Finding low cost TSP and 2-matching solutions using certain half integer subtour vertices

Transcription

1 Finding low cost TSP and 2-matching solutions using crtain half intgr subtour vrtics Sylvia Boyd and Robrt Carr Novmbr 996 Introduction Givn th complt graph K n = (V, E) on n nods with dg costs c R E, th symmtric travling salsman problm (hncforth TSP) is to find a Hamilton cycl (or tour) in K n of minimum cost. This problm is known to b NP-hard, vn in th cas whr th costs satisfy th triangl inquality, i.. whn c ij + c jk c ik for all i, j, k V (s [5]). Whn th costs satisfy th triangl inquality, w say that this instanc is a mtric TSP. For any dg st F E and x R E, lt x(f) dnot th sum F x. For any nod st W V, lt δ(w) dnot {uv E u W, v W }. An intgr linar programming (ILP) formulation for th TSP is as follows: minimiz cx () subjct to x(δ(v)) = 2 for all v V, (2) x(δ(s)) 2 for all S V, (3) 3 S n 3, 0 x for all E, (4) x intgr. (5) Constraints (2) ar calld th dgr constraints, and constraints (3) ar calld th subtour limination constraints. W dnot by TOUR th optimal solution valu of ().

2 Thr ar svral wll-known rlaxations of th TSP which can b solvd in polynomial tim and whos solutions provid lowr bounds for (). Such lowr bounds ar usful in assssing TSP solutions found huristically, and in numration schms usd for solving th TSP. It would b intrsting to obtain rsults on how clos ths lowr bounds ar to TOUR, and how clos thy ar to ach othr. Som rsults of this natur ar known (s [3, 7, 0, ]), but thr ar still many opn qustions. In this papr w focus on two rlaxations of th TSP. Th first is calld th subtour limination problm, and w dnot th valu of its optimal solution by SUBT. This problm is obtaind from th ILP formulation () for th TSP by rlaxing th intgr rquirmnt, i.. SUBT = min{cx : cx satisfis (2), (3), (4)}. Th associatd polytop S n, calld th subtour polytop, is dfind by S n := {x R E : x satisfis (2), (3), (4)}. If c satisfis th triangl inquality, th bst known bound for th ratio TOUR/SUBT is 3 (s [0, ]). This bound is not known to b tight. In 2 fact an intrsting conjctur which has bn around for som tim and is discussd in [4] is th following: Conjctur If th cost function c satisfis th triangl inquality, thn TOUR SUBT 4 3. Th scond rlaxation of th TSP which w focus on is calld th 2- matching problm, which can b dscribd as follows. Givn K n = (V, E) and dg costs c R E, find a minimum cost st of dgs M E such that ach nod in V is th ndpoint of xactly two dgs in M. Such a st of dgs is calld a 2-matching. Not that a 2-matching forms a st of disjoint cycls in K n which span V. W dnot by 2M th cost of an optimal 2-matching. Th ILP formulation for th 2-matching problm is obtaind from th ILP formulation for th TSP () by rmoving th subtour limination constraints (3), i.. 2M = min{cx : x satisfis (2), (4), (5)}. (6) 2

3 Not that th lowr bounds providd by 2M for th TSP ar in gnral poor (s [6]). Our main intrst in this rlaxation is to look at how big th ratio 2M/SUBT can b. If c satisfis th triangl inquality, th bst known bound for this ratio is 4/3 (s [2]). This bound is not known to b tight. In fact, a conjctur advancd by Gomans in [4] implis th following: Conjctur 2 If th cost function satisfis th triangl inquality, thn 2M SUBT 0 9. In this papr w ar intrstd in classs of subtour vrtics for which w can construct tours or 2-matchings whos cost is within som factor tims th cost of th minimum cost mmbr for ach of ths classs of subtour vrtics. W call a vrtx x of S n /2-intgr if x {0,, } for all E. 2 For such a vrtx, w call th dgs E for which x = /2 th /2-dgs, and th dgs E for which x = th -dgs. Not that th -dgs in x form a st of disjoint paths in K n which w call th -paths of x. Th two classs of subtour vrtics in which w ar intrstd ar both subclasss of th /2-intgr subtour vrtics. Th first class, which w call triangl vrtics, ar /2-intgr vrtics in which th /2-dgs form a union of triangls in K n. Th scond class, which w call k-prisms, ar /2-intgr vrtics for which th /2-dgs form two cycls of k dgs ach for k 3 and odd, and th -paths hav on ndpoint in ach of ths cycls. W call such a vrtx a planar k-prism if its support graph is planar. In Figur, w giv an xampl of a triangl vrtx and in Figur 2, w giv an xampl of a planar 5-prism. In Sction 2, w show that on can construct a Hamilton cycl whos cost is within 4/3 tims th cost of th minimum cost triangl vrtx. In Sction 3, w show that on can construct a Hamilton cycl whos cost is within k+ tims th cost of th minimum cost planar k-prism. Ths rsults both k lnd support to Conjctur. In Sction 4, w show that on can construct a 2-matching whos cost is within 3k+ tims th cost of th minimum cost /2-3k intgr subtour vrtx that satisfis crtain connctivity conditions involving th paramtr k 3. Evn whn k = 3, this supports Conjctur 2. In all of ths analyss, th structur of th subtour vrtx in qustion is havily xploitd. Morovr, w show how to construct ths Hamilton cycls and 2-matchings in th tim ndd to solv a minimum cost prfct matching 3

4 x = x = /2 Figur : A triangl vrtx. x = x = /2 Figur 2: A planar 5-prism. 4

5 problm on a graph with fwr than n nods. Hnc, ths proofs also yild approximation algorithms whn th optimal subtour solution is a triangl vrtx or k-prism vrtx or whn on can othrwis find a triangl vrtx or k-prism vrtx which on can prov costs no mor than th optimal Hamilton cycl or 2-matching in qustion. Th rmaindr of this sction is dvotd to background and dfinitions which w rquir latr in th papr. For gnral graph thory background, w rfr th radr to []. For gnral polyhdral thory background, w rfr th radr to [9]. Givn a graph G, w lt E(G) dnot th dg st for G, and V (G) th nod st. Whn w writ G = (V, E), thn V is th nod st of G and E is th dg st of G. For a wightd graph whr vry dg E(G) has a wight x, w us th notation G = (V, E, x). Considr th vctor x R E indxd on th dg st E of a graph G. Dnot by E x th st of dgs E for which x 0. Dfin th wightd support graph of x as G x = (V, E x, x), and th support graph of x as G = (V, E x ). W call M E(G) a matching in G if no two dgs in M ar adjacnt. If in addition, vry nod v V (G) is an ndpoint of som dg in M, thn M is calld a prfct matching in G. W call G a multigraph if thr xist pairs of vrtics in G which hav multipl dgs btwn thm. Givn a graph G (usually a multigraph), w say that a st of dgs M E(G) is a graphical 2-matching for G if vry nod in G is th ndpoint of an vn, non-zro numbr of dgs in M, and vry componnt of G[M] := (V, M) has at last 3 vrtics. This is a slightly diffrnt dfinition of graphical 2-matching than what was givn in [2] du to th prsnc of this last condition. Finally, an incidnc vctor of a graphical 2-matching is a A 4 approximation algorithm for TSP using 3 triangl vrtics Lt x b a triangl vrtx of S n of cost cx = SUBT. In this sction w show how to us th structur of x to construct an Eulrian multigraph H of K n whos cost is lss than or qual to 4 SUBT. If th cost function c satisfis th 3 triangl inquality, thn w can short-cut any Eulrian tour of H to obtain a Hamilton cycl whos cost is still lss than or qual to 4 SUBT. Thus 3 5

6 this construction givs a 4/3 approximation algorithm for any mtric TSP problm for which th optimal solution to th subtour problm is a triangl vrtx. In this and othr sctions of this papr, w will mak us of th following usful thorm of Naddf and Pullyblank, which is found in [8]. Thorm 3 Lt G = (V, E, c) b a k-rgular k -dg connctd wightd multigraph with cost function c. Thn thr is a prfct matching M such that c(m) k c(e). W now prov our main rsult for this sction. Thorm 4 Lt x b a triangl vrtx. If th cost vctor c satisfis th triangl inquality, thn on can construct a Hamilton cycl whos cost is within 4/3 tims SUBT. Proof: Lt x b any triangl vrtx of S n. By th following rasoning, w can assum without loss of gnrality that th /2-triangls in G x ar disjoint. Suppos w can construct a Hamilton cycl of cost lss than or qual to 4SUBT whnvr th /2-triangls in G 3 x ar disjoint. Lt x S n b a triangl vrtx with p nods at which two /2-triangls intrsct. On by on, tak ach nod q at which /2-triangls T and T 2 intrsct, split q into two nods q and q 2, with q i blonging only to /2-triangl T i for i =, 2, and add a -dg q q 2 of cost c q q 2 = 0. This procss rsults in a triangl vrtx x S n+p in which th /2-triangls in G x ar disjoint. Thn w construct our Hamilton cycl H on K n+p, and us H to crat our dsird Hamilton cycl H on our original n nods by simply bypassing q 2 for ach nod q that was split in two, at no xtra cost whn on assums that th costs satisfy th triangl inquality. W bgin by constructing a 3-rgular wightd multigraph H = (V, E, y) from G x as follows. Rprsnt ach /2-triangl in G x by a nod in V. Rprsnt ach -path P in G x by an dg P E whos ndpoints ar th two nods rprsnting th triangls which path P joins in G x. In Figur 3, w illustrat th graph H that would b obtaind from th triangl vrtx x shown in Figur. For ach -path P in G x, th cost y P of th corrsponding dg P in H is dfind using a pattrn vctor x P R E. Lt T and T 2 b th two triangls 6

7 Figur 3: An xampl of graph H. which P joins in G x. Thn x P is dfind by x P = if E(P), /2 if E(T ) E(T 2 ) and is adjacnt to P, /2 if E(T ) E(T 2 ) and is not adjacnt to P, 0 othrwis. An xampl of x P is shown in Figur 4. In this figur, th dgs for which x P = 0 ar not shown. Givn x P, w thn dfin y P as y P := cx P. Not that for ach dg in a /2-triangl T, w hav x P is /2 for two of th -paths incidnt with T and -/2 for th othr. Hnc it follows from th dfinition of x P that (x P : P is a -path of x) = x, (7) and thus th cost y(e ) of H satisfis y(e ) = cx = SUBT. (8) 7

8 -/2 T /2 /2 P /2 /2 T 2 -/2 Figur 4: Th pattrn vctor x P. W now wish to find a minimum cost prfct matching M in H. Clarly, H is 3-rgular and 2-dg connctd. So, using Thorm 3 and (8), w dduc that thr xists a prfct matching M in H such that Our final stp is to considr th vctor y(m) 3 y(e ) = SUBT. (9) 3 x := x + (x P : P M P ), (0) whr M P rprsnts th st of -paths P of x such that P M. W hav that 2 if P for som P M P, if P for som P M x = P, () or if is a /2-dg of x adjacnt to a -path P M P, 0 othrwis. Lt H := (V, E) b th multigraph obtaind by taking x copis of ach dg E. In Figur 5 w show th multigraph H which is obtaind using th triangl vrtx x and th graph H shown in Figurs and 3, and th matching M := {, 2 }. Not that th solid dgs in Figur 5 dnot thos in E := E(H), whil th dashd dgs rprsnt thos that ar prsnt in G x but absnt in H. 8

9 Figur 5: An xampl of th graph H. By (), it is asy to s that all nods in H hav dgr 2 or 4, and H is connctd. Thus H forms an Eulrian multigraph of G x. Morovr, it follows from (0) and (9), that th cost c(e) of H satisfis c(e) = cx 4 3 SUBT. Sinc th cost function c is assumd to satisfy th triangl inquality, w can short-cut any Eulrian tour of H to obtain a Hamilton cycl whos cost is within 4 SUBT, as dsird. 3 3 A k+ k approximation algorithm for TSP using planar k-prisms Using tchniqus similar to thos usd in Sction 2, w can show th following. Thorm 5 Lt x b any planar k-prism of S n of cost cx = SUBT. If th cost vctor c satisfis th triangl inquality, thn on can construct a Hamilton cycl whos cost is at most k+ k SUBT. Proof: Lt x S n b any planar k-prism, and labl th two /2-cycls in G x as C and C 2. By th sam argumnt as was usd in Thorm 4, w may assum without loss of gnrality that th /2-cycls of this planar prism ar 9

10 P 2P Figur 6: An xampl of graph H. vrtx disjoint. W construct a wightd multigraph H = (V, E, y) from G x as follows. Rprsnt ach /2-cycl in G x by a nod in V, and rprsnt ach -path P in G x by two dgs P E and 2P E whos ndpoints ar th two nods in V. In Figur 6 w show th graph H that would b obtaind from th planar 5-prism x shown in Figur 2. In ordr to dfin th costs y for H, w first dfin for ach -path P of x two pattrn vctors x P, x 2P R E. For ach /2-cycl, numbr th dgs from to k, starting with an dg adjacnt to P. Lt A P b th st of dgs with an odd labl in C and an vn labl in C 2, and lt B P b th st of dgs with an vn labl in C and an odd labl in C 2. Thn x P and x 2P ar dfind as follows: x P = x 2P = if E(P), /2 if A P, /2 if B P, 0 othrwis, if E(P), /2 if A P, /2 if B P, 0 othrwis. 0

11 /2 -/2 -/2 C /2 /2 P -/2 -/2 C 2 /2 /2 -/2 Figur 7: Th pattrn vctor x P. In Figur 7, w show x P for th planar 5-prism shown in Figur 2. Now w ar rady to dfin th cost function y for H. Lt y P = cx P and lt y 2P = cx 2P. Clarly w hav that y P + y 2P = 2c(E(P)), and thus th cost y(e ) of H satisfis y(e ) 2SUBT. (2) Sinc H is clarly 2k-rgular and 2k-dg connctd, it follows from Thorm 3 that thr xists an dg m in H such that and thus by (2) w hav y m 2k y(e ), (3) y m SUBT. (4) k Without loss of gnrality, lt m = P for som -path P of x. Our final stp is to considr th vctor x = x + x P. (5)

12 x = x = 0 Figur 8: An xampl of th graph H. W hav that x = 2 if P, if is in a -path othr than P, or if A P, 0 othrwis. (6) Lt H = (V, E) b th multigraph obtaind by taking x copis of ach dg E. In Figur 8 w show th multigraph H which is obtaind using th 5-prism and multigraph H shown in Figurs 2 and 6, and th matching dg P, whr P is th middl path in th 5-prism. Not that solid dgs in Figur 8 dnot thos in E := E(H), whil th dashd dgs rprsnt thos that ar prsnt in G x but absnt in H. By (6) it follows that all nods in H hav dgr 2 or 4, and that H is connctd sinc x is a planar k-prism (not that in th cas of a non-planar k-prism, H may not b connctd). Hnc H is an Eulrian subgraph of G x. Sinc it follows from (5) that c(e) = cx = cx+y m, it follows from (4) that c(e) k + k SUBT. Sinc th cost function c is assumd to satisfy th triangl inquality, w can short-cut any Eulrian tour of H to obtain a Hamilton cycl whos cost 2

13 is within k+ SUBT, as dsird. k 4 Approximation algorithms for 2-matching using crtain /2-intgr vrtics Lt x b a /2-intgr vrtx of S n of cost cx = SUBT such that all th /2-cycls in G x ar k-cycls for som odd k 3, and such that no -path in G x has its nds blonging to th sam /2-cycl. W call such a vrtx a k-cycl vrtx of S n. In this sction w show that if a k-cycl vrtx x of S n satisfis a crtain connctivity proprty, thn w can us th structur of x to construct a graphical 2-matching multigraph H of K n whos cost is lss than or qual to 3k+ SUBT. Rcall that ach componnt of a graphical 2-matching 3k multigraph is an Eulrian tour of at last 3 vrtics. If th cost function c satisfis th triangl inquality, w can thn short-cut th Eulrian tour of ach componnt of H to obtain a 2-matching of K n whos cost is still lss than or qual to 3k+ 3k+ SUBT. Thus, this construction givs a approximation 3k 3k algorithm to th 2-matching problm for any mtric TSP problm for which th optimal solution to th subtour problm is such a subtour vrtx. As in Sction 2, w bgin by constructing a multigraph H = (V, E, y) from G x as follows. Rprsnt ach /2-cycl in G x by a nod in V. For ach -path P in G x, lt its ndpoints b u P and v P, and lt C up and C vp b th /2-cycls containing u P and v P rspctivly. Rprsnt ach -path in G x by an dg P E whos ndpoints ar th two nods in V rprsnting C up and C vp. Finally, ach tim two /2-cycls intrsct at a vrtx w V, put in an dg P E whos ndpoints ar th two nods in V rprsnting th two intrscting /2-cycls. Th symbol P that w usd hr is said to dnot a dummy -path, which is a usful but actually fictitious objct that can b thought of as bing a -path consisting of no dgs. Mor will b said about this latr. Also, th wights y on th dgs of E will b dfind latr. W hav th following thorm: Thorm 6 If x is a k-cycl vrtx of S n for som odd k 3 such that H is k -dg connctd, and th cost function c satisfis th triangl inquality, thn on can construct a 2-matching of K n of cost lss than or qual to 3k+ 3k SUBT. 3

14 Proof: Th construction mthod w us for this thorm is vry similar to th mthod usd in Sction 2, xcpt that w will attmpt to dfin a cost function y for th st of dgs E of H which will tak advantag of th fact that our objctiv this tim is to obtain a 2-matching rathr than a Hamilton cycl. W bgin by gnralizing th dfinition of x P givn in Sction 2 as follows. Starting at ach ndpoint u P and v P of a -path P, arbitrarily choos a dirction about C up and C vp in which to travrs th dgs in ach of ths cycls, labling th dgs in ach of ths cycls from to k as w travrs thm. Thn dfin x P R E as if E(P), x P /2 if E(C = up ) or E(C vp ) and has an odd labl, /2 if E(C up ) or E(C vp ) and has an vn labl, 0 othrwis. An xampl of x P is givn in Figur 9. W also wish to hav x P dfind for a dummy -path P, in which cas E(P) :=. Not that, as in Sction 2, this mor gnral x P also satisfis (x P : P is a -path in G x ) = x. (7) Lt P b any -path in G x. To obtain y P, w will dfin thr possibl pattrn vctors x P,, x P,2, and x P,3, whr th scond suprscript is calld th pattrn of th vctor. W thn will lt y P = min{cx P,i : i =, 2, 3} and ind(p) b th pattrn for which this minimum is attaind. Ths pattrn vctors will ach b similar to x P, but with th signs on som of th componnts flippd in ach of ths pattrn vctors so that th following quation is satisfid: x P, + x P,2 + x P,3 = x P. (8) This nsurs that w ar abl to find a matching M in H of small nough cost just lik what was don in our prvious proofs. W wish to nd up with th incidnc vctor of a graphical 2-matching whn w add P M x P,ind(P) to x. To nsur this, our thr pattrn vctors will b dfind so that thy satisfy th following. Lt D P = E(P) E(C up ) E(C vp ), and lt x = x + x P,i for i =, 2, or 3. Thn w nd i)x {0,, 2} for D P. 4

15 /2 -/2 -/2 /2 /2 P /2 /2 x P = x P = /2 x P = -/2 -/2 -/2 /2 Figur 9: Th pattrn vctor x P. 5

16 ii) x (δ(w)) {2, 4} for w V (P). iii)x (δ(w)) = for w V (C up ) \ {u P } or w V (C vp ) \ {v P }. iv)if x = 2, thn x f 0 for som dg f adjacnt to. Lt l P b th lngth of path P. Th dfinitions for x P,i, i =, 2, 3 will dpnd on th valu of l P mod 3. For instanc, lt l P = (mod 3). Starting at u P, labl th dgs in P from to l P. Thn w dfin x P, R E as follows: x P, = if E(P) has a labl which is not 0 (mod 3). if E(P) has a labl which is 0 (mod 3). /2 if E(C up ) has a labl which is vn. /2 if E(C up ) has a labl which is odd. /2 if E(C vp ) has a labl which is odd. /2 if E(C vp ) has a labl which is vn. 0 othrwis. With this dfinition of x P,, x := x+x P, clarly satisfis conditions i) and iii), and also condition ii) xcpt possibly for th cas whn w = u P or w = v P. In ordr to nsur that x actually dos satisfy condition ii) for this cas as wll, and that x satisfis condition iv), on nds th /2-dgs incidnt to v P to b prsnt in x. Howvr, on dos not nd th /2-dgs incidnt to u P to b prsnt in x. This distinction is capturd by th following concpt. Call an ndpoint w of a -path P anchord in P for pattrn i if w is an ndpoint of a path of two -dgs which hav valu 2 in x. Thn in this cas u P is anchord, so on dos not nd its incidnt /2-dgs to b prsnt in x. Howvr, v P is not anchord, so on dos nd its incidnt /2-dgs. In light of this, w dfin x P,i in gnral as follows: x P,i = if E(P) has a labl which is not i (mod 3). if E(P) has a labl which is i (mod 3). /2 if E(C w ) has a labl which is odd and w is not anchord. /2 if E(C w ) has a labl which is vn and w is not anchord. /2 if E(C w ) has a labl which is odd and w is anchord. /2 if E(C w ) has a labl which is vn and w is anchord. 0 othrwis. 6

17 Howvr, th cass l P = and l P = 0 hav to b tratd diffrntly, with u P dfind to b anchord for Pattrn and v P dfind to b anchord for Pattrn 3. It can now b vrifid that x := x + x P,i satisfis conditions i)-iv). It can also b vrifid that (8) holds. Exampls of x P,, x P,2, and x P,3 whn l P = (mod 3) ar shown in Figur 0. In Figur, w show x + x P,i, i =, 2, 3 for th dgs in D P. Finally, in Figurs 2 and 3, w show x P,i for l P = 2 (mod 3) and l P = 0 (mod 3). W ar now rady to construct our graphical 2-matching whos incidnc vctor w call m. First, dtrmin th minimum cost matching M in th multigraph H. Thn dfin m by m := x + x P,ind(P) P M W wish to show that m is th incidnc vctor of a graphical 2-matching. But, with P, Q M and P Q, w hav that D P D Q =. Hnc, m = x + x P,ind(P) for any dg D P. Thrfor, our prvious analysis stablishs that m taks valus of 0,, or 2 on all th dgs of D P for all P M, which includs all th /2-dgs in x. Clarly, m taks a valu of on th rmaining -dgs. Hnc, m is a non-ngativ intgral vctor. Now, lt v V. Not that x P,ind(P) (δ(v)) is always 0 or 2. Sinc x(δ(v)) = 2, it follows that m(δ(v)) is always positiv and vn ( in fact it is always ithr 2 or 4). Finally, thr is no isolatd dg whr m = 2 sinc on of th ndpoints of would hav had to hav bn an unanchord ndpoint u P of a path P, but thn th /2-dgs incidnt to u P would hav non-zro valus for m. Thrfor, m is a graphical 2-matching. Th only rmaining thing to show is that th cost cm is lss than or qual to k SUBT. k+ As a consqunc of th dfinition of y togthr with (8) and (7), w hav that: y(e ) = y P 3 cx P,i = cx P = P E P 3 E i= 3 P 3 cx. E Sinc H is k-rgular and k -dg connctd, by Thorm 3, w hav that y(m) k y(e ) cx. (9) 3k 7

18 Pattrn Pattrn 2 Pattrn 3 -/2 /2 /2 /2 /2 -/2 -/2 -/2 -/2 -/2 -/2 /2 /2 /2 / /2 /2 /2 /2 -/2 -/2 -/2 -/2 -/2 -/2 /2 /2 /2 /2 Figur 0: Pattrn vctors for l P = (mod 3). -/2 8

19 Pattrn Pattrn 2 Pattrn 3 = dg is absnt Figur : Th x + x P,i vctors. 9

20 Pattrn -/2 Pattrn 2 /2 Pattrn 3 /2 /2 /2 -/2 -/2 -/2 -/2 -/2 -/2 /2 /2 /2 / /2 -/2 /2 /2 /2 /2 /2 /2 -/2 -/2 -/2 -/2 -/2 /2 Figur 2: Pattrn vctors for l P = 2 (mod 3). /2 20

21 Pattrn -/2 Pattrn 2 /2 Pattrn 3 /2 /2 /2 -/2 -/2 -/2 -/2 -/2 -/2 /2 /2 /2 / /2 /2 -/2 -/2 /2 /2 -/2 -/2 /2 /2 -/2 -/2 /2 -/2 Figur 3: Pattrn vctors for l P = 0 (mod 3). /2 2

22 But w hav y(m) = y P = cx P,ind(P). (20) P M P M Thrfor, by (9) and (20), th cost cm is lss than or qual to 3k+cx. 3k Hnc, this thorm follows. W hav th following corollaris to Thorm 6. Corollary 7 Lt x b a triangl vrtx. On can construct a 2-matching whos cost is lss than or qual to 0 9 cx. Corollary 8 Lt x b a k-prism. On can construct a 2-matching whos cost is lss than or qual to 3k+ 3k cx. Rfrncs [] J.A. Bondy and U.S.R. Murty, Graph Thory with Applications (Macmillan, London, 976). [2] S. Boyd and R. Carr [3] N. Christofids, Worst cas analysis of a nw huristic for th travling salsman problm, Rport 388, Graduat School of Industrial Administration, Carngi Mllon Univrsity, Pittsburgh (976). [4] M.X. Gomans, Worst-cas comparison of valid inqualitis for th TSP, Mathmatical Programming 69 (995) [5] D.S. Johnson and C.H. Papadimitriou, Computational Complxity, in: E.L. Lawlr, J.K. Lnstra, A.H.G. Rinnooy Kan, and D.B. Shmoys, ds., Th Travling Salsman Problm (John Wily & Sons, Chichstr, 985, 37-85). [6] M. Jüngr, G. Rinlt, and G. Rinaldi, Th Travling Salsman Problm, in: M.O. Ball t al., ds., Handbooks in OR & MS, Vol. 7 (Elsvir Scinc B.V., 995). 22

23 [7] C.L. Monma, B. Munson, and W.R. Pullyblank, Minimum-wight twoconnctd spanning ntworks, Mathmatical Programming 46 (990) [8] D. Naddf and W.R. Pullyblank, Matchings in Rgular Graphs, Discrt Mathmatics 34 (98) [9] W.R. Pullyblank, Polyhdral Combinatorics. in: A. Bachm t al., ds., Mathmatical Programming Th Stat of th Art (Springr-Vrlag, 983, ). [0] D.B. Shmoys and D.P. Williamson, Analyzing th Hld-Karp TSP bound: A monotonicity proprty with application, Inf. Procss. Ltt. 35 (990) [] L.A. Wolsy, Huristic analysis, linar programming and branch and bound, Math. Program. Study 3 (980)