n

→ ∞

, where

C

and

α

are positive onstants.Hen e, theset

T

log n,log n−1

hasanumberofelementspolynomialon

n

. Inaddition, one an e iently generateall theelements of

T

log n,log n−1

, sin eby Proposi-tion3.1 anyunlabeled tree on

log n

nodes an berepresentedusing

2 log n + o(log n)

bits with adja en ybeingsupported in

O(log n)

time. Finally, the set

T

log n,d

is obtainedfrom

T

log n,log n−1

byremoving alltheelements

T

with

∆(T ) > d

, where

∆(T )

is themaximum

degreeofthetree

T

.

2

Themain ingredientof therstalgorithm isapowerfulresultof[2℄, whi h usesthe olor- oding method.

Theorem 3.1( [2℄) If a graph

G = (V, E)

ontains a subgraph isomorphi to a graph

H = (V

H

, E

H

)

whose treewidth isatmost

t

, thensu hasubgraph anbefound in

2

O(|V

H

|)

·

|V |

t+1

_{· log |V |}

time.

Inparti ular,treeson

log

|V |

verti es anbefoundintime

|V |

O(1)

_{· log |V |}

.Wearereadyto des ribeouralgorithmforunweightedgraphs.

Algorithm

A

:

(1) Generatealltheelementsof

T

log n,d

.Denetheset

F := {}

.

(2) For ea h

T

∈ T

log n,d

, test if

G

ontains a subgraph isomorphi to

T

. If su h a subgraphisfound,additto

F

.

(3) If

F = ∅

or

d > log n

,outputanarbitrary onne tedsubgraphof

G

with

d

edges. Otherwise,outputanyelementin

F

.

Theorem 3.2 Forall

d

≥ 2

,algorithm

A

providesa

ρ

-approximationalgorithmfor MDBCS

d

inunweightedgraphs, with

ρ =

min{m,nd/2}

log n

.

Proof: Letus rstsee that therunning timeof algorithm

A

is polynomialon

n

.Indeed, steps (1) and (2) an be exe uted in polynomial time by Lemma 3.1 and Theorem 3.1, respe tively.Step(3)takes onstanttime.Algorithm

A

is learly orre t,sin ebydenition oftheset

T

log n,d

theoutputgraphisasolutionof MDBCS

d

in

G

.

Finally,letus onsidertheapproximationratioofalgorithm

A

.Let

OP T

bethenumber ofedgesofanoptimalsolutionof MDBCS

d

in

G

, andlet

ALG

bethenumberofedgesof thesolutionfoundbyalgorithm

A

.Wedistinguishtwo ases:

•

If

OP T

≥

d·log n

2

,thenanyoptimalsolution

H

ˆ

hasatleast

log n

verti es.Inparti ular,

ˆ

H

ontains a tree on

log n

verti es, and so does

G

. Hen e, this tree will be found in step (2), and therefore

ALG

≥ log n − 1

. (We an assume that

ALG = log n

by repla ing everywhere

T

log n,d

with

T

log n+1,d

.) On the other hand, we know that

OP T

≤ min{m, nd/2}

.

•

Otherwise, if

OP T <

d·log n

2

, then

ALG

≥ d

. Note that su h a onne ted subgraph with

d

edges anbegreedilyfoundstartingfrom anynodeof

G

.

In both ases,

OP T

ALG

≤ max



min

{

m,

nd

2

}

log n

,

log n

2



=

min{m,nd/2}

_{log n}

(sin e

log n =

O(

√

_n)

), as

laimed.

2

Inparti ular,if

d = 2

,algorithm

A

redu esto theLongest Pathalgorithmof[2℄. Theorem 3.3 The MDBCS

d

problem admits a

ρ

-approximation algorithm in weighted graphs, with

ρ = min

{n/2, m/d}

.

Proof: Letusdes ribethealgorithm.Let

F

bethesetof

d

heaviestedgesintheinputgraph

G

,andlet

W

bethesetofendpointsofthoseedges.Wedistinguishtwo asesa ordingto the onne tivityofthesubgraph

H = (W, F )

.Let

ω(F )

denotethetotalweightoftheedges in

F

.

If

H

is onne ted,thealgorithmreturns

H

.We laimthatthisyieldsa

ρ

-approximation. Indeed,ifanoptimalsolution onsistsof

m

∗

edgesoftotalweight

ω

∗

,then

ALG = ω(F )

≥

ω

∗

m

∗

· d

,sin ebythe hoi eof

F

theaverageweightoftheedgesin

F

annotbesmallerthan theaverageweightoftheedgesofanoptimalsolution.As

m

∗

_{≤ m}

and

m

∗

_{≤ dn/2}

,weget that

ALG

≥

ω

∗

m

· d

and

ALG

≥

ω

∗

dn/2

· d =

ω

∗

n/2

.

Nowsuppose

H = (W, F )

onsistsof a olle tion

F

of

k

onne ted omponents. Then wegluethese omponentstogetherin

k

− 1

phases.Inea hphase,wepi ktwo omponents

C, C

′

_{∈ F}

,and ombine theminto anew onne ted omponent

C

ˆ

byaddinga onne ting path,withouttou hinganyother onne ted omponentof

F

.Wethenset

F ← F \{C, C

′

_}∪

{ ˆ

C

_}

.

Ea h phase operates as follows. For every two omponents

C, C

′

_{∈ F}

, ompute their distan e, dened as

d(C, C

′

_{) = min}

_{{dist(u, u}

′

_{, G)}

_{| u ∈ C, u}

′

_{∈ C}

′

_}

. Takeapair

C, C

′

_{∈ F}

attaining thesmallest distan e

d(C, C

′

₎

. Let

u

∈ C

and

u

′

_{∈ C}

′

be twoverti esrealizing thisdistan e,i.e.su hthat

dist(u, u

′

_{, G) = d(C, C}

′

₎

.Let

p(u, u

′

₎

beashortestpathbetween

u

and

u

′

in

G

.Let

C

ˆ

bethe onne ted omponentobtainedbymerging

C

,

C

′

andthepath

p(u, u

′

₎

.

Forthe orre tnessproof,weneedthefollowingtwoobservations: First,observethatineveryphase,thepath

p(u, u

′

₎

usedtomergethe omponents

C

and

C

′

does notgo through any other luster

C

′′

, sin e otherwise,

d(C, C

′′

₎

would be stri tly smallerthan

d(C, C

′

₎

, ontradi ting the hoi e of thepair

(C, C

′

₎

. Moreover,

p(u, u

′

₎

does notgothroughanyothervertex

v

inthe luster

C

ex eptforitsendpoint

u

,sin eotherwise,

dist(v, u

′

_{, G) < dist(u, u}

′

_{, G)}

, ontradi ting the hoi e of the pair

u, u

′

. Similarly,

p(u, u

′

₎

doesnotgothroughanyothervertex

v

′

in

C

′

.

Wenow laimthatafter

i

phases,themaximumdegreeof

H

satises

∆

H

≤ d − k + i + 1

. This is proved by indu tion on

i

. For

i = 0

, i.e. for the initial graph

H = (W, F )

, we observe that as

F

onsists of

d

edges arranged in

k

separate omponents, the largest omponentwill haveno more than

d

− k + 1

edges, hen e

∆

H

≤ d − k + 1

, as required. Nowsupposethe laimholdsafter

i

− 1

phases,and onsiderphase

i

.Allnodesotherthan thoseof thepath

p(u, u

′

₎

maintaintheirdegreefrom thepreviousphase.Thenodes

u

and

u

′

(d

_{− k + (i − 1)+ 1)+ 1 = d − k + i + 1}

,asrequired.Finally,theintermediatenodesof

p(u, u

′

₎

havedegree

2

≤ d − k + i + 1

(sin e

i

≥ 1

and

k

≤ d

).

Itfollowsthatbytheendofphase

k

− 1

,

∆

H

≤ d − k + k − 1 + 1 = d

.Also,atthatpoint

H

is onne ted.Hen e

H

isavalidsolution.

Finally, the approximation ratio of the algorithm is still at most

ρ = min

{n/2, m/d}

, sin ethisratiowasguaranteedfortheoriginallysele ted

F

,andthenalsubgraph ontains

theset

F

.

2

Let us now ompare the algorithm of Theorem 3.2 (algorithm

A

) and the algorithm of Theorem3.3(namely,algorithm

B

)forunweightedgraphs.Comparingbothapproximation ratios,we on ludethatalgorithm

A

performsbetterwhen

d < 2 log n

, whilealgorithm

B

isbetterwhen

d

≥ 2 log n

.Runningbothalgorithmsandsele tingthebestsolutionweget thefollowing

Corollary3.1 TheMDBCS

d

problem admitsa

ρ

-approximation algorithm in unweighted graphs, with

ρ = min

{n/2, nd/(2 log n), m/d, m/ log n}

.

4 Hardness of Approximating MSMD

d

Themain theorem ofthis se tion,Theorem 4.2,showsthatMSMD

d

doesnotadmit a onstant-fa torapproximation ongeneral graphs,for

d

≥ 3

. Werst provein Se tion 4.1 that MSMD

d

does notadmit aPTAS and then, using the errorampli ation te hnique, weprovethemainresult.Our redu tionis obtainedfrom thewell knownVertex Cover (VC)problem(seeAppendixA.1).

4.1 MSMD

d

does not admit a PTAS for any

d

≥ 3

WeproveTheorem4.1for

d = 3

,movingtheprooffor

d

≥ 4

toAppendixEduetola k ofspa e.

Theorem 4.1 MSMD

d

,

d

≥ 3

, isnot in PTAS,unlessP

=

NP.

Proof: Wegiveagap-preservingredu tionfromVertex Cover.Let

H

beaninstan eof Vertex Cover on

n

verti es. We onstru t aninstan e

G = f (H)

of MSMD

3

. Without lossofgenerality,we ansupposethat

H

ontains

3

· 2

m

edgesforsomeinteger

m

,andalso thateveryvertexof

H

hasdegreeatleastthree.

Let

T

bethe ompleteternaryrootedtreewithroot

r

andheight

m + 1

.Thenumberof leavesof

T

is

3

· 2

m

,and

T

ontains

3

· 2

m+1

− 2

verti es.Letusidentifytheleavesof

T

with edgesof

H

,and allthisset

E

(notethat

E

⊆ V (T )

).Weaddanother opyof

E

, alled

F

, andaHamiltonian y leon

E

∪ F

indu ingabipartitegraphwith partition lasses

E

and

F

asshowninFigure??.Letusalsoidentifytheverti esof

F

withedgesin

H

.Nowweadd

n

new verti es

A

identiedwith verti es of

H

,and join them tothe leavesof

T

a ording to the adja en y relations between the edges and verti es in

H

, i.e. an element

ℓ

∈ T

is

T

E(H)

E(H)

V(H)

E

A

F

onne tedto

v

∈ A

iftheedge orrespondingto

ℓ

in

H

isadja enttothevertex

v

of

V (H)

. Thegraph

G

builtin thiswayisdepi tedinFigure??.

We laimthatminimumsubgraphsof

G

ofminimumdegreeatleastthree orrespondto minimumvertex oversof

H

andvi e versa.Toseethis, rstnote thatifsu hasubgraph

U

of

G

ontainsavertexof

T

∪ F

,thenitshould ontainalltheverti esof

T

∪ F

,be ause of thedegree onstraints. Obviously

U

annot onsist just of verti esof

A

, hen e

U

must ontainalltheverti esof

T

∪ F

.Notethatalltheverti esof

F

havedegreetwoin

G[T

∪ F ]

. Therefore,the problem redu es to ndingthe smallestsubsetof verti esin

A

overingall theverti esin

F

.Thisisexa tlytheVertex Cover problemfor

H

.Thus,wehavethat

OP T

MSMD

3

(G) = OP T

VC

(H) +

|V (T )| + |V (F )| = OP T

VC

(H) + 9

· 2

m

− 2 .

Usingthisformula,itisstraightforwardto he kthat

f

is agap-preservingredu tion[25℄. To ompletetheproof,notethatVertex CoverisApx-hard,evenrestri tedtographs

H

ofsize linear in

OP T

VC

(H)

. Theexisten eof aPTAS forMSMD

3

provides aPTAS for Vertex Cover,whi hisa ontradi tion(underassumptionApx

6=

PTAS).

2

4.2 MSMD

d

is not in APX for any

d

≥ 3

Wearenowreadytoprovethefollowingtheorem:

Theorem 4.2 MSMD

d

,

d

≥ 3

, does notadmit any onstant-fa tor approximation, unless P

=

NP.

Proof: We give again the details for

d = 3

, and prove the result for the ase

d

≥ 4

in Appendix F. The proof is by appropriately applying the standard error ampli ation

te hnique.Let

G

1

=

{G}

bethefamilyofgraphswe onstru tedabove(Figure??)fromthe instan es

H

ofvertex over,

G

beingatypi almemberofthis family,andlet

α > 1

bethe fa torofinapproximabilityof MSMD

3

,thatexists byTheorem 4.1.

We onstru t a sequen e of families of graphs

G

k

, su h that MSMD

3

is hard to ap-proximate withinafa tor

θ(α

k

₎

in thefamily

G

k

.ThisprovesthatMSMD

3

doesnothave any onstant-fa torapproximation.Inthefollowing

G

k

willdenoteatypi alelementof

G

k

onstru tedusing theelement

G

of

G

1

.Wedes ribethe onstru tionof

G

2

,andobtainthe resultbyrepeatingthesame onstru tionindu tivelytoobtain

G

k

.Foreveryvertex

v

in

G

(denotingitsdegreeby

d

v

),we onstru tagraph

G

v

asfollows.First,takea opyof

G

,and hoose

d

v

other arbitraryverti es

x

1

, . . . , x

d

_v

of degree three in

T

⊂ G

. Then, werepla e ea h of these verti es

x

i

with a y le of length four, and join three of the verti esof the y letothethree neighborsof

x

i

,

i = 1, . . . , d

v

.Let

G

v

bethegraphobtainedin thisway. Notethatit ontainsexa tly

d

v

verti esofdegreetwoin

G

v

.

Nowwetakea opyof

G

,andrepla eea hvertex

v

with

G

v

.Then,wejointhe

d

v

edges in identto

v

tothe

d

v

verti esofdegreetwoin

G

v

.This ompletesthe onstru tionofthe graph

G

2

,illustratedin FigureGofAppendixG.

Wehavethat

|V (G

2

)

| = |V (G)|

2

_{+ o(}

|V (G)|

2

₎

,be auseea hvertexof

G

isrepla edwith a opyof

G

wherewehadrepla edsomeof theverti eswitha y leoflengthfour.

Tondasolutionof MSMD

3

in

G

2

,note that forany

v

∈ V (G)

, on e avertexin

G

v

is hosen,wehaveto look forMSMD

3

in

G

, whi h ishardupto a onstantfa tor

α

.But approximatingthenumberof

v

'sforwhi hweshouldtou h

G

v

isalsoMSMD

3

in

G

,whi h isharduptothesamefa tor

α

.Thisprovesthat approximatingMSMD

3

in

G

2

ishardup toafa tor

α

2

.Theproofofthetheoremis ompletedbyrepeatingthispro edure,applying thesame onstru tiontoobtain

G

3

,andindu tively

G

k

.

2

5 Approximating MSMD

d

Inthisse tion,itisshownthatforxed

d

,MSMD

d

isinPforgraphswhosetreewidth is

O(log n)

.Thisisdonebygivingapolynomialtimealgorithmbasedondynami program-ming.WerefertoAppendixA.2forthedenitionsoftree-de ompositionandtreewidth.

Thisdynami programmingalgorithmisthenusedinSe tion5.2toprovidean

O(n/ log n)

-approximationalgorithm of MSMD

d

forall lassesof graphsex ludingaxed graphas a minor. This algorithm relies on a partitioning result for minor-ex luded lass of graphs, provedbyDemaineet al.in [8℄.

5.1 MSMD

d

is in P for Graphs with Small Treewidth

In order to proveour resultsweneed the following lemma, whi h givesthe time om-plexity of nding a smallest indu ed subgraph of degree at least

d

in graphs of bounded treewidth. The proof is based on standarddynami programmingte hniques,and an be foundinAppendix H.

Lemma5.1 Let

G

be a graph on

n

verti es with a tree-de omposition of width atmost

t

, andlet

d

beapositiveinteger.Thenintime

O((d+1)

t

_(t+1)

d

2

n)

we aneitherndasmallest indu edsubgraphofminimumdegreeatleast

d

in

G

,oridentifythatnosu hsubgraphexists. A graph

G

is

q

-degenerated if everyindu ed subgraph of

G

hasa vertex of degree at most

q

.Itiswellknownthatthereisa onstant

c

su hthatforevery

h

,everygraphwithno

K

h

minoris

ch

√

_{log h}

-degenerated[9℄.Thisimpliesthat

M

-minor-freegraphswith

|M| = h

are

ch

√

log h

-degeneratedandhen ethelargestvalueof

d

forwhi hMSMD

d

isnon-empty is

ch

√

log h

,a onstant.Theabovedis ussion, ombinedwiththetime omplexityanalysis mentionedin Lemma5.1,implythefollowing

Corollary5.1 Let

G

be an

n

-vertex graph ex luding a xed graph

M

as minor, with a tree-de omposition of width

O(log n)

, and let

d

be apositive integer (a onstant). Then in polynomialtimeone aneithernd asmallestindu edsubgraphofminimum degreeatleast

d

in

G

, or on lude thatno su hsubgraphexists.

5.2 Approximation Algorithm for

M

-minor-Free Graphs

ThefollowingresultofDemaineetal.[8℄providesawayforpartitioningtheverti esof agraphex ludingaxedgraphas aminorinto subsetswithsmall treewidth.

Theorem 5.1( [8℄) Foraxedgraph

M

, thereisa onstant

c

M

su hthatfor anyinteger

k

_{≥ 1}

and for every

M

-minor-free graph

G

, the verti es of

G

(or the edges of

G

) an be partitionedinto

k + 1

sets su hthat any

k

of the sets indu e a graph of treewidth atmost

c

M

k

. Furthermore, su hapartition anbe foundin polynomial time.

Onemayassumewithoutlossofgeneralitythattheminimumdegreeoftheminor-freeinput graph

G = (V, E)

is at least

d

(by removing all the verti es of lower degree), and also that

|V (G)| = n = 2

p

forsome integer

p

≥ 0

(otherwise,repla e

log n

with

⌈log n⌉

in the des riptionofthealgorithm).

Des riptionof thealgorithm:

(1) Relying on Theorem 5.1, partition

V (G)

in polynomial time into

log n + 1

sets

V

0

, . . . , V

log n

su h that any

log n

of the sets indu e a graph of treewidth at most

c

M

log n

,where

c

M

isa onstantdepending onlyontheex ludedgraph

M

.

(2) Runthedynami programmingalgorithmofSe tion5.1onallthesubgraphs

G

i

=

G[V

_{\ V}

i

]

of

log n

sets,

i = 0, . . . , log n

.

(3) Thispro edure ndsall thesolutionsofsize at most

log n

. Ifno solutionisfound, outputthewholegraph

G

.

This algorithm learly provides an

O(n/ log n)

-approximation for MSMD

d

in minor-free graphs,for all

d

≥ 3

. The runningtime of thealgorithm is polynomial in

n

, sin ein step (2),forea h

G

i

,thedynami programmingalgorithmrunsin

O((d + 1)

t

i

_(t

i

+ 1)

d

2

n)

time, where

t

i

isthetreewidthof

G

i

,whi hisatmost

c

M

log n

.

6 Approximating DDD

k

S

Weprovideadeterministi approximationalgorithmfortheDDD

k

SprobleminTheorem 6.1 (strongly based on the algorithm for D

k

S of [11℄), and a randomized approximation algorithminAppendixI.Eveniftheperforman eoftherandomizedalgorithmisworse,we in ludeitbe ausetheideabehindthealgorithmisquitesimple.

Theorem 6.1 TheDDD

k

Sproblemadmitsadeterministi

O(n

δ

₎

-approximationalgorithm, for someuniversal onstant

δ < 1/3

.

Proof: Givenaninputgraph

G

,let

ρ

OP T

k

betheoptimalaveragedegreeofasubgraphof

G

onexa tly

k

verti es(i.e.theoptimumof D

k

S),andlet

δ

OP T

k

betheoptimalminimum degreeofasubgraphof

G

withat most

k

verti es(i.e.theoptimumofDDD

k

S).Let

C

be theapproximationratioofthealgorithmforD

k

Sof[11℄,i.e.

C = O(n

δ

₎

forsomeuniversal onstant

δ < 1/3

.Givenagraph

H

,let

ρ(H)

denotetheaveragedegreeof

H

,andlet

δ(H)

denotetheminimumdegreeof

H

.

Weknow,by[11℄,thatwe anndasubgraph

H

k

of

G

on

k

verti essu hthat

ρ(H

k

)

≥

ρ

OP T

k

/C

.Removingre ursivelytheverti esof

H

k

withdegreestri tlysmallerthat

ρ(H

k

)/2

we obtain a subgraph

H

′

k

of

H

k

on at most

k

verti es su h that

δ(H

′

k

)

≥ ρ(H

k

)/2

≥

ρ

OP T

k

/(2C)

.

Thenextstep onsistsin provingthatthere existsaninteger

k

0

,

1

≤ k

0

≤ k

, su h that

ρ

OP T

k

0

≥ δ

OP T

k

,sowe anruntheD

k

Salgorithmforea h

k

′

_{≤ k}

,removelow-degreeverti es ea h time,andtakethebest solutionofDDD

k

Samong

H

′

2

, H

3

′

, . . . , H

_k−1

′

, H

k

′

.

Finally, letusprovethat

k

0

exists. Let

H

bethe optimalsolutionof DDD

k

S,

δ(H) =

δ

OP T

k

.Let

k

0

=

|V (H)|

(

k

0

≤ k

).Thisisthe

k

0

wearelookingfor,be ause

ρ

OP T

k

0

≥ ρ(H) ≥

δ(H) = δ

OP T

k

.

Theabovepro edure learly onstitutesa

(2C)

-approximationforDDD

k

S.

2

7 Con lusions

This paper onsideredthree Degree-Constrained Subgraphproblems and studied their behavior in terms of approximation algorithms and hardnessof approximation.Our mainresultsandseveralinterestingquestions thatremainopenaredis ussedbelow.

We proved that the MDBCS

d

problem is not in Apx for any

d

≥ 2

, and we pro-videdadeterministi approximationalgorithmwithratio

min

{m/ log n, nd/(2 log n)}

(resp.

min

{n/2, m/d}

)forgeneralunweighted(resp.weighted)graphs.Finally,wegavea onstant-fa torapproximationwhentheinputgrapha eptsalow-degreespanningtree.Closingthe hugegap betweenthehardnessbound andthe approximationratioofouralgorithm looks likeapromising resear hdire tion.Itwas provedin[16℄ that ifanypolynomialtime algo-rithm anapproximatetheLongest Pathproblemtoaratioof

2

O(log

1−ε

_n)

,forany

ε > 0

, thenNPhasaquasi-polynomialdeterministi timesimulation.Nevertheless,thisresultdoes

notapplydire tly totheMDBCS

d

problemforall

d

≥ 2

,soadierentstrategyshould be devised.

WeprovedthattheMSMD

d

problemisnotinApxforany

d

≥ 3

.Itwouldbeinteresting tostrengthenthishardnessresultusingthepowerofthePCPtheorem.Onthepositiveside, we gave an

O(n/ log n)

-approximation algorithm for the lass of graphsex luding a xed graph

H

as a minor. Finally, nding an approximation algorithm for MSMD

d

in general graphsseemsto bea hallengingopenproblem. Itseemsthat MSMD

d

remainshardeven forproperminor- losed lassesofgraphs.

Weprovideda

O(n

δ

₎

-approximationalgorithmforthe DDD

k

S problem,for some uni-versal onstant

δ < 1/3

.Itwould beinterestingtoprovidehardnessresults omplementing this approximation algorithm. Anotheravenuefor further resear h ould be to onsider a mixedversionbetweenDDD

k

SandMSMD

d

,that wouldresultin atwo- riteria optimiza-tionproblem.Namely,givenagraph

G

,thegoalwouldbetomaximizetheminimumdegree whileminimizingthesizeofthesubgraph,bothparametersbeingsubje ttoalowerandan upperbound,respe tively.

Référen es

[1℄ L.Addario-Berry,K.Dalal,andB.Reed.Degree onstrainedsubgraphs.Dis reteAppl. Math., 156(7):11681174,2008.

[2℄ N. Alon, R. Yuster, and U. Zwi k. Color- oding : a new method for nding simple paths, y lesandothersmallsubgraphswithinlargegraphs.InPro eedings ofthe26th annual ACM symposium on Theory of Computing, pages 326335, New York, USA, 1994.

[3℄ O.Amini, S.Pérennes,andI.Sau. HardnessandApproximationofTra Grooming. In Pro eedings of the 18th International Symposiumon Algorithms and Computation, volume4835ofLNCS,pages561573,volume4835ofLNCSseries,2007.

[4℄ O. Amini,I.Sau, andS. Saurabh. ParameterizedComplexityoftheSmallest Degree-Constrained SubgraphProblem. InPro eedings ofthe International Workshop on Pa-rameterizedandExa tComputation, volume5008ofLNCS,pages1329,volume5008 of LNCSseries,2008.

[5℄ R.P.Anstee.Minimumvertexweightedde ien yof

(g, f )

-fa tors:agreedyalgorithm. Dis reteAppl.Math.,44(1-3):247260,1993.

[6℄ A. Björklundand T.Husfeldt. Finding aPathof Superlogarithmi Length. SIAM J. Comput.,32(6):13951402,2003.

[7℄ W. Cook, W. Cunningham, W. Pulleyblank, and A. S hrijver. Combinatorial Opti-mization. JohnWileyandSons,NewYork,1998.

[8℄ E.Demaine,M.Hajiaghayi,andK.C.Kawarabayashi.Algorithmi GraphMinor The-ory : De omposition,ApproximationandColoring. In46th Annual IEEE Symposium on FoundationsofComputer S ien e (FOCS),pages637646,O tober2005.

[9℄ R. Diestel. GraphTheory. Springer-Verlag,2005.

[10℄ P.Erd®s,R.Faudree,C.C.Rousseau,andR.H.S help. Subgraphsofminimaldegree

k

. Dis reteMath.,85(1):5358,1990.

[11℄ U. Feige, D. Peleg, and G. Kortsarz. The Dense

k

-Subgraph Problem. Algorithmi a, 29(3) :410421,2001.

[12℄ M. Fürerand B.Raghava hari. Approximatingtheminimum-degree spanning treeto within one from the optimal degree. In Pro eedings of the 3rd annual ACM-SIAM Symposiumon Dis reteAlgorithms,pages317324,USA,1992.

[13℄ M. Fürer and B. Raghava hari. Approximating the minimum-degree steiner tree to within one of optimal. InSODA : sele ted papers from the third annual ACM-SIAM symposiumonDis retealgorithms,pages409423,USA,1994.

[14℄ H. Gabow. An e ientredu tionte hniquefordegree- onstrainedsubgraphand bidi-re ted networkowproblems. InPro eedings of the 15th annual ACMsymposiumon Theory ofComputing,pages448456,USA,1983.ACM Press.

[15℄ M.GareyandD.Johnson.ComputersandIntra tability.W.H.Freeman,SanFran is o, 1979.

[16℄ D. Karger,R. Motwani, andG. Ramkumar. Onapproximatingthe longestpath in a graph. Algorithmi a,18(1):8298,1997.

[17℄ S. Khot. Ruling out PTAS for graph min-bise tion, densest subgraph and bipartite lique.In45thAnnualIEEESymposiumonFoundationsofComputerS ien e(FOCS), pages136145,2004.

[18℄ L. Lovász and M. Plummer. Mat hing Theory. Annals of Dis rete Math. 29, North-Holland, 1986.

[19℄ C. Lund and M. Yannakakis. The Approximation of Maximum Subgraph Problems. Automata, Languages and Programming : 20th International Colloquium, ICALP 93, Lund, Sweden, July5-9, 1993 : Pro eedings,1993.

[20℄ J.MunroandV.Raman. Su in tRepresentationofBalan edParenthesesandStati Trees. SIAMJournalonComputing,31(3):762776,2001.

[21℄ C. Papadimitriouand M.Yannakakis. Thetravelingsalesmanproblem withdistan es one andtwo. Mathemati sof OperationsResear h,18(1):111,1993.

[22℄ R.Ravi,M.Marathe,S.Ravi,D.Rosenkrantz,andH.H.III.Approximationalgorithms fordegree- onstrainedminimum- ostnetwork-designproblems.Algorithmi a,31(1):58 78, 2001.

[23℄ O. Ri hard. TheNumberofTrees. Annals of Mathemati s,Se ondSeries49(3):583 599,1948.

[24℄ Y. Shiloa h. Anotherlook at thedegree onstrained subgraphproblem. Inf. Pro ess. Lett.,12(2):8992,1981.

[25℄ V. Vazirani. Approximation Algorithms. Springer-Verlag,2003.

[26℄ S. Win. On a Conne tion Betweenthe Existen e of

k

-Trees and the Toughness of a Graph. GraphsandCombinatori s, 5(1):201205,1989.

A Basi Denitions

A.1 Approximation Algorithms and Gap-preserving Redu tions GivenanNP-hardminimization(resp.maximization)problem

Π

andapolynomialtime algorithm

A

, let

OP T

Π

(I)

bethe optimal valueof the problem

Π

for theinstan e

I

, and let

ALG(I)

be the value given by algorithm

A

for the instan e

I

. We say that

A

is an

α

-approximationalgorithmfor

Π

ifforanyinstan e

I

of

Π

,

OP T

Π

(I)/ALG(I)

≥ α

(resp.

OP T

Π

(I)/ALG(I)

≤ α

).

The lassApx onsistsofallNP-hardoptimizationproblemsthat anbeapproximated within a onstant fa tor. The sub lass PTAS (Polynomial Time Approximation S heme) ontainstheproblemsthat anbeapproximatedinpolynomialtimewithinaratio

1 + ε

for any onstant

ε > 0

. Assuming P

6=

NP, there is a stri t in lusion of PTAS in Apx(for instan e,Vertex CoverisinApx

\

PTAS),hen eanApx-hardnessresultforaproblem impliesthenon-existen eofaPTAS.

Forourinapproximabilityresults,wemakeuseof thefollowingredu tions ( f.[25℄). DenitionA.1 (Gap-preserving redu tion) For two minimization problems

Π

1

and

Π

2

, a gap-preserving redu tion from

Π

1

to

Π

2

, parameterized by (

f

1

,

α

) and (

f

2

,

β

), is a pro edure thatgiven aninstan e

x

of

Π

1

, omputes inpolynomial timeaninstan e

y

of

Π

2

su hthat:

 if

OP T (x)

≤ f

1

(x)

, then

OP T (y)

≤ f

2

(x)

.

 if

OP T (x) > α(

|x|)f

1

(x)

, then

OP T (y) > β(

|x|)f

2

(x)

.

Theusefulnessofgap-preservingredu tions stemsfrom thefollowingknownfa t:

LemmaA.1 If there is a gap-preserving redu tion from

Π

1

to

Π

2

and it is NP-hard to approximate

Π

1

within afa torstri tlylessthan

α

,then itisalso NP-hardtoapproximate

Π

2

withina fa torstri tlylessthan

β

.

Finally, let us re all thedenition of the vertex overproblem, from whi h weobtainthe hardnessredu tionofSe tion4.

Vertex Cover (VC)

Input:An undire tedgraph

G = (V, E)

. Output : Asubset

V

′

_{⊆ V}

of theminimum size su h that for everyedge

e =

(u, v)

,either

u

∈ V

′

or

v

∈ V

′

.

A.2 Tree-de omposition and Treewidth

DenitionA.2 (Tree-de omposition,treewidth) Atree-de ompositionofagraph

G =

(V, E)

isapair

(T,

X )

,where

T = (I, F )

isatree,and

X = {X

i

}, i ∈ I

isafamilyofsubsets of

V (G)

, alledbagsandindexedby thenodesof

T

, su hthat

1. ea h vertex

v

∈ V

appearsinatleastone bag,i.e.

S

i∈I

X

i

= V

;

2. for ea h

v

∈ V

the setof nodesindexedby

{i | i ∈ I, v ∈ X

i

}

forms asubtreeof

T

; 3. Forea h edge

e = (x, y)

∈ E

, thereisan

i

∈ I

su hthat

x, y

∈ X

i

.

Thewidthofatree-de ompositionisdenedas

max

i∈I

{|X

i

|−1}

.Thetreewidthof

G

,denoted by

tw(G)

, isthe minimum widthofatree-de ompositionof

G

.

B Proof of Claim 1

Weprovethe laiminaseriesofobservationsimprovingthesolution,andapplythemin orderoftheirappearan e.Foragivenedgeset

F

,let

X(F )

bethesetofverti es ontaining theend-pointsoftheedgesin

F

.

(a) Suppose

E

1

∩E

′

₌

_∅

.Then

H = (X(E

1

), E

1

)

is onne tedandeveryvertex

v

∈ X(E

1

)

hasdegreeatmost

d

in

H

.Thisimpliesthat

H

ontainsa y le,soremovinganyedge from this y le will not break onne tivity. So we an remove any edge

(u, v)

from this y le andaddtheedges

(u

1

, u)

and

(v

1

, v)

,obtainingasolutionoflargerweight. Therefore,weassumehen eforththat

E

1

∩ E

′

_{6= ∅}

.

(b) Suppose

V

\ X(E

1

)

6= ∅

, that is there is a vertex

v

∈ V

whi h is not ontained in

X(E

1

)

. In this ase, by Observation (a), there exists a vertex

u

∈ X(E

1

)

su h that one of theedges

(u

i

, u)

,

1

≤ i ≤ d − 2

,is in

E

1

.We thenset

E

1

← E

1

− {(u

i

, u)

} +

{(u, v), (v, v

i

)

| 1 ≤ i ≤ d − 2}

. Clearly wemaintain onne tivity(as removingedges from

E

′

doesnotbreak onne tivity)and theweightofsolutionin reasesbyatleast

1

.Werepeatthispro edureuntilthe urrentsolution ontainsalltheverti esof

G

. ( ) Suppose

H

′

_{= (V, E}

_{∩ E}

1

)

is neitheraspanningtreenorahamiltonian y le.Noti e that

H

′

is onne ted,asremovingdegree

1

verti esof

V

′

doesnotbreak onne tivity. This implies that there is a y le

C

in

H

′

su h that there is avertex

v

onit whose degree is at least

3

in

H

′

(otherwise,

H

′

would be dis onne ted). This implies that thereexistsanedge

e = (v, v

i

)

su hthat

e /

∈ E

1

.Let

(u, v)

beanedgeon

C

.Wethen set

E

1

← E

1

− {(u, v)} + {(v, v

i

)

}

. Clearlywemaintain onne tivity(asremovingan edgefroma y ledoesnotbreak onne tivity)andtheweightofthesolutionin reases byatleast

1

.

(d) Suppose

H

′

_{= (V, E}

_∩E

1

)

isaspanningtree.If

H

′

isapaththentheend-pointsofthis path,say

u

and

v

,havedegre

1

in

H

′

,hen ewe an addtheedge

(u, v)

andobtaina solutionofhigherweight.Soletus supposethat

H

′

isnotapath,hen ethere exists avertex

v

ofdegreeatleast

3

in

H

′

.Thisimpliesthatthereexistsanedge

e = (v, v

i

)

su hthat

e /

∈ E

1

.Let

(u, v)

beanedgein identto

v

inthespanningtree

H

′

.Consider thespanningforest

H

′

_{− {(u, v)}}

, onsistingoftwosub-tress

H

′

u

and

H

′

v

ontaining

u

and

v

respe tively.Wesele taleaf

w

1

∈ H

′

u

andaleaf

w

2

∈ H

′

v

(

w

2

6= v

),andweset

E

1

← E

1

− {(u, v)} + {(v, v

i

), (w

1

, w

2

)

}

.Clearlytheresultantgraphis onne tedand hashigherweight.

We anapply theaboverulesin polynomialtime toobatinagraph

G

3

whi h isasolution of MDBCS

d

in

G

′

C Proof of Theorem 2.2

Werststatethefollowingte hni allemma.

LemmaC.1 Forall

d

≥ 2

,MDBCS

d

restri tedtothe lassofgraphsforwhi hanyoptimal solution ontainsatleast

2

verti esof degreeatmost

d

− 1

is NP- omplete.

Proof: WeknowthatMDBCS

d

isNP- ompleteingeneralgraphsforall

d

≥ 2

[15℄,even whenalltheweightsoftheedgesareequalto1.Let

G = (V, E)

beageneralinputgraphwith allweightsequalto 1.For ea h (unordered)pair ofverti es

u, v

∈ V

,

u

6= v

, we onstru t agraph

G

u,v

in thefollowingway:

G

u,v

isobtainedfrom

G

byaddingtwonewverti es

u

′

and

v

′

,plustheedges

{u

′

_{, u}

_}

and

{v

′

_{, v}

_}

withweight

W

≥ 2|E(G)|

.Itis learthat,forea h pair

{u, v}

,anyoptimalsolutionfor

G

u,v

ontainstheedges

{u

′

_{, u}

_}

and

{v

′

_{, v}

_}

,hen eany optimalsolution ontainsthe2verti es

u

′

and

v

′

ofdegreeone,

1

≤ d − 1

.Letusseethatif we ouldsolveMDBCS

d

in

G

u,v

in polynomialtimeforea hpair

u, v

,then we ouldalso ndanoptimalsolutionfor

G

inpolynomialtime, whi hwouldbea ontradi tion.Indeed, let

OP T

u,v

betheweightofanoptimalsolutionof MDBCS

d

in

G

u,v

.Let

OP T

G

u,v

=



OP T

u,v

− 2W + 1,

if

{u, v} ∈ E(G)

anditisnotintheoptimalsolutionin

G

u,v

,

OP T

u,v

− 2W,

otherwise.

ThenthenumberofedgesofanoptimalsolutionofMDBCS

d

in

G

isexa tly

max

u,v

OP T

G

u,v

, that an be omputedinpolynomialtime. Thelemmafollows.

2

Again, we provetheresult for

d

≥ 3

,the result for

d = 2

followingfrom [16℄.We will usetheerrorampli ationte hnique.Let

α > 1

betheinapproximability onstantgivenby Theorem2.1.Givenafamilyofgraphs

G

withatypi alelement

G = (V, E)

with

|V (G)| = n

and

|E(G)| = m

,su hthat MDBCS

d

ishardtoapproximatein thisfamilywithin afa tor

α > 1

,wewillbuildasequen eof familiesofgraphs

G = G

1

_,

G

2

_{, . . .}

,su hthat MDBCS

d

is hardtoapproximatein

G

k

withinafa tor

α

k

.ThisimpliesthatMDBCS

d

isnotinApx.In thefollowing

G

i

willbeatypi alelementof

G

i

.Letussupposethatthereexistsanalgorithm

C

forapproximatingtheoptimalvalueof MDBCS

d

onanygraphwithina onstantfa tor of

ρ > 1

,andderivea ontradi tion.

Assume without loss of generality that all the weights of the edges of

G

are equal to 1(this an be assumedby repla ingthe edgesof weight

W

by aternarytree oftotal size

W

− 1

,for

d > 2

, andadding twoedges ofweight1to

u

i

and

v

i+1

). Combinining Lemma C.1 andthe proof ofTheorem 2.1, itis easyto see that MDBCS

d

doesnot admitPTAS restri tedtographsforwhi hany optimalsolution ontainsat leasttwoverti esof degree

d

_{− 1}

.(Indeed, note that in thefamily ofgraphs ontru ted in Theorem 2.1, anyoptimal solutionof MDBCS

d

ontains

n(d

− 2)

verti es of degree1.) So an also assume without lossof generality that anyoptimal solution in

G

(and indu tivelyalso in

G

k

) ontains at least2verti esofdegree

d

− 1

.

Let

OP T

k

and

H

k

betheweightof anoptimalsolutionand anoptimal onne ted sub-graphin

G

k

,respe tively.Wepro eedtoillustrateindetailthe onstru tionof

G

2

pairofverti es

{u, v} ∈ V

2

,

u

6= v

,webuildthegraph

G

2

u,v

in thefollowingway:wetake thegraph

G

andwerepla eea h edge

e

i

= (x, y)

∈ E(G)

,

i = 1, . . . , m

,with a opy

G

i

of

G

(again,the opyofthevertex

u

∈ V (G)

in

G

i

islabeled

u

i

),andweaddtheedges

(x, u

i

)

and

(y, v

i

)

with weight

ε

2

,

0 < ε

2

<< 1

for

i = 1, . . . , m

.We dene

G

2

as the graph

G

2

u,v

forwhi halgorithm

C

givesthebest solution. Claim2

OP T

2

= OP T

2

1

+ 2ε

2

· OP T

1

≈ OP T

1

2

.

Sin eanyoptimalsolutionin

G

ontainsat least2verti eswith degreeat most

d

− 1

,the best solutionin

G

2

ontains

OP T

1

opiesof

H

1

,oneforea hedgeof

H

1

,plus2edgeswith weight

ε

2

forea h opyof

H

1

.

Claim3 Given any solution

S

2

in

G

2

with weight

x

, itispossible tond asolution

S

1

in

G

withweight atleast

√

_x

.

Toprovethe laim,wedistinguishtwo ases:

•

Case a :

S

2

interse tsatleast

√

_x

opies of

G

.

Let

S

1

bethesubgraphof

G

indu edbytheedges orrespondingtothese opies of

G

in

G

2

.

•

Case b :

S

2

interse tsstri tly fewerthan

√

_x

opies of

G

. Let

S

1

be

S

2

∩ G

i

, with

G

i

being the opy of

G

in

G

2

su h that

|E(S

2

∩ G

i

)

|

is maximized.

Inboth ases

S

1

is onne ted,hasmaximumdegreeatmost

d

, andhasat least

√

_x

edges.

ρ

-approximationin

G

2

,then itis possibleto ndasolutionfor

G

withweightat least

q

OP T

2

ρ

≥

OP T

_√

1

ρ

.Thatis,there existsa

√

_ρ

-approximationin

G

. Indu tively,to build

G

k

wetakeasequen e ofweightsfortheedges onne tingthe opies of

G

k−1

su hthat

0 < ε

k

<< ε

k−1

<< . . . << ε

3

<< ε

2

<< 1.

Claim2be omes

OP T

k

≈ OP T

2

k−1

+ 2ε

k

· OP T

k−1

≥ OP T

k−1

2

≥ OP T

k−2

4

≥ . . . ≥ OP T

2

k

1

, and the sameargumentsapply. As thesize of

G

k

is a polynomialfun tion on the size of

G

, this means that given a

ρ

-approximation algorithm for MDBCS

d

for

G

k

in

G

k

with runningtimepolynomialinthesizeof

G

k

,one anobtaina

ρ

1/2

k

-approximationalgorithm for MDBCS

d

for

G

in

G

and with running time polynomial in

|G|

. But there exists an integer

k

su hthat

ρ

2

−k

< α

, ontradi tingTheorem2.1. Thetheoremfollows.

D Approximating MDBCS

d

in Graphs with Low-degree Spanning Trees

We rst state asimple lemma about the optimalsolutions of the polynomial problem MDBS

d

(the denitionisthesameas theMDBCS

d

problem, ex eptthatthe onne tivity oftheoutputsubgraphisnotrequired).