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Laboratoire d'Analyse et Modélisation de Systèmes pour

l'Aide à la Décision

CNRS UMR 7024

CAHIER DU LAMSADE

172

Juin 2000

Differential approximation results for traveling

salesman problem

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Résumé ii

Abstra t ii

1 Introdu tion 1

2 Preserving dierential approximation for several min_TSP versions 3

3 2_OPT anddierential approximationforthe generalminimumtraveling

sales-man 4

4 Approximating min_TSP12 7

4.1 Spe i ation of the min_TSP12-algorithm and evaluationof(K

n

) . . . 8

4.1.1 Constru tion andevaluationof C . . . 9

4.1.1.1 q=1 . . . 10

4.1.1.2 q=k . . . 10

4.1.1.3 q=k+1 . . . 10

4.1.2 Constru tion andevaluationof D . . . 10

4.1.3 Constru tion andevaluationof ~ T . . . 11

4.1.4 Overallspe i ationofthemin_TSP12-algorithm, onstru tionand eval-uation ofT . . . 11

4.2 Abound for!(K n ) . . . 11

4.2.1 Disjoint elementary paths onV(C) . . . 12

4.2.2 Disjoint elementary paths onV(D) . . . 12

4.2.3 Disjoint elementary paths onV( ^ T). . . 12

4.2.3.1 q>0 . . . 12

4.2.3.2 q=0 . . . 13

4.2.4 Disjoint elementary paths onV(K n ). . . 13 4.2.4.1 q>0;jE2j<jC p+1 j . . . 13 4.2.4.2 q>0;jE2j=jC p+1 j . . . 13 4.2.4.3 q=0;jE2j<jC p+1 j . . . 14

4.3 Thedierential approximationratio of TSP12 . . . 15

4.4 Ratio3/4 istight for TSP12 . . . 15

5 Further results for minimum traveling salesman 15 5.1 Bridgesbetween dierential andstandard approximation . . . 15

5.2 An inapproximability result . . . 17

6 Dierentialapproximation of maximum traveling salesman 18

7 An improvement of the standard ratio for the maximum traveling salesman

with distan es 1 and 2 19

8 Towardsstrongerdierential-inapproximabilityresultsforthetraveling

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ommer e

Résumé

Nous ommençonspardémontrerquelesversionsmaximisationetminimisationdu pro-blème duvoyageurde ommer e sont approximablesà rapport diérentiel 1/2.Nous pré-sentons ensuite une 3/4-approximationpolynomiale du as parti ulier à distan es 1et 2; e résultatnouspermet notammentderamener lerapportstandard onnu pourlaversion maxisation de e sous-problème de 5/7 à 7/8. Nous proposons enn un résultat négatif:

approximer le voyageur de ommer e, à oût minimum omme maximum, à mieux que

3475=3476+ estNP-di ilepourtout>0.

Mots- lé: algorithmed'approximation,rapportd'approximation,problèmeNP- omplet, omplexité,rédu tion,voyageurdu ommer e.

Dierential approximation results for traveling salesman problem

Abstra t

Weprovethatbothminimumandmaximumtravelingsalesmanproblems anbe approx-imatelysolved,in polynomialtime withinapproximationratioboundedaboveby1/2. We nextprovethat,whendealingwithedge-distan es1and2,bothversionsareapproximable within 3/4. Based upon this result, we then improve the standard approximation ratio knownformaximumtravelingsalesmanwithdistan es1and2from5/7to7/8. Finally,we provethat,forany>0,itisNP-hard toapproximate bothproblems withinbetterthan 3475=3476+.

Keywords: approximationalgorithm,approximationratio,NP- omplete problem, om-plexity,redu tion,travelingsalesman.

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Given a omplete graph onn verti es,denoted byK n

, with positive distan es on itsedges, the minimum travelingsalesman problem(min_TSP) onsistsinminimizing the ostofa Hamilto-nian y le, the ostof su h a y lebeingthe sum of the distan es onits edges. Themaximum traveling salesmanproblem(max_TSP) onsistsinmaximizingthe ostofaHamiltonian y le. Furtherspe ialbutverynatural asesofTSParetheoneswhereedge-distan esaredenedusing the `

2

norm(Eu lideanTSP), orwhere edge-distan esverifytriangleinequalities(metri TSP); an interesting sub- ase of the metri TSP is the one in whi h edge-distan es are only 1 or 2 (TSP12). Bothmin_andmax_TSP,evenintheirrestri tedversionsmentionedjustmentioned

above, arefamousNP-hard problems.

In general,NP optimization (NPO) problemsare ommonly dened asfollows.

Denition 1. An NPO problem isasa four-tuple(I;S;v I

;opt )su h that:

1. I isthe setof instan es of andit anbe re ognizedin polynomialtime;

2. given I 2 I (let jIj be the size of I), S(I) denotes the set of feasible solutions of I; moreover, there exists a polynomial P su h that, for every S 2 S(I) (let jSj be the size ofS),jSj=O(P(jIj)); furthermore,givenanyI andanyS with jSj=O(P(jIj)),one an de ide in polynomial time ifS2S(I);

3. given I 2 I and S 2 S(I), v

I

(S) denotes the value of S; v I

is integer, polynomially omputable andis ommonly alled obje tive fun tion;

4. opt2fmax;ming.

Given an instan e I of an NPO problem and a polynomial time approximation algorithm A

feasibly solving , we will denote by !(I),  A

(I) and (I) the values of the worst solution of I, of the approximated one (provided by A when running on I), and the optimal one for I, respe tively. Thereexistmainly twothought pro essesdealing with polynomialapproximation.

Commonly ([13℄), the quality of an approximation algorithm for an NP-hard minimization

(resp., maximization) problem  is expressed by the ratio ( alled standard in what follows) 

A

(I)=(I)= (I),andthequantity A

=inffr: A

(I)<r;I instan eof g(resp., A

=supfr: 

A

(I)>r;I instan eof g) onstitutestheapproximationratioofAfor. Re entworks([9,8℄), strongly inspired by[3 ℄ (see also[12, 23℄), bring to the foreanother approximationmeasure, as powerfulasthetraditionalone( on erningthetype,thediversityandthequantityoftheresults produ ed), the ratio ( alled dierential in what follows) Æ

A

(I) = (!(I) (I))=(!(I) (I)). ThequantityÆ

A

=supfr: Æ A

(I)>r;I instan eof gisthedierential approximationratioof A for. Inwhatfollows,weusenotationwhendealingwith standardratioandnotationÆ when dealing with the dierential one. Moreover () (resp., Æ()) will denote the best standard (resp., dierential) approximation ratio for .

In [3℄, the term trivial solution is used to denote what in [9, 8℄ and here is alled worst solution. Moreover,alltheexamplesin[3 ℄ arryoverNP-hardproblemsforwhi hworstsolution an be trivially omputed. This is for example the ase of maximum independent set where, givenagraph,the worstsolutionisthe emptyset, orofminimumvertex over,where theworst solution isthe vertex-set ofthe input-graph, or evenof the minimum graph- oloring whereone an trivially olor the verti es of the input-graph using a distin t olor per vertex. On the ontrary, for TSP things are very dierent. Let us take for example min_TSP. Here, given a

graph K

n

, the worst solution for K n

is a maximum total-distan e Hamiltonian y le, i.e., the

optimalsolutionofmax_TSPinK

n

. The omputationofsu hasolutionisveryfar frombeing

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on ept ofthe worst solution, the following denition, proposedin [9℄,will be usedhere.

Denition 2. Givenatypi alinstan eI ofanNPO problem,the worstsolutionofI isthe

optimal solution of a new NPO problem



 where items 1 to 3 of denition 1 areidenti al for

both and  ,and opt (  )= 

max opt ()=min

min opt()=max

One of the features of the dierential ratio is to be stable under ane transformation of the obje tive fun tion of aproblem and soitdoesnot reate a dissymmetrybetween minimization andmaximization problems. Thisisvery lear in the aseofTSP. Dealingwith min_TSP itis very well-known that its general version is not approximable in polynomial time within better than 2

p(n)

for a polynomial p. On the other hand, its maximization version, max_TSP, the NP-hardness of whi h is immediately proved if one repla es distan e d(i;j) for min_TSP by M d(i;j) in max_TSP, for an M greater than the largest edge distan e in the input graph of min_TSP, an be approximated in polynomial time within 5/7([20 ℄).

Letusre allsomestandard terminologyfromthe theoryofthepolynomialapproximationof theNP-hardproblems(forthestandardapproximationframework). GivenanNPminimization (resp.,maximization)problem,a onstant-ratioapproximationalgorithmforisapolynomial timeapproximationalgorithm(PTAA)guaranteeing approximationratio boundedabove (resp., below) by a xed onstant, i.e., by a onstant that does not depend on any input-parameter

of . APX is the lass of the NP optimization problems solved by onstant-ratio PTAAs.

A polynomial time approximation s hema (PTAS) for  is a sequen e of PTAAs (re eiving as inputs any instan e of  and a xed onstant ) guaranteeing approximation ratio bounded above (resp., below) by 1+ (resp., 1 ), for every  > 0. If a PTAS is polynomial in bothnand1=, then itis alledfully polynomialtime approximation s hema (FPTAS).Forthe dierential approximation, the ratio a hieved by polynomial time approximation s hemata is 1 for both minimizationandmaximization. Finally, APX- omplete isthe lassof problems inAPX,whi h, inaddition,are ompletewithrespe ttotheexisten eofaPTASsolvingthem,

in other words, if any APX- omplete problem ould be solved by a PTAS, then any other

APX- omplete problem ould be so.

Asitisshownin [9 , 8℄,manyproblems behave in ompletely dierent ways regarding tradi-tional or dierential approximation. This is, for example, the ase of minimum graph- oloring or, even, of minimum vertex- overing. This paper deals with another example of the diversity in the nature of approximation results a hieved within the two frameworks, the TSP. For this problem and its versions mentioned above, a bun h of standard-approximation results (posi-tive or negative) have been obtained until nowadays. The rst inapproximability result is the one of [21 ℄ (see also [13 ℄) arming that it is NP-hard to approximate min_TSP within any onstant fa tor; with the same proof, one an easily rene the result of [21 ℄ to dedu e the in-aproximability of min_TSP within any ratio of the form 2

p(n)

for any polynomial p. On the otherhand, the metri min_TSP isapproximable within 3/2 ([5℄), the symmetri min_TSP12 within7/6([18 ℄)(re allthattheoriginalproofoftheNP- ompletenessofthe min_TSPisdone

by redu tion to min_TSP12), while the asymmetri version of min_TSP12 is approximable

within 17/12 ([22℄). Moreover, min_TSP12 is APX- omplete ([18 ℄), onsequently, given the result of [2 ℄, it annot be solved by a PTAS unless P=NP; in other words, 9 > 0 for whi h approximation ofmin_TSP12 within ratio smallerthan 1+is NP-hard. Furthermore, even in graphswhere thedensityofthe subgraphspannedbythe edgesof length1isboundedbelow by a onstant 2℄0;1=2[, min_TSP12 annot be solved by a polynomial time approximation s hema ([11℄). The works of [10 ℄ and more re ently of [4℄ rene the result of [18℄ spe ifying

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smallerthan, or equal to, 3475=3476 ; in otherwordsthe resultof [10℄ gives avalue equal

to 1=3476 

0 , 8

0

> 0  for the hardness threshold  of min_TSP12 rening so the negative results of [18 , 10 ℄. Finally another restri tive version of the metri min_TSP, the Eu lidean min_TSP an besolved bya standard PTAS ([1 ℄). A omplete listof standard-approximation results formin_TSP isgivenin [6℄.

Inwhatfollows,weshowthat,inthedierentialapproximationframeworkthe lassi al2_OPT algorithm, originally devised in [7 ℄ and revisited in numerous works (see, for example, [15℄), approximately solves min_TSP with edge-distan es bounded by a polynomial of n within dif-ferential approximation ratio 1/2. In other words, 2_OPT provides for these graphs solutions fairly lose to the optimal and, simultaneously, fairly far from the worst one. We also prove that, in the oppositeof what happensin the standard framework,metri min_TSP and

general min_TSP are equi-approximable in the dierential framework. Moreover we prove

thatmin_TSP12 isapproximable within 3/4.

Formax_TSPthingsaremu hmoreoptimisti for standardapproximation, sin ethis prob-lem isin APX. Bythe end of70sithasbeenproved in [12 ℄ that2_OPT guarantees approxima-tion ratio 1/2 for max_TSP. Morere ently, in [20℄ is proved that max_TSP an be solved by a standard PTAA within ratio 5/7, if the distan e-ve tor is symmetri and within 38/63, ifit

is asymmetri . The dissymmetry in the approximability of min_ and max_TSP an be

on-sideredassomewhat uriousgiven the stru tural symmetryexistingbetween them. Infa tthe transformationd7!M dmentionedaboveandrevisitedindetailinse tion6isane. Sin e dif-ferentialapproximationisstableunderanetransformationoftheobje tivefun tion,min_TSP

and max_TSPareequi-approximable.

In what follows, we will denote by V = fv 1 ;:::;v n g the vertex-set of K n , by E its edge-set and, for v

i v j 2 E, we denote by d(v i ;v j

) (or by d(i;j) when no ambiguity o urs) the distan e of the edge v

i v

j

2 E; we onsider that the distan e-ve tor is symmetri and integer. Given a feasible TSP-solution T(K

n ) of K

n

(both min_ and max_TSP have the same set

of feasible solutions), we denote by d(T(K n

)) its (obje tive) value; T will be indexed by min or max dependingon whether itdeals with min_or max_TSP. Whenne essary, the values of

the worst asesolution, the approximated one and the optimal one for min_TSP (max_TSP)

will be denoted by ! min (K n ),  min A (K n ) and min (K n ) (! max (K n ),  max A (K n ) and max (K n )), respe tively. Given agraph Gindu edbyK

n

, we denote byV(G)itsvertex-set. Finally, given anysetC ofedges, we denotebyd(C)the totaldistan eofC, i.e.,the quantity

P vivj2C

d(i;j).

2 Preserving dierential approximation for several min_TSP versions

Thisse tionisapreliminaryone ontainingseveralresultsabouthowdierentialapproximation ispreserved between several restri tive versions ofmin_TSP.

Proposition 1. Metri min_TSPandgeneral min_TSPare dierentially equi-approximable.

Proof. Obviously,metri min_TSPbeingaspe ial aseofthegeneralone, anbesolvedwithin the same dierential approximation ratio with the latter.

Supposenowthatmetri min_TSPisapproximatelysolvedwithindierentialratioÆ. Given

an instan e I = (K

n ;

~

d) of general min_TSP (set d max

= maxfd(i;j) : v i

v j

2 Eg), one an

transform it into a new one I 0 = (K n ; ~ d 0 ) by hanging, 8v i

vj 2 E, distan e d(i;j) of the former to d

0

(i;j) = d max

+d(i;j). It is easy to see that I 0

is metri , and that every feasible tour T(I) of(K

n ;

~

d ) remains feasible for (K n

; ~ d 0

). The ost of su h a tourbe omesd(T(I 0

))= d(T(I))+nd

max

. Thenthe Æ-PTAA for metri min_TSP will a hieve

Æ= !(I 0 ) d(T(I 0 )) !(I 0 ) (I 0 ) = !(I)+nd max d(T(I)) nd max !(I)+nd max (I) nd max = !(I) d(T(I)) !(I) (I) :

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within the samedierential approximationratio. Let d min = minfd(i;j) : v i v j

2 Eg. Then, if one transforms every distan e d(i;j) into d(i;j) d

min

+1,oneobtains a omplete graphwhere d min

=1 andwith arguments ompletely analogous to the onesof proposition1,the following holds.

Proposition 2. General min_TSPandmin_TSPwithd

min

=1 are differentially equi-appro-ximable.

We next onsider another lass of instan es, the one where the edge-distan es are either a,

or b (notorious member of this lass of min_TSP-problems, denoted by min_TSPab, is the

min_TSP12). Suppose, without loss of generality that a < b. Then, by proposition 2,

min_TSPab is equi-approximable with min_TSP1b. Consider now an instan e of the latter

problem. If one sets b = 2 for all the b-edges (edges of distan e b), then by arguments om-pletely similar to the ones of the proof of proposition 1 (and sin e for a tour T ontaining k

b b-edges,d(T)=n+(b 1)k

b

),the following resultholds.

Proposition 3. min_TSPab and min_TSP12 are dierentially equi-approximable.

Note thatresults analogous to the onesof propositions 1, 2 and 3 do not hold in the standard

approximation framework.

3 2_OPT and dierential approximation for the general minimum traveling

salesman

In what follows, we denote by D-APX the analogous of the lass APX, the lass of NPO

problems solved bya onstant-ratio PTAA, for the dierential approximation framework.

Theorem 1. min_TSP isdierentiallyapproximable withinapproximation ratio 1/2and this ratio istight.

Proof. In what follows, suppose that a tour is listed as the set of its edges and onsider the following algorithm of[7 ℄.

BEGIN/2_OPT/

(1) start from any feasible tour T;

(2) REPEAT

(3) pi k a new set fv

i v j ;v i 0v j 0 gT; (4) IFd(i;j)+d(i 0 ;j 0 )>d(i;i 0 )+d(j;j 0 )THENT (Tnfv i v j ;v i 0v j 0g) [fv i v i 0 ;v j v j 0 g FI

(5) UNTIL no improvement of d(T) is possible;

(6) OUTPUT T;

END. /2_OPT/

Suppose now that, starting from a vertex denoted by v

1

, the rest of the verti es is ordered followingthetourT nally omputedby2_0PT(so,givenavertexv

i

,i=1;:::;n 1,v i+1

isits su essorwith respe ttoT; v

n+1

=v

1

). LetusxoneoptimaltouranddenoteitbyT  . Given a vertex v i , denote by v s  (i) its su essor in T  (remark that v s  (i)+1 is the su essor of v s  (i) in T; in other words, edge v

s  (i) v s  (i)+1

2 T). Finally let us x one (of the eventually many) worst- ase(maximum total-distan e)tourT

! .

The tourT omputed by 2_OPTis alo al optimum for the 2-ex hange of edgesin the sense thateveryinter hange between two non-interse ting edgesof T and two non-interse ting edges of EnT will produ e a tourof total distan eat least equal to d(T). This impliesin parti ular that, 8i2f1;:::;ng, d(i;i+1)+d(s  (i);s  (i)+1)6d( i;s  (i))+d( i+1;s  (i)+1);

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n X i=1 (d(i;i+1)+d( s  (i);s  (i)+1))6 n X i=1 ( d( i;s  (i))+d( i+1;s  (i)+1) ) (1)

Moreover, itis easyto see thatthe following holds: [ i=1;:::;n fv i v i+1 g = [ i=1;:::;n  v s  (i) v s  (i)+1 = T (2) [ i=1;:::;n  v i v s  (i) = T  (3) [ i=1;:::;n  v i+1 v s  (i)+1

= some feasibletourT

0

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LetusshowthatT 0 =[ i=1;:::;n fv i+1 v s  (i)+1

gisfeasible. Re allthatana y li permutation isa bije tive fun tionf :f1;:::;ng!f1;:::;ngsu h that, 8i2f1;:::;ng:

 f (k) (i)6=i k<n f (n) (i)=i

Every feasible tour T, oriented as mentioned above, an be seen as an a y li permutation. Consider now the following mappings

s  : i7!s  (i) f : i7!i+1 h : i7!s  (i 1)+1:

Itiseasyto seethatifs 

isana y li permutationandf isapermutation,then h=fÆs 

Æf 1 isana y li permutation. Moreover,itisnothardtoseethatpairs(i;h(i)) orrespond(modn) to the edgesof T

0 .

Combining expression(1) with expressions (2),(3)and (4), one gets:

(2) =) n P i=1 d(i;i+1)+ n P i=1 d(s  (i);s  (i)+1) = 2 2_ OPT (K n ) (3) =) n P i=1 d(i;s  (i)) = (K n ) (4) =) n P i=1 d(i+1;s  (i)+1) = d(T 0 ) 6 !(K n ) (5)

and expressions(1) and(5) leadto

2 2_ OPT (K n )6 ( K n )+!( K n )() !( K n )  2_ OPT ( K n ) !(K n ) (K n ) > 1 2 : Consequently, Æ 2_ OPT >1=2. Consider nowa K 4n+8 , n>0,set V =fv i :i=1;:::;4n+8g,let d(2k+1;2k+2) =1 k =0;1;:::;2n+3 d(4k+2;4k+4) =1 k =0;1;:::;n+1 d(4k+3;4k+5) =1 k =0;1;:::;n d(4n+7;1) =1

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T = fv i v i+1 :i=1;:::;4n+7g[fv 4n+8 v 1 g T  = fv 2k+1 v 2k+2 :k =0;:::;2n+3g[f v 4k+2 v 4k+4 :k=0;:::;n+1g [fv 4k+3 v 4k+5 : k=0;:::;ng[f v 4n+7 v 1 g T w = fv 2k+2 v 2k+3 :k =0;:::;2n+2g[f v 2k+1 v 2k+4 :k=0;:::;2n+1g [fv 2k+2 v 2k+5 : k=0;:::;2n+1g[f v 4n+8 v 1 g: Ingure1,T  andT w

areshownforn=1(T =f1;:::;11;12;1g). Hen e,Æ 2_ OPT

(K 4n+8

)=1=2 and this ompletes the proof of the theorem.

PSfrag repla ements 1 2 3 4 5 6 7 8 9 10 11 12 T  T w

Figure1. Tightness ofthe 2_OPT approximationratio for n=1.

Fromthe proofof the tightness ofthe ratio of 2_OPT,the following orollaryis immediately dedu ed.

Corollary 1. Æ

2_ OPT

=1=2 is tight even for min_TSP12.

A rst ase of polynomial omplexity for algorithm 2_OPT (even ifedge-distan es of the graph areexponentialinn)isforgraphswherethenumberof(feasible)tour-values,denotedby(K

n ), ispolynomialinn. Here,sin ethereexistsapolynomialnumberofdierentmin_TSP solution-values,a hievementofalo allyminimalsolution(starting,atworstfortheworst-valuesolution) will need apolynomialnumber ofsteps (atmost (K

n

))for 2_OPT. Theorem1obviouslyworksinpolynomialtimewhend

max

isboundedabovebyapolynomial ofn. However,evenwhenthis onditionisnotsatised,thereexistrestri tive asesofmin_TSP for whi h2_OPT remains polynomial.

Consider now omplete graphs with a xed number k 2 IN of distin t edge-distan es,

d 1 ;d 2 ;:::;d k

. Then, any tour-value an be seen as k-tuple (n

1 ;n 2 ;:::n k ) with n 1 +n 2 + ::: +n k = n, where n 1

edges of the tour are of distan e d 1

, ..., n k

edges are of distan e d k ( P k i=1 n i d i

= d(T)). Consequently, the onse utive solutions retained by 2_OPT (in line (4)) beforeattainingalo alminimumare,atmost,asmanyasthenumberofthearrangementswith repetitions of k distin t items between n items (in other words, the number of all the distin t k-tuplesformed byallthe numbers in f1;:::;ng),i.e., bounded above byO(n

k ).

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n

apolynomialofn. Re allthat,fromproposition1,generalandmetri min_TSParedierentially equi-approximable. Consequently, givenan instan eK

n where (K n ) ispolynomial, K n anbe transformed into agraph K

0 n

asin proposition1. Then,ifone runsthe algorithm of[5℄in order to obtain an initial feasible tour T (line (1) of algorithm 2_OPT), then its total distan e, at most 3/2times the optimalone,will be ofpolynomialvalueand, onsequently, 2_OPT willneed apolynomial number ofsteps until attaininga lo alminimum.

Letus notethatthe rstand thefourthofthe above ases annotbede ided inpolynomial time. However, ifone systemati ally transforms general min_TSP into a metri one (proposi-tion 1) and then uses the algorithm of [5℄ in line (1) of 2_OPT, then all instan es meeting the se ond item of orollary2 will be solved in polynomial time even ifwe annotre ognize them.

Corollary 2. The following versions of min_TSP are in D-APX (solved by 2_OPT within

ratio 1/2):

 on graphs where the optimaltour-value is polynomialin n;

 on graphs where the number of feasible tour-values is polynomialin n (examples of these graphs are theones where edge-distan es are polynomiallybounded, oreven theones where there exists a xed number of distin t edge-distan es).

4 Approximating min_TSP12

Let us rst re all that, given a graph G, a 2-mat hing is a a set M of edges of G su h that if V(M) is the set of the endpoints of M, the verti es of the graph (V(M);M) have degree at most 2, in other words, the graph (V(M);M) is a olle tion of y les and simple paths. A 2-mat hing is optimal ifit is the largest over all the 2-mat hings of G. Asit is shown in [14℄, an optimal triangle-free 2-mat hing an be omputed inpolynomialtime.

Our min_TSP12 PTAA is based upon a spe ial kind of triangle-free 2-mat hing in K

n , the y les of whi h will be progressively pat hed in order to produ e a Hamiltonian tour. In what follows, we deal with optimal triangle-free 2-mat hings, i.e., with triangle-free olle tions of y les.

Theorem 2. min_TSP12 is approximable within dierential approximation ratio Æ > 3=4. Thisratio is tight for the algorithm devised.

Proof. Let M = (C

1 ;C

2

;::: ) be any maximal triangle-free 2-mat hing of K n

. In the sequel, we all by value of a 2-mat hing the sum of the distan es of its edges. For any mat hing M, we will denote its value by d(M). Also, let us all y le-pat hing (see also [18℄) the operation onsisting in taking two y les C

i and C j of M, in pi king edges v k v l 2 C i , v p v q 2 C j and in transforming C i , and C j

into a unique y le C = C

i \C j nfv k v l ;v p v q g\fe ij ;e 0 ij g, where fe ij ;e 0 ij g = fv k v p ;v l v q g, or fe ij ;e 0 ij g = fv k v q ;v l v p

g. This spe ies the following pro edure, polynomial in n, omputing, in addition, the total distan e of the y le resulting from y le pat hing. BEGIN /CYCLE_PATCH/ take edges v k v l 2C i and v p v q 2C j ; C 1 ij C i [C j nfv k v l ;v p v q g[fv k v p ;v l v q g; C 2 ij C i [C j nfv k v l ;v p v q g[fv k v q ;v l v p g; OUTPUT C ij argminfd(C 1 ij );d(C 2 ij )g ; END. /CYCLE_PATCH/

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n In the sequel, we will rst spe ify a PTAA min_TSP12 and estimate the value 

TSP12 (K n ) = d(T(K n

)) ofthe Hamiltonian tour omputed. Next, we will ompute a lower bound for !(K n

). Asfor theorem1,wewill exhibit afeasible tourof a ertainvalue. Sin e worst solution'svalue is larger than the valueof every other Hamiltonian tourof K

n

, the value ofthe tourexhibited will be the bound laimed.

Let ^

M be an optimal triangle-free 2-mat hing of K n

(re all that, as we have mentioned, su h amat hingis maximal, i.e,it does only ontains y les). Startingfrom

^

M, one an easily

onstru t an optimal 2-mat hing M



where every pat hing of two y les stri tly in reases its value. Inwhatfollows,wewill allM



2-minimal. Constru tionofM 

anbedoneinpolynomial time bythe following pro edure.

BEGIN /2_MIN/

M p

;; REPEAT

pi k a new set fC

i ;C j g ^ M; FOR all v k v l 2C i ;v p v q 2C j DO M p ^ MnfC i ;C j g[CYCLE_PATCH(C i ;C j ) IF d( ^ M)>d(M p ) THEN ^ M M p FI OD UNTIL no improvement of d( ^ M) is possible; OUTPUT M  ^ M; END. /2_MIN/

Remark 1. In any 2-minimal mat hing M there exists at most one y le C ontaining

2-edges(edges ofdistan e2). Infa t, ifnot, pro edure CYCLE_PATCHING an always be appliedin ordertopat htwodistin t y les ontaining2-edges intoone y lewith totaldistan enolonger than the sum of the distan es of the two y les pat hed. Moreover, if M = C, then M is an optimal solution for min_TSP (ingeneral, a Hamiltonian y lebeinga parti ular triangle-free

2-mat hing, d(M)6

min (K

n )).

Fixa 2-minimal triangle-free mat hingM  =(C 1 ;C 2 ;:::;C p+1 ) (re all thatM  is a minimum

total-distan e triangle-free 2-mat hing) and suppose, without loss of generality, that p > 0 and that y les C

1 ;:::;C

p

ontain only 1-edges (edges of distan e 1) and thatonly y le C p+1 ontains,eventually, some2-edges. Finally,re allthatitisassumedthatjC

i

j>4. Thefollowing fa ts anbe on ludedregarding M

 . Fa t 1. 8(C ;C 0 )2M  M  su h thatC 6=C 0 , 8uv 2C, 8u 0 v 0 2C 0 , max fd(u;u 0 );d(v;v 0 )g= maxfd(u;v 0 );d(v;u 0 )g =2.

Fa t 2. Ifvertexu is adja ent to a2-edge in C p+1 , then,8u 0 = 2V(C p+1 ), d(u;u 0 )=2. Fa t 3. Ifuu 0 and vv 0

are two distin t non-adja ent 2-edges of C p+1

, then d(u;v)=d(u;v 0 )= d(u 0 v)=d(u 0 v 0 )=2. Given M  =(C 1 ;:::;C p+1

), we rstperform the following prepro essingon C 1 ;:::;C p . BEGIN /PREPROCESS/ PR ;; WHILEpossible DO arbitrarily pi k C i ;C j 2M  nfC p+1

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i j M  M  nfC i ;C j g; OD OUTPUT PR; END. /PREPROCESS/

Suppose the WHILE loop of PREPROCESS exe uted q times and denote by fC

s 1 ;C s 2 g the y les onsidered during the sth exe ution of the loop, s =1;:::;q. Then PR = [

q s=1 fC s 1 ;C s 2 g. Set r=p 2q anddenote byD t , t=1;:::;r, the y lesin fC 1 ;:::;C p gn[ q s=1 fC s 1 ;C s 2 g. Underall this, M  = ( q [ s=1 f C s 1 ;C s 2 g ) [ ( r [ t=1 D t ) [ C p+1 :

Thefollowing fa ts holdand omplete the abovedis ussion.

Fa t 4. 2q+r>1;if2q+r=1 then C p+1 6=;. Fa t 5. 8s2f1;:::;qg,8`2f1;2g,8e2C s ` , d(e)=1. Fa t 6. 8t2f1;:::;rg,8e2D t , d(e)=1. Fa t 7. 8s2f1;:::;qg,9i s 2V(C s 1 ),9I s 2V(C s 2

) su h thatd(i s ;I s )=1. Fa t 8. 8(t;t 0 )2f1;:::;rgf1;:::;rg,t6=t 0 , 8(u;v)2V(D t )V(D t 0 ), d(u;v) =2.

In the sequel,for s=1;:::;q, we denote by a s andb s (resp., A s and B s

) the verti es adja ent to i s (resp., I s ) in C s 1 (resp., C s 2 ). We set = P q s=1 (jC s 1 j+jC s 2 j), d = P r t=1 jD t j, E2 = fe 2 C p+1

:d(e)=2g. Followingthesenotations,n= +d+jC p+1

jand,denotingbyjE2j ardinality of the setE2,

d( M 

)=n+jE2j (6)

We are well-prepared now to des ribe the algorithm proposed. Informally, it rst pat hes y- lesC s 1 andC s 2

intoasingle y leC s

, s=1;:::;q. Next,itpat hes y leC 1

withC 2

toprodu e

a y le C whi h will be pat hed with C

3

, and so on, nally produ ing a single y le C. It does sofor the y les D

t

, t = 1;:::;r, produ ing a single y le D. Then it pat hes C and D in order to produ e a partial tour

~

T and nally it pat hes ~

T and C

p+1

obtaining so the nal

TSP-tour T(K

n ).

4.1.1 Constru tion and evaluation of C

Constru tion ofC is performed bymeans of the following pro edure.

BEGIN /C/

FORs 1 to q DOusing edge i

s I s C s CYCLE_PATCH(C s 1 ;C s 2 );OD C C 1 ; FORs 1 TO q 1 DO

repla ing as many 2-edges as possible C CYCLE_PATCH(C;C

s+1 ) ; OD

OUTPUT C;

END. /C/

The all of algorithm CYCLE_PATCH in the rstFOR-loop of C isa very slightly dierent variant of the orresponding pro edure presented above where one imposes to the 1-edge i

s I

s

(fa t 7) to be one ofthe ross-edges entering y leC

s .

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Lemma 1. The 2-mat hing (C ;C ;:::;C ) produ ed during the q exe utions of the rst FOR-loopof algorithm C has value d(C

1 ;C 2 ;:::;C q )= +q.

Proof of lemma 1. Pat hing of C

s 1 and C s 2 into C s

is done using 1-edge i s

I s

(fa t 7), s = 1;:::;q. Consequently, only one 2-edge has been in luded in C

s

(the one used with i s I s to pat hC s 1 andC s 2

). Su hanedgealwaysexistsbe auseoffa t1. So,fors=1;:::;q,exe utionof CYCLE_PATCH(C s 1 ;C s 2

)intherstFOR-loopofCwillprodu einallexa tlyq2-edgesrepla ingandq 1-edges repla ing 2q 1-edges. Consequently, d(C

1 ;C 2 ;:::;C q )= P q s=1 (jC s 1 j+jC s 2 j)+q = +q and this ompletes the proof of lemma1.

During the exe utions of CYCLE_PATCH in the se ond FOR-loop of C, we try that the total distan e of the resulting y le is no longer than the sum of the total distan es of the y les pat hed. In other words, we try to not produ e additional 2-edges in the resulting y le. Here the following lemmaholds.

Lemma 2. The y le C produ ed during the se ond FOR-loopof algorithm C does notin rease d(C 1 ;C 2 ;:::;C q ).

Proof of lemma 2. Theproof isdone by indu tionon q.

4.1.1.1 q=1

Theproof ofthis ase isan immediateappli ation of lemma1 with q=1.

4.1.1.2 q=k

Supposethatduring the k rst exe utionsofthe FOR-loop,the number of2-edges is atmost k.

4.1.1.3 q=k+1

Supposenowthat there exists at leastone 2-edge in C (note also thatC s+1

, sin e it hasbeen not pro essedyet, always ontains the 2-edge produ ed bythe exe ution of the rst FOR-loop). Sin e the pat hing with C

s+1

is done by algorithm C using two 2-edges, there is no additional 2-edge reated. On the other hand, ifno 2-edge existsin C, the pat hing ofC with C

s+1 will produ eatmost 26k+1new2-edges andthis on ludesindu tionandtheproofoflemma2.

Lemmata 1and 2 indu e

d(C)6 +q (7)

4.1.2 Constru tion and evaluation of D

Thefollowing pro edure isusedto onstru t D.

BEGIN /D/

D D

1 ;

FORt 1 TO r 1 DO

repla ing as many 2-edges as possible D CYCLE_PATCH(D;D

t+1 ) ; OD

OUTPUT D;

END. /D/

Exa tly analogous arguments to the ones of the proof of lemma 2 applied to algorithm D and thanksto fa t8indu e

d(D)6d+r (8)

Also,letus notethatanypat hingof y lesD i

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4.1.3 Constru tion and evaluation of T

BEGIN /

~ T/

repla ing as many 2-edges as possible

~ T CYCLE_PATCH(C;D); OUTPUT ~ T; END. / ~ T/

Withthe same argumentsasin lemma2, thefollowing holds for j ~ Tj:  d( ~ T) 6 +d+q+r 2q+r>2 d( ~ T) = d (q;r)=(0;1) (9)

4.1.4 Overall spe i ation of the min_TSP12-algorithm, onstru tion and

evalu-ation of T On e

~

T onstru ted, all of CYCLE_PATCH( ~ T;C

p+1

) , hanging as many as 2-edges (at most 2) as possible, onstru ts the nalTSP-solution T(K

n

) and the whole min_TSP12-PTAA is the

following. The 2-mat hing ^

M produ ed in the rst line of the algorithm below is supposedto beoptimal andwithout y les onlessthan, or equal to,four edges.

BEGIN /TSP12/

all the algorithm of [14℄ to produ e

^ M; ^ M=(C 1 ;:::;C p+1 ) 2_MIN( ^ M); M  PREPROCESS( ^ M)[ r t=1 fD t g[C p+1 ; C C(M  ) ; D D(M  ) ; ~ T CYCLE_PATCH(C;D); OUTPUTT(K n ) CYCLE_PATCH( ~ T;C p+1 ) ; END. /TSP12/

Itis easyto seethat, sin eall thealgorithms alled arepolynomial, TSP12 worksin polynomial time.

If(q;r)=(0;1)(inthis ase,byfa t4,C p+1 6=;),thenpat hingofD 1 withC p+1 onstru ts atour with d(T(K n ))=d(D 1 )+d(C p+1 )+1=d(M  )+1=d(M  )+q+r. Suppose2q+r>2. Then,byexpression(9),d(

~ T)6 +d+q+r. Ifd( ~ T)< +d+q+r,i.e., d( ~ T)6 +d+q+r 1,evenif ~

T doesnot ontainany2-edge,pat hingof ~

T withC p+1

will reate onlyoneadditional2-edgeso,nally, d(T(K

n ))6d( ~ T)+d(C p+1 )6 +d+q+r+jC p+1 j+jE2j and,byexpression(6), d(T(K n ))6d(M  )+q+r. Ifd( ~ T)= +d+q+r,we simplyex hange two 2-edges andthe same expression ford(T(K

n

))always holds.

Thedis ussionaboveleadstothefollowing on ludingexpressionforthequantityd(T(K n )): d(T( K n ))= TSP12 (K n )6v(M  )+q+r (10) 4.2 A bound for !(K n )

In what follows,we will exhibit a TSP12-solution, the obje tive value of whi h will provide us withalower boundforthevalue!(K

n

)oftheworstTSP12-solutiononK n

. Forthiswedenea setW ofdisjointelementarypaths(d.e.p.),anyone ofthem ontainingonly2-edges. Obviously,

if W = fw

1 ;:::;w

jWj

g, one, by properly linking w i

's, an easily onstru t a tour T 0 verifying d(T 0 )>n+ P w i 2W jw i

j whi h isalower bound for !(K n

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Re all that, for q 6= 0, d(i s

;I s

) = 1; hen e, by fa t 1, d(a s ;A s ) = d(a s ;B s ) = d(b s ;A s ) = d(b s ;B s

)=2,s=1;:::;q. Alwaysbyfa t1,either d(i s ;B s )=2,ord(I s ;b s )=2. Withoutloss of generality, we supposeallover the rest of the proofof theorem2that d(i

s ;B

s )=2. Consequently, for s= 1;:::;q, set W

C s =fb s A s ;A s a s ;a s B s ;B s i s

g and the set of d.e.p. on the verti es ofC isW C =[ q s=1 W C s with jW C j=4q (11)

4.2.2 Disjoint elementary paths on V(D)

If r>1,we hoose, for t=1;:::;r, a sequen e fw t ;x t ;y t ;z t g2V(D t

). Then, the setof d.e.p. and its ardinality onV(D) is

W D = f w 1 ;w 2 ;:::;w t ;w t+1 ;:::;w r ;x 1 ;:::;x t ;:::;x r ;y 1 ;:::;y r ;z 1 ;:::;z r g (12) jW D j = 4(r 1)+3=4r 1 (13) Ifr 61,then we setW D =;.

4.2.3 Disjoint elementary paths on V(

^ T) 4.2.3.1 q>0 Suppose rst r6=1. If r =0, then W ^ T = W C

. Suppose now r > 1. Then, by fa t 1, there existsvertex v

1

2V(D 1

) su h that either d(v 1 ;B 1 ) =2,or d(v 1 ;I 1

)=2. Lete be this 2-edge. Without loss of generality, we an suppose v

1

=w

1

(see the paragraph just above). Then the setof d.e.p.on V( ^ T) isW ^ T =W C [W D

[fegwith (see expressions (11)and (13)) W ^ T =4q+4r 1+1=4(q+r) (14)

Suppose r=1. Consider y les C

1 1 , C 1 2 and D 1 and denote by a 01 (resp., A 01 ) the vertex, distin t from i 1 (resp., I 1 ) in C 1 1 (resp., C 1 2 ) adja ent to a 1 (resp., A 1 ). If for any u 2V(D 1 ) andfor anyv2fa

1 ;b 1 ;A 1 ;B 1

g,d(u;v) =2,thenletw;x;y;z befour verti esofV(D 1 ) andset W 1 =fwa 1 ;a 1 x;xb 1 ;b 1 y;yA 1 ;A 1 z;zB 1 ;B 1 i 1 g. Suppose now thatthere exist x 2 v(D

1 ) and v 2 fa 1 ;b 1 ;A 1 ;B 1 g with d(x;v) =1; assume v=a 1 (so,d(xa 1

)=1). Letw;y;zbethreeverti esinV(D 1

)su hthatw;x;y;zaresubsequent in D 1 . Then, byfa t1,d(w;a 0 1 )=d(w;i 1 )=d(y;a 0 1 )=d(y;i 1 )=2. If d(y;I 1 )=d(x;A 1 ) =2, then W 1 =fI 1 y 1 ;ya 0 1 ;a 0 1 w;wi 1 ;a 1 B 1 ;B 1 b 1 ;b 1 A 1 ;A 1 xg. If not, we ansuppose(uptorenamingof y lesC

1 1 ,C 1 2 andD 1

inthedis ussionthatfollows)d(y;I 1

)= 1. Then, byfa t 1one of the edges i

1

x and a

1

y is a2-edge; letus denote it bye. Set f =a 0 1 y if e = i 1 x, or f = i 1 w if e = a 1 y. Then, W 1 = fa 01 w;i 1 y;a 1 A 1 ;A 1 b 1 ;b 1 B 1 ;B 1 zg [fe;fg. Figure2illustratesthis ase. Inall the above ases setW

^ T =(W C nW C 1 )[W 1 isasetof d.e.p (remark thatthe hypothesis d(i

s ;B

S

)=2doesnotintervene in thespe i ation ofthesetW ^ T ) of ardinality W ^ T =4q+4+8=4(q+r) (15)

From expressions(14) and(15) we on lude for the aseq>0: W ^ T =4(q+r) (16)

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PSfragrepla ements C 1 1 a 0 1 a 1 i 1 b 1 1 w x y z C 1 2 A 0 1 A 1 I 1 B 1 2-edge 1-edge 2-edgeinW 1 Figure2. Constru tion of W 1 . 4.2.3.2 q=0 ConsiderrstjE2j=jC p+1

j. Remarkthatr>1(ifnotT(K n

)=C

p+1

isaminimum-distan e Hamiltoniantour); notealsothatC

p+1

aneventually beempty. Fromfa ts 2,3 and8,for any y leD

t

, t=1;:::;r the only1-edges (other than the onesofD t ) in ident to verti es ofV(D t ) are pairs of V(D t )V(D t ) not in luded in D t

. However, in any feasible Hamiltonian tour,

one annot use more than

P t=r t=1

(jD t

j 1) = d r of them and, onsequently, no less than

n (d r)=jC p+1 j+r2-edges. Hen e, (K n )>n+jC p+1 j+r =d(M  )+r=jT(K n )jandthe solution omputed byalgorithm TSP12 is optimal. For ase jE2j<jC

p+1 j, we set W ^ T =W D ifr>1 , andW ^ T =;,ifr =1.

4.2.4 Disjoint elementary paths on V(K

n ) 4.2.4.1 q>0;jE2j<jC p+1 j Consider setW =W ^ T

[E2. Using expression (16),wegetjWj=4(q+r)+jE2j.

4.2.4.2 q>0;jE2j=jC p+1 j Let uv 2 C p+1 and u 0 be a vertexwith j W ^ T (u 0 )j =1, where by W ^ T (u 0

) we denote the set of neighborsof u

0

belonging alsoto V(W ^ T

). Remarkthatsu ha vertexu 0

existsbe ause W ^ T

is a simple setof paths. Fa t 2ensuresd(u;u

0 ) =2. Wethen setW =W ^ T [(C p+1 nfuvg)[fuu 0 g with (see expression(16)) jWj=4(q+r)+jE2j.

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p+1

Let us rst suppose r =1. Then, let H =fe 1 ;e 2 ;e 3 ;e 4

g be an elementary path on four edges in C

p+1

with endpoints u and vand su h thatd(e 1

)=1 and d(e 2

)=2;letM =fw;x;y;zg be a sequen e offour su essive verti es in V(D

1

) and set H2=fe2H :d(e)=2g. Then, using fa ts1 and2,we an onstru t(see gure3),between pathsH and M, a pathP ontaining at least4+jH2j2-edges where j

P

(v)j1. We set W =P [(E2nH2) that onstitutes a d.e.p with jWj=(4+jH2j)+(jE2j jH2j)=jE2j+4.

PSfragrepla ements C p+1 C p+1 C p+1 D 1 D 1 D 1 e 1 e 1 e 1 e 2 e 2 e 2 e 3 e 3 e 3 e 4 e 4 e 4 u u u v v v v 0 w w w x x x y y y z z z 2-edge 2-edge 1-edge in ludedinW

Figure3. Constru tion ofW supposing vz=argmax fd(v 0

;y);d(v;z)g.

Let us now suppose r > 1 and let uv be an 1-edge of C p+1

. Moreover, from the previous paragraph, for the ase we deal with, W

^ T = W D , where W D is given by expression (12). By fa t 1 we have that either x

1

u, or y 1

v is a 2-edge; let us suppose d(x 1

;u)= 2. Then, the set

W =W

^ T

[E2[fx

1

ug forms a d.e.p. omposed of jWj =(4r 1)+jE2j+1 = 4r+jE2j =

4(q+r)+jE2j2-edges.

Consequently, dealing with W, we always have jWj > 4(q+r)+jE2j. One an obtain a tour T

w (K

n

) by properly linking d.e.ps by edges (at worst by 1-edges) in order that they

form a Hamiltonian y le on K

n

. The so obtained T w

(K n

) has obje tive value d(T w

(K n

)) > n+4(q+r)+jE2j; so,using expression(6)

!( K n )>d( T w (K n ))>n+4(q+r)+jE2j=d(M  )+4(q+r): (17)

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We have already seen thatif q =0 and jE2j=jC p+1

j, then Æ(min_TSP12)=1. So, for q >0 or q=0and jE2j<jC

p+1

j expressions(10), (17) andthe fa tthat (K n )>d(M  ), we get Æ TSP12 (K n )= !(K n )  TSP12 (K n ) !(K n ) (K n ) > d( M  )+4(q+r) ( d( M  )+(q+r)) d( M  )+4(q+r) d( M  ) = 3(q+r) 4(q+r) = 3 4 :

4.4 Ratio 3/4 is tight for TSP12

PSfragrepla ements 1 2 3 4 5 6 7 8 9 10 T  T w T  andT w

Figure4. Tightness ofthe TSP12 approximationratio.

Consider two liques and number their verti es by f1;:::;4g and by f5;6;:::;n+8g, re-spe tively. Edges ofboth liqueshave all distan e 1. Cross-edges ij, i=1;3, j =5;:::;n+8, areall ofdistan e 2,whileeveryother ross-edge isof distan e1.

Unraveling of TSP12 will produ e:

T = f1;2;3;4;5;6;:::;n+7;n+8;1g y le-pathingonedges(1;4)and(5;n+8) T

w

= f1;5;2;6;3;7;4;8;9:::;n+7;n+8;1g using2-edges(1;5);(6;3);(3;7)and(n+8;1) T  = f1;2;n+8;n+7;:::;5;4;3;1g using1-edges(4;5);(2;n+8) i.e., (K n+8 ) = n+9, (K n+8 ) = n+8 and !(K n+8 ) = n+12 (in gure 4, T  and T w are shownfor n=2;T =f1;:::;10;1g). Consequently, Æ

TSP12 (K

n+8

)=3=4 andthis ompletes the proofof theorem2.

Let us note that the dierential approximation ratio of the 7/6-algorithm of [18 ℄, when

running on K

n+8

, is also 3/4. The authors of [18℄ bring also to the fore a family of worst- ase instan es for their algorithm: one has k y les of length four arranged around a y le of length 2k. We have performed a limited omparative study between their algorithm and the our one, for k =3;4;5;6 (on 24 graphs). The average dierential and standard approximation ratiosfor the two algorithms arepresented in table 1.

5 Further results for minimum traveling salesman

5.1 Bridges between dierential and standardapproximation

Let us onsider the following approximation-preserving redu tion proposed in [16 ℄, strongly inspiredbytheA-redu tionof[17 ℄betweenpairs(;R ),where isanNPOproblemandRan

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3 0,931100364 0,846702091 4 0,9000002 0,833333 5 0,920289696 0,833333 6 0,9222222 0,833333 Dieren tial ratio 3 0,923350955 0,87013 4 0,9094018 0,857143 5 0,92646313 0,857143 6 0,928178 0,857143 Standard ratio

Table 1: A limited omparison between TSP12 and the algorithm of [18℄ on some worst- ase instan esof the latter.

approximationmeasure. Inwhatfollows,wedenotebyR [℄(I;S)thevalueoftheapproximation measureR relativeto asolutionS of aninstan eI of . WesupposethatR hasvaluesin [0;1℄ (forthe standardapproximation,weinversetheapproximationratiointhe aseofminimization problems).

Denition 3. AG-redu tionofthepair( 1 ;R 1 )to( 2 ;R 2 ),denotedby( 1 ;R 1 )6 G ( 2 ;R 2 ), isa triplet (/;g; ) su hthat:

 /:I 1

!I

2

polynomially transformsinstan es of 1

into instan es of 2

;

 g:S(/(I))!S(I)polynomially transformssolutions for  2

into solutions for  1 ;  :[0;1℄![0;1℄( 1 (0)=f0g)is su hthat, 82[0;1℄;8I 2I 1 ;8S2S(/(I)), R 2 [ 2 ℄(/(I);S) >=)R 1 [ 1 ℄(I;g(S))> ():

Thefollowing easy lemmaholds.

Lemma 3. Consider an NPO problem  = (I;S;v

I

;opt). If 9t > 0 su h that, 8I 2 I, j!(I) (I)j6tminf!(I); (I)g, then (;)6

G (;Æ) with t ()=  t+1 t+1 opt=max 1 t+1 t opt=min

Remarkthatfor min_TSPab we have

!( K n ) (K n )6bn an6(b a)n6 b a a ( K n )

and byappli ation of lemma3 the followingtheorem holds.

Theorem 3. (min_TSPab;) 6 G (min_TSPab;Æ) with ()= a b (b a) (min_TSP12;) 6 G (min_TSP12;Æ) with ()= 1 2  :

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TSP12 TSP12

one of [5℄ for this parti ular ase, but with no operational impa tsin e it isdominated bythe resultof [18℄.

Re allthat min_TSP12 andmin_TSPab areequi-approximable in the dierential approxi-mation framework. Consequently, usingtheorem3 with Æ=3=4,the following orollaryholds.

Corollary 3. min_TSPab isapproximable within

 6 3 4 + 1 4 b a in the standard framework. Thisratio tends to1 with b.

Let us now denote by A

max

and A

min

a maximum and a minimum spanning trees of K

n , re-spe tively, and by (A

max

) and (A

min

) their respe tive osts. Then, the following proposition holds. Proposition 4. If (A max )= (A min ) 6 ,  > 1, then (min_TSP;)6 G (min_TSP;Æ) with ()=1=((1 )+). Proof. Let T w (K n ) and T  (K n

) be a worst-value tour and an optimal tour of K n , respe -tively. Set d w = min vivj2Tw(Kn) fd(i;j)g and d = max vivj2T  (Kn) fd(i;j)g. Sin e T w (K n ) n fargmin v i v j 2T w (K n ) fd(i;j)gg and T  (K n )nfargmax v i v j 2T  (K n )

fd(i;j)gg areobviouslyspanning treesofK n : (A max )>!(K n ) d w , (A min )6 (K n ) d

. Remarkalsothatd w 6!(K n )=nand d > (K n )=n. So, (A max )>!(K n )(1 1=n) and (A min )6 (K n )(1 1=n). Consequently, !(K n ) ( K n ) 6 ( A max ) 1 1 n  ( A min ) 1 1 n  6 ( A max ) ( A min )  : Hen e, !(K n ) (K n )  ( 1) (K n

), and using lemma 3 for t = ( 1) we get () =

((1 )+)

1 .

5.2 An inapproximability result

Werstnotethatone an proveveryeasily(withargumentssimilarto theonesoftheorem6.13 in[13 ℄)thatmin_TSP annotbe solvedby adierential FPTASunlessP=NP.Wenowrestri t ourselvesto min_TSP12 and revisittheorem3. It iseasy to seethat itdoesnot onlyestablish linksbetween theapproximabilitiesofmin_TSP12 instandardanddierentialframeworks,but it also establishes limits on its approximability in the two frameworks. Plainly, sin e approxi-mation ofmin_TSP12withinÆ =1 impliesitsapproximationwithin=2 (1 )=1+, 0 6  6 1, if there exists an 

0

su h that, under a very likely omplexity theory hypothesis,

min_TSP12 isinapproximable within

0

61+ 0

, then itisinapproximable withinÆ 0

>1  0

. Inother words, the hardnessthresholds for standard anddierential frameworksareidenti al.

Theorem 4. If under a omplexity theory hypothesis min_TSP12 is inapproximable within

1+ 0

, then, under the same hypothesis, min_TSP12 is dierentially inapproximable within

1 

0 .

Re allthe negative resultof [4 ℄: 8>0,no PTAA an guaranteestandard approximationratio lessthan,or equal to,3477=3476 unlessP=NP. Using theorem4,8>0,itis NP-hardto

approximate min_TSP12 with dierential ratio better than 3475=3476+. Sin e min_TSP12

isa spe ial aseof general min_TSP, the following orollaryholds on ludingthe se tion. Corollary 4. min_TSP annotbe approximatedwithindierentialratiogreater than,or equal to,3475=3476+, forevery positive ,unless P=NP.

Finally, let us note that the inapproximability result of [11℄ for dense graphs holds also in the dierential approximation framework with the same hardnessthreshold.

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Wehavealsomentionedthatintheoppositeofmin_TSP,max_TSP( ertainlylesspopularthan its ousin), although it is APX-hard ([20, 18℄), an be solved by a PTAA a hieving standard approximation ratio  =5=7 (this ratio is somewhat worst  38/63  when the input-graph is dire ted).

Thepurposeofthisse tionisto showthat, inthe dierentialapproximationframework,the two ousinsareequi-approximable establishing soa kindof natural symmetrybetween the two problems at hand.

Theorem 5. max_TSP isequi-approximable withmin_TSP; onsequently it isin D-APX.

Proof. Observe rst that, given a graph K

n

, there exists a very interesting symmetry be-tween min_and max_TSPwith respe tto worst- aseand best obje tive values:

 min ( K n ) = ! max (K n ) max ( K n ) = ! min (K n ) (18)

Expression(18) onrmswhat we said in the introdu tion of the paper thatthe worst value of aproblem an be ashard to ompute asthe optimalone.

Givena ompletegraph K

n

,letus denoteby  K n

the ompletegraph onnverti eswhenone repla es distan e d(i;j) by



d (i;j) =M d(i;j), i;j = 1;:::;n, for M =max vivj2E fd(i;j)g+ min v i v j 2E

fd(i;j)g. It iseasyto seethat   K n =K n

. Moreover, anyTSP-feasible solutionforK n isTSP-feasible for  K n .

Given a Hamiltonian y le T, we use notation T min

(resp., T max

) in order to indi ate that we dealwith a solutionof min_TSP (resp., max_TSP). We then have

jT min (K n ) j = Mn T max (  K n ) jT max (K n ) j = Mn T min  K n 

and,more parti ularly,

! min ( K n ) = Mn min  K n  = Mn ! max  K n  (19) min ( K n ) = Mn ! min  K n  = Mn max  K n  (20)  min A ( K n ) = Mn  max A  K n  (21)

Bythedis ussionabove,one animmediately on ludethatforeveryPTAAAandforeveryK n , Æ min A (K n ) =Æ max A (  K n

) (where, on e again, indi es min and max are used to denote min_TSP

and max_TSP, respe tively). Consequently, Æ min A = Æ max A , 8A. Sin e Æ min 2_ OPT > 1=2, the same holds forÆ max 2_ OPT

and this ompletes the proof ofthe theorem. Ford(i;j)2fa;bg,max

v i v j 2E fd(i;j)g+min v i v j 2E

fd(i;j)g d(i;j) 2fa;bg8v i

v j

2E;so,the proofoftheorem5establishes alsoequi-approximabilitybetween min_TSPabandmax_TSPab and the following theorem summarizes dierential approximationresults for max_TSP.

Theorem 6.

 max_TSP isapproximable within dierential approximation ratio1/2;

 max_TSP12 and max_TSPab are approximable within dierential approximation

ra-tio3/4;

 for every >0, max_TSP annot be approximated within dierential ratio greater than, or equal to,5379/5380+, unlessP=NP.

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with distan es 1 and 2

Appli ation of lemma3 in the aseof max_TSPabwith t=(b a)=agets

= b a a (Æ)= b a b Æ+ a b and for Æ=3=4 we have

= b a a  3 4  = 3 4 + 1 4 a b (22)

The above ratio is always bounded below by 3=4. Here we see another impa t of the

asym-metrybetween minimization andmaximization versionsof TSPin the standard approximation framework. Re all that, as we have seen in se tion 5.1, the standard approximation ratio for min_TSPab tendsto 1 with band this obviously holdsfor every PTAA.

Set now a=1 and b=2 and revisit expression(22). Then, the following theorem immedi-atelyholds.

Theorem 7. max_TSP12is polynomiallyapproximable within

max

>7=8.

Su h an improved ratio (7=8 > 5=7) for max_TSP12 seems that it annot be immediately

a hieved bythe interesting work of [20 ℄.

Consider nowthe following algorithm for max_TSP.

BEGIN /MTSPALG/ onstru t  K n ;

all the algorithm of [18℄ to ompute a tour T

min (  K n ) ; OUTPUT T max (K n ) T min (  K n ); END.

Re allthat the algorithm alled in the rst line of the algorithm justabove guarantees  min

6 7=6. Then, using expressions (20),(21) and someeasy algebra, onegets 

max MTSPALG

(K n

)>2=3.

8 Towards stronger dierential-inapproximability results for the traveling

salesman

Re allthatanNPOproblemis alledsimple ([19℄)ifitsrestri tion k

toinstan esverifying, for every xed onstant k 2 IN, (I) 6 k an be solved in polynomial time. Analogously, we will all  D-simpleifits restri tion 

k

to instan es verifying, for every xed onstant k 2IN, j!(I) (I)j6k ispolynomial. Then the following propositionholds.

Proposition 5. If  is notD-simple, then there exists k 0 2IN su h that Æ()<k 0 =(k 0 +1). Proof. Suppose not D-simpleand Æ()>k=(k+1), 8k2IN. Then, 8I 2I,

!(I) (I)>!(I) (I)

!(I) (I) k+1

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Consider now an instan e I 0

2I su h that!(I 0

) (I

0

) 6k. Then, sin e obje tive fun tion's values are integer (item 3 of denition 1), expression (23) gives !(I

0 ) (I 0 ) = !(I 0 ) (I 0 ). Consequently, it su es to set k

0

= minfk : 

k

non polynomial g in order to omplete the proof.

In other words, problems whi h are not D-simple do not admit dierential PTAS (this is the dierential-equivalent oftheresultof[19℄forthe standardapproximation). Theproposition above allows a hievement of stronger hardness thresholds provided that k

0

isa xed onstant. We onje ture thatTSPis not D-simpleand this for a smallk

0

. If this wastrue, the hardness thresholdof orollary4 ouldbe meaningfullyimproved.

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bound on the approximability of metri TSP and approximation algorithms for the TSP

with sharpened triangle inequality. InPro . STACS'2000. To appear.

[5℄ N.Christodes. Worst- aseanalysisofa newheuristi for the traveling salesman problem. Te hni al Report388,Grad.S hoolofIndustrial Administration, CMU,1976.

[6℄ P. Cres enzi and V. Kann. A ompendium of NP optimization problems. Available on

www_address: http://www.nada.kth.se/~viggo/problemlist/ ompendium.html,1995.

[7℄ A. Croes. A methodfor solving traveling-salesman problems. Oper. Res.,5:791812, 1958.

[8℄ M.Demange,P.Grisoni,andV.T.Pas hos. Dierentialapproximationalgorithmsforsome ombinatorial optimization problems. Theoret. Comput.S i.,209:107122, 1998.

[9℄ M.DemangeandV.T.Pas hos.Onanapproximationmeasurefoundedonthelinksbetween optimization and polynomial approximation theory. Theoret. Comput. S i., 158:117141, 1996.

[10℄ L. Engebretsen. An expli it lower bound for TSP with distan es one and two. Te hni al ReportTR98-046, revised,Ele troni ColloquiumonComputational Complexity, January

1999. Availableon www_address: http://www.e .uni-trier.de/e /.

[11℄ W. FernandezdelaVegaandM. Karpinski. OnapproximationhardnessofdenseTSPand other path problems. Inform.Pro ess. Lett., 70:5355,1999.

[12℄ M.L.Fisher,G.L.Nemhauser,andL.A.Wolsey. Ananalysisofapproximationsfornding a maximum weight Hamiltonian ir uit. Oper. Res.,27(4):799809, 1979.

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[14℄ D. B.Hartvigsen. Extensions of mat hing theory. PhDthesis, Carnegie-Mellon University, 1984.

[15℄ S. Lin and B. W. Kernighan. An ee tive heuristi algorithm for the traveling salesman problem. Oper. Res.,21:498516, 1973.

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approximations. Theoret. Comput. S i.,15:251277, 1981.

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Figure

Figure 1. Tightness of the 2_OPT approximation ratio for n = 1.
Figure 3. Constru
tion of W supposing vz = argmax fd(v 0
Figure 4. Tightness of the TSP12 approximation ratio.

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