A POLYNOMIAL-SPACE EXACT ALGORITHM FOR TSP IN DEGREE-5 GRAPHS

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1 A POLYNOMIAL-SPACE EXACT ALGORITHM FOR TSP IN DEGREE-5 GRAPHS Norhazwani Md Yunos 1,2, Alksandar Shurbski 1, Hiroshi Nagamochi 1 1 Dpartmnt of Applid Mathmatics and Physics, Graduat School of Informatics, Kyoto Unirsity, , Kyoto, Japan 2 Unirsiti Tknikal Malaysia Mlaka, Durian Tunggal, Mlaka, Malaysia {wani, shurbski, nag}@amp.i.kyoto-u.ac.jp Kywords: Traling Salsman Problm, Exact Exponntial Algorithm, Branch-and-rduc, Masurand-conqur. Abstract Th Traling Salsman Problm (TSP) is on of th most wll-known NP-hard optimization problms. Following a rcnt trnd of rsarch which focuss on dloping algorithms for spcial typs of TSP instancs, namly graphs of limitd dgr, and thus alliating a part of th tim and spac complxity, w prsnt a polynomial-spac branching algorithm for th TSP in graphs with dgr at mos, and show that it has a running tim of O ( n ). To th bst of our knowldg, this is th first xact algorithm spcializd to graphs of such high dgr. Whil th bas of th xponnt in th running tim bound is gratr than two, our algorithm uss spac mrly polynomial in an input instanc siz, and thus by far outprforms Gurich and Shlah s O (4 n n log n ) polynomial-spac xact algorithm for th gnral TSP (Siam Journal of Computation, Vol. 16, No. 3, pp , 1987). In th analysis of th running tim, w us th masur-andconqur mthod, and w dlop a st of branching ruls which fostr th analysis of th running tim. 1 Introduction Th Traling Salsman Problm (TSP) is on of th most xtnsily studid problms in optimization. It has bn formulatd as a mathmatical problm in th 1930s. Many algorithmic mthods ha bn instigatd to bat th challng of finding th fastst algorithm in trms of running tim. On th othr hand, it has pron n mor challnging to dis fast algorithms that would us a managabl amount of computation spac, boundd by a polynomial in an input instanc s siz. W will riw prious algorithmic attmpts, making a distinction btwn thos which rquir spac xponntial in th siz of a problm instanc, and thos rquiring spac mrly polynomial in th input siz. W us th O notation, which supprsss polynomial factors. Th first non-triial algorithm for th TSP in an n- rtx graph is th O (2 n )-tim dynamic programming algorithm discord indpndntly by Bllman [1], and Hld and Karp [9] in th arly 1960s. This dynamic programming algorithm howr, rquirs also an xponntial amount of spac. Er sinc, this running tim has only bn improd for spcial typs of graphs. Primarily, instigation fforts ha bn focusd on graphs in which rtics ha a limitd dgr. Hncforth, lt dgr-i graph stand for a graph in which rtics ha maximum dgr at most i. A rcnt impromnt of th tim bound to O ( n ) for dgr-3 graphs has bn prsntd by Bodlandr t al. [2]. Thy ha usd a gnral approach for spding up straightforward dynamic programming algorithms. For TSP in dgr-4 graphs, Gbaur [7] has shown a tim bound of O (1.733 n ), by using a dynamic programming approach. In th in of polynomial spac algorithms, Gurich and Shlah [8] ha shown that th TSP in a gnral n-rtx graph is solabl in tim O ( 4 n n log n). Eppstin [4] has startd th xploration into polynomial spac TSP algorithms spcializd for graphs of boundd dgr by dsigning an algorithm for dgr-3 graphs that runs in O (1.260 n )-tim. H introducd a branch-and-sarch mthod by considring a gnralization of th TSP calld th forcd TSP. Iwama and Nakashima [10] ha claimd an impromnt of Eppstin s tim bound to O (1.251 n )-tim for TSP in dgr-3 graphs. Latr, Liskiwicz and Schustr [11] ha uncord som orsights mad in Iwama and Nakashima s analysis, and prod that thir algorithm actually runs in O (1.257 n )-tim. Liskiwicz and Schustr thn mad som minor modifications of Eppstin s algorithm and showd that this modifid algorithm runs in O ( n )-tim, a slight impromnt or Iwama and Nakashima s algorithm. Xiao and Nagamochi [14] ha rcntly prsntd an O ( n )-tim algorithm for TSP in dgr-3 graphs, and this improd prious tim bounds for polynomial-spac algorithms. Thy usd th basic stps of Eppstin s branch-and-sarch algorithm, and introducd a branching rul basd on a cut-circuit structur. In th procss of improing th tim bound, thy usd simpl analysis of masur and conqur, and ffctily analyzd thir algorithm by introducing an amortization schm or th cut circuit structur, stting wights to both rtics and connctd compo ISORA IET 45 Luoyang, China, Augus1 24, 2015

2 nnts of inducd graphs. For TSP in dgr-4 graphs, Eppstin [4] dsignd an algorithm that runs in O (1.890 n )-tim, basd on a branch-and-sarch mthod. Latr, Xiao and Nagamochi [13] showd an improd alu for th uppr bound of th running tim and showd that thir algorithm runs in O (1.692 n )-tim. Currntly, this is th fastst algorithm for TSP in dgr-4 graphs. Basically, th ida bhind thir algorithm is to apply rduction ruls until no furthr rduction is possibl, and thn branch on an dg by ithr including it to a solution or xcluding it from a solution. This is similar to most prious branch-and-sarch algorithms for th TSP. To ffctily analyz thir algorithm, Xiao and Nagamochi usd th masur and conqur mthod by stting a wight to ach rtx in a graph. From ach branching opration, thy drid a branching ctor using th assignd wight and aluat how much wight can b dcrasd in ach of th two instancs obtaind by branching on a slctd dg. In this way, thy wr abl to analyz by how much th total wight would dcras in ach branch. Moror, thy indicatd that th masur will dcras mor if w slct a good dg to branch on, and ga a st of simpl ruls, basd on a graph s topological proprtis, for choosing such an dg. Howr, th analysis of th running tim itslf is not as straightforward [13]. To th bst of our knowldg, thr xist no rports in th litratur of xact algorithms spcializd to th TSP in dgr-5 graphs. Thrfor, this papr prsnts th first algorithm for th TSP in dgr-5 graphs, and prsnts an uppr bound on th running tim of O ( n ). In this xploration, w us a dtrministic branch-and-sarch algorithm for TSP in dgr-5 graphs. Basically, our algorithm mploys similar tchniqus to most prious branching algorithms for th TSP. Whn thr ar no rtics of dgr 5 in an input graph, w call an xisting algorithm for TSP in dgr-4 graphs, and sol th rmaining instanc. In th analysis, w us th masur and conqur mthod as a tool to gt an uppr bound of th running tim. Th rmaindr of this papr is organizd as follows; Sction 2 oriws th basic notation usd in this papr and prsnts an introduction to th branching algorithm and masur and conqur mthod. Sction 3 dscribs our polynomial-spac branching algorithm. W stat our main rsult in sction 4, whr w procd with th analysis of th proposd algorithm. Finally, Sction 5 concluds th papr. 2 Mthods 2.1 Prliminaris For a graph G, lt V (G) dnot th st of rtics in G, and lt E(G) dnot th st of dgs in G. A pair of rtics and u ar calld nighbors if and u ar adjacnt by an dg u. W dnot th st of all nighbors of a rtx by N(), and dnot by d() th cardinality N() of N(), also calld th dgr of. For a subst of rtics W V (G), lt N(; W ) = N() W. For a subst of dgs E E(G), lt N E () = N() {u u E }, and lt d E () = N E (). Analogously, lt N E (; W ) = N E () W, and d E (, W ) = N E (, W ). Also, for a subst E of E(G), w dnot by G E th graph (V, E \ E ) obtaind from G by rmoing th dgs in E. W considr a gnralization of th TSP, namd th forcd Traling Salsman Problm. W dfin an instanc I = (G, F ) that consists of a simpl, dg wightd, undirctd graph G, and a subst F of dgs in G, calld forcd. A rtx is calld forcd if xactly on of its incidnt dgs is forcd. Similarly, it is calld unforcd if no forcd dg is incidnt to it. A Hamiltonian cycl in G is calld a tour if it passs through all th forcd dgs in F. Undr ths circumstancs, th forcd TSP rqusts to find a minimum cost tour of an instanc (G, F ). In this papr, w assum that th maximum dgr of a rtx in G is at mos. W dnot a forcd (rsp., unforcd) rtx of dgr i by fi (rsp., ui). W ar intrstd in six typs of rtics in an instanc of (G, F ), namly, u5, f5, u4, f4, u3 and f3-rtics. As shall b sn in Subsction 2.4.1, forcd and unforcd rtics of dgr 2 and 1 ar tratd as spcial cass. Lt V fi (rsp., V ui ), i = 3, 4, 5 dnot th st of firtics (rsp., ui-rtics) in (G, F ). 2.2 Essntials on Branching Algorithms W hr riw how to dri an uppr bound on th numbr of instancs that can b gnratd from an initial instanc by a branching algorithm. W can rprsnt th solution spac in our branching algorithm as a sarch tr. This is a ry usful way to illustrat th xcution of th branching ruls, and to aid th tim analysis of th branching algorithm. Th sarch tr is obtaind by assigning th input instanc of a problm as a root nod, and rcursily assigning a child to a nod for ach smallr instanc obtaind by applying th branching ruls. For a singl nod of th sarch tr, th algorithm taks tim polynomial in th siz of th nod instanc, which in turn, is smallr than or qual to th original instanc siz. Thus, w can conclud that th running tim of th branching algorithm is qual to th numbr of nods of th sarch tr tims a polynomial of th original input instanc siz. Lt I b a gin instanc with siz µ, and lt I and I b instancs obtaind from I by a branching opration. W us T (µ) to dnot th maximum numbr of nods in th sarch tr of an input of siz µ whn w xcut our branching algorithm. Lt a and b b th amounts of dcras in siz of instancs I and I, rspctily; ths alus dirctly dtrmin th prformanc of th algorithm. Thn, w call (a, b) th 2015 ISORA IET 46 Luoyang, China, Augus1 24, 2015

3 branching ctor of th branching ruls, and this implis th linar rcurrnc: T (µ) T (µ a) + T (µ b). (1) To aluat th prformanc of this branching ctor, w can us any standard mthod for linar rcurrnc rlations. In fact, it is known that T (µ) is of th form O (τ µ ), whr τ is th uniqu positi ral root of th function f(x) = 1 ( x a + x b) [6]. Th alu τ is calld th branching factor (of a gin branching ctor), and th running tim of th algorithm dcrass with th alu of this branching factor. 2.3 Th Masur-and-Conqur Mthod To ffctily analyz our sarch tr algorithm, w us th masur and conqur mthod. A complt dscription of this mthod is byond th scop of this papr, and th intrstd radr might rfr to th book of Fomin and Kratsch [6]. Th basic ida bhind th masur and conqur mthod is to assign a masur to an instanc, as opposd to using simply its siz whn analyzing th branching ctors of th branching oprations. A good choic for a masur might lad to a significantly improd analysis on th uppr bound of th running tim of a branching algorithm. For xampl, Fomin t al. [5] ha prsntd simpl polynomial-spac algorithms for th Maximum Indpndnt St and th Minimum Dominating St Problm, and obtaind an imprssi rfinmnt of th tim analysis by using th masur and conqur mthod. This shows that a good choic of masur is ry important to th tim bounds achiabl. For a gin problm instanc I of siz µ, lt W (I) b th masur of I. Whn considring a branch and rduc algorithm for th concrnd problm, intuitily w sk for a masur which satisfis th following proprtis (i) W (I) = 0 if and only if I can b sold in polynomial tim; (ii) If I is a sub-instanc of I obtaind through a rduction or a branching opration, thn W (I ) W (I). W call a masur W satisfying conditions (i) and (ii) abo a propr masur. 2.4 A Polynomial-Spac Branching Algorithm W assum that th maximum dgr of a rtx in a gin graph G is at mos. Basically, our algorithm contains two major stps. In th first stp, th algorithm applis rduction ruls until no furthr rduction is possibl. In th scond stp, th algorithm applis branching ruls in a rducd instanc to sarch for a solution. Ths two stps ar rpatd itratily. As a rsult of th rduction and branching oprations, th dgr of som rtics will dcras, whil th dgr of othr rtics will rmain unchangd. A forcd dg will nr disappar, nithr by th rduction nor branching oprations, but an unforcd dg may b rasd by ithr of th rduction or branching opration. Throughout th procss of th rduction and branching oprations, th masur of an instanc will nr incras. Dtails about th rduction and branching procdurs will b discussd in th following sub-sctions Rduction Ruls Rduction is a procss of transforming an instanc to a smallr instanc. It taks polynomial-tim to obtain a solution of an original instanc from a solution of a smallr instanc that has bn obtaind by a rduction procdur from th original instanc. Not all forcd TSP instancs ha a tour. If an instanc has no tour, w calld it infasibl. Lmma 1 gis two sufficint conditions for an instanc to b infasibl. Lmma 1 If on of th following conditions holds, thn th instanc (G, F ) is infasibl. (i) d() 1 for som rtx V (G). (ii) d F () 3 for som rtx V (G). In this papr, thr ar two rduction ruls applid in ach of th branching opration. Ths rduction ruls prsr th minimum cost tour of an instanc, as statd in Lmma 2. Lmma 2 Each of th following rductions prsrs th fasibility and a minimum cost tour of an instanc (G, F ). (i) If d() = 2 for a rtx, thn add to F any unforcd dg incidnt to rtx ; and (ii) If d() > 2 and d F () = 2 for a rtx, thn rmo from G any unforcd dg incidnt to rtx. Proof. Statmnts (i) and (ii) immdiatly follow from th dfinition of tours. From Lmma 1 and Lmma 2, w form our rduction algorithm as dscribd in Figur 1. An instanc (G, F ) which dos not satisfy any of th conditions in Lmma 1 and Lmma 2 is calld rducd Branching Ruls Our algorithm itratily branchs on an unforcd dg in a rducd instanc I = (G, F ) by ithr including into F, forc(), or xcluding it from G, dlt(). By applying a branching opration, th algorithm gnrats two nw instancs, calld branchs, by adding an unforcd dg to F, or by rmoing it from G ISORA IET 47 Luoyang, China, Augus1 24, 2015

4 Input: An instanc (G, F ) such that th maximum dgr of G is at mos. Output: A mssag for th infasibility of (G, F ); or a rducd instanc (G, F ) of (G, F ). Initializ (G, F ) := (G, F ); whil (G, F ) is not a rducd instanc do If thr is a rtx in (G, F ) such that d() 1 or d F () 3 thn Rturn mssag Infasibl Elsif thr is a rtx in (G, F ) such tha = d() > d F () thn Lt E b th st of unforcd dgs incidnt to all such rtics; St F := F E Elsif thr is a rtx in (G, F ) such that d() > d F () = 2 thn Lt E b th st of unforcd dgs incidnt to all such rtics; St G := G E End whil; Rturn (G, F ). Figur 1: Algorithm Rd(G, F ) To dscrib our branching algorithm, lt (G, F ) b a rducd instanc such that th maximum dgr of G is at mos. In (G, F ), an unforcd dg = t incidnt to a rtx of dgr 5 is calld optimal, if it satisfis a condition (c-i) blow with minimum indx i, or all unforcd dgs t in (G, F ): (c-1) V f5 and t N U (; V f3 ) such that N U () N U (t) = ; (c-2) V f5 and t N U (; V f3 ) such that N U () N U (t) ; (c-3) V f5 and t N U (; V u3 ); (c-4) V f5 and t N U (; V f4 ) such that N U () N U (t) = ; (c-5) V f5 and t N U (; V f4 ) such that N U () N U (t) ; (I) N U () N U (t) = 1; and (II) N U () N U (t) = 2; (c-6) V f5 and t N U (; V u4 ); (c-7) V f5 and t N U (; V f5 ) such that N U () N U (t) = ; (c-8) V f5 and t N U (; V f5 ) such that N U () N U (t) ; (I) N U () N U (t) = 1; (II) N U () N U (t) = 2; and (III) N U () N U (t) = 3; (c-9) V f5 and t N U (; V u5 ); (c-10) V u5 and t N U (; V f3 ); (c-11) V u5 and t N U (; V f4 ); (c-12) V u5 and t N U (; V u3 ); (c-13) V u5 and t N U (; V u4 ); and (c-14) V u5 and t N U (; V u5 ). W rfr to th abo conditions for choosing an optimal dg to branch on, c-1 to c-14, as th branching ruls. Th collcti st of branching ruls ar illustratd in Figur 2. For conninc in th analysis of th algorithm, cas (c-5) and cas (c-8) ha bn subdiidd into subcass according to th cardinality of th nighborhood intrsction. Intrsctions of lowr cardinality tak prcdnc or highr ons. Gin a rducd instanc I = (G, F ), our algorithm first chcks whthr thr xists a rtx of dgr 5, and if it dos, chooss an optimal dg according to th branching ruls. If thr xists no optimal dg according to th branching ruls, thn th rducd instanc has no mor rtics of dgr 5, and th maximum dgr of th rducd instanc at this point is at mos. Thn, w can call a polynomial spac xact algorithm for th TSP that is spcializd for dgr-4 graphs,.g., th algorithm spcializd for dgr-4 graphs by Xiao and Nagamochi [13]. Our branching algorithm is dscribd in Figur Wight Stting In ordr to obtain a masur which will imply th sam running tim bound as a function of th siz of a TSP instanc, w rquir that th wight of ach rtx is not gratr than 1. In what follows, w xamin som ncssary constraints which th rtx wights should satisfy in ordr to obtain a propr masur. For i = {3, 4, 5}, w dnot w i to b th wight of a ui-rtx, and w i to b th wight of an fi-rtx. Th conditions for a propr masur rquir that th masur of an instanc obtaind through a branching or a rduction opration will not b gratr than th masur of th original instanc. Thus, rtx wights should satisfy th following rlations w 5 1, (2) w 5 w 5, (3) w 4 w 4, (4) w 3 w 3, (5) w 3 w 4 w 5, and (6) w 3 w 4 w 5. (7) Th rtx wight for rtics of dgr lss than 3 is st to b 0. W procd to show in th algorithms gin in Figurs 1 and 3, stting rtx wights which satisfy th conditions of Eqs. (3) to (7) is sufficint to obtain a propr masur. Lmma 3 If th wights of rtics ar chosn as in Eqs. (3) to (7), thn th masur W (I) nr incrass as a rsult of th rduction or th branching oprations of Figur 1 and Figur 3. Proof. Lt I = (G, F ) b a gin instanc of th forcd TSP. Du to our dfinition of th masur W (I) of 2015 ISORA IET 48 Luoyang, China, Augus1 24, 2015

5 c-1 c-2 t 6 t 6 c-3 c-4 c-5(i) c-5(ii) t 7 t6 t6 c-6 c-7 t 7 t2 t3 t 6 c-8(i) c-8(ii) c-8(iii) t 8 t6 t 7 c-9 t 6 c-10 t 6 t 7 c-11 t 6 t 7 c-12 t 6 t7 t 8 c-13 t t 6 9 t7 t 8 c-14 : unforcd dgs : forcd dgs Figur 2: Illustration of th Branching Ruls Eq.(18), it suffics to show that non of th indiidual rtx wights will incras as a rsult of a rduction or a branching opration in th algorithms of Figurs 1 and 3. Th branching ruls stat that for an unforcd dg in E(G)\F, two subinstancs ar gnratd by ithr 2015 ISORA IET 49 Luoyang, China, Augus1 24, 2015

6 Input: An instanc (G, F ) such that th maximum dgr of G is at mos. Output: A mssag for th infasibility of (G, F ); or th minimum cost of a tour of (G, F ). Run Rd(G, F ); If Rd(G, F ) rturns mssag Infasibl thn Rturn mssag Infasibl Els Lt (G, F ) := Rd(G, F ); If V u5 V f5 thn Choos an optimal unforcd dg Rturn min{tsp5(g, F {}), tsp5(g {}, F )} Els /* thr is no rtx of dgr 5 in (G, F ) */ Rturn tsp4(g, F ). Not: Th input and output of algorithm tsp4(g, F ) ar as follows Input: An instanc (G, F ) such that th maximum dgr of G is at mos. Output: A mssag for th infasibility of (G, F ); or th minimum cost of a tour of (G, F ). Figur 3: Algorithm tsp5(g, F ) stting F := F {}, trmd as forc(), or by stting G := G {}, trmd dlt(). In fact, w bring to attntion that a rduction opration, if it dos not rturn a mssag Infasibl, is in fact a rpatd application of th abo two stps, forc() or dlt(), for som unforcd dg, idntifid by th conditions in Lmma 2. Thrfor, w procd with analyzing th ffcts of applying th forc() or th dlt() opration. Lt = u b an unforcd dg to which on of th forc() or dlt() oprations will b applid. Both u and must ha dgr mor than 2, othrwis by Lmma 1, th instanc is infasibl. Without loss of gnrality, w obsr th ffct of th opration on th rtx wight ω(). In th cas that opration forc() is applid, th following cass may aris. If is an unforcd rtx, thn will bcom forcd. By Eqs. (3) to (5), th wight ω() will not incras. If is a forcd rtx, thn ω() will bcom 0. If d F () = 2, thn by Lmma 1 th instanc will bcom infasibl. On th othr hand, if opration dlt() is applid, w obsr th following cass. If is ithr forcd or unforcd, and d() 3, thn th dgr of will dcras by on, and by Eqs. (6) and (7) ω() will not incras. If is ithr forcd or unforcd, and d() 2, thn by Lmma 1 th instanc will bcom infasibl. Following th abo obsrations, w conclud that th complt masur W (I) of a gin instanc I = (G, F ) of th forcd TSP will not incras as a rsult of th rduction and branching oprations of th algorithms in Figurs 1 and 3. To simplify som argumnts, w introduc th following notation: 3 = w 3 w 3, 4 = w 4 w 4, 5 = w 5 w 5, 4 3 = w 4 w 3, 5 4 = w 5 w 4, 5 3 = w 5 w 3, 4 3 = w 4 w 3, 5 4 = w 5 w 4, and 5 3 = w 5 w 3. To simplify th list of our branching ctors, w us th following notation: m 1 = min{w 3, w 3, 4 3, 4 3, 5 4, 5 4 }, (8) m 2 = min{w 3, 4 3, 4 3, 5 4, 5 4 }, (9) m 3 = min{w 3, 3, w 4, 4, w 5, 5 }, (10) m 4 = min{ 4 3, 4 3, 5 4, 5 4 }, (11) m 5 = min{w 4, w 4, 5 3, 5 3 }, (12) m 6 = min{ 4 3, 5 4, 5 4 }, (13) m 7 = min{ 5 4, 5 4 }, (14) m 8 = min{ 5 3, 5 3 }, (15) m 9 = min{w 3, w 3, 4 3, 4 3, 5 4 }, and (16) m 10 = min{w 3, 3, w 4, 4, 5 }. (17) 3 Rsults In ordr to adopt th masur and conqur mthod in our algorithm, w nd to st a masur W for a gin an instanc I = (G, F ) of th forcd TSP. To this ffct, w st a non-ngati rtx wight function ω : V R + in th graph G, and w us th sum of wights of all rtics in th graph as th masur W (I) of instanc I. That is, W (I) = (ω()). (18) V (G) W bring to attntion th fact that th numbr n of rtics in th graph G rmains unmodifid throughout th procss of th rduction and branching oprations. In addition to sking a propr masur, w also rquir that th wight of ach rtx is not gratr than 1, and thrfor, th masur W (I) will not b gratr than th numbr n of rtics in G. As a consqunc, a 2015 ISORA IET 50 Luoyang, China, Augus1 24, 2015

7 running tim bound as a function of th masur W (I) implis th sam running tim bound as a function of n. Th wight assignd to ach rtx typ plays an important rol, sinc th alu of th branching factor dpnds solly on ths wights. Lt th rtx wight function ω() b chosn as follows: w 3 = for an f3-rtx w 3 = for a u3-rtx w 4 = for an f4-rtx ω() = w 4 = for a u4-rtx (19) w 5 = for an f5-rtx w 5 = 1 for a u5-rtx 0 othrwis. Lmma 4 If th rtx wight function ω() is st as in Eq. (19), thn ach branching opration in Figur 3 has a branching factor not gratr than A proof of Lmma 4 will b drid analytically in th sral subsctions which follow. From th lmma, w gt our main rsult: Thorm 1 Th TSP in an n-rtx graph G with maximum dgr 5 can b sold in O ( n )-tim and polynomial-spac. In th rmaindr of th analysis, for an optimal dg =, w dnot N U () by {,,..., t a }, a = d U (), and N U ( ) \ {} by {t a+1, t a+1,..., t a+b }, b = d U ( ) 1. W assum without loss of gnrality that +i = t a+i for i = 1, 2,..., c, whr c = N U () N U ( ), th numbr of good nighbors that and ha in common. 3.1 Branching on Edgs Around f5- rtics (c-1 to c-9) This sction dris branching ctors for th branching opration on an optimal dg =, incidnt to an f5-rtx, distinguishing nin cass for conditions c-1 to c-9. Cas c-1: Thr xist rtics V f5 and N U (; V f3 ) such that N U () N U ( ) = (s Figur 4): W branch on dg. Not that N U ( ) \ {} = { }. In th branch of forc( ), dg will b addd to F by th branching opration, and dgs,, and will b dltd from G by th rduction ruls. Both and will bcom rtics of dgr 2. From Eq. (19), th wight of rtics of dgr 2 is 0. So, th wight of rtx dcrass by w 5 and th wight of rtx dcrass by w 3. Each of th rtics, and can b ithr a typ f3, u3, f4, u4, f5, or u5-rtx, and ach of thir wights would dcras by at last m 1 = min { } w 3, w 3, 4 3, 4 3, 5 4, 5 4. If rtx t5 is (a) forc( ) in c-1 (b) dlt(t1) in c-1 : unforcd dgs : forcd dgs : nwly dltd dgs : nwly forcd dgs Figur 4: Illustration of branching rul c-1, whr rtx V f5 and rtx N U (; V f3 ) such that N U () N U ( ) =. an f3-rtx (rsp., u3, f4, u4, f5, or a u5-rtx), thn th wight dcras α of rtx would b w 3 (rsp., w 3, 4 3, 4 3, 5 4, and 5 4 ). Thus, th total wight dcras in th branch of forc( ) is at last (w 5 + w 3 + 3m 1 + α). In th branch of dlt( ), dg will b dltd from G by th branching opration, and dg will b addd to F by th rduction ruls. Th wight of rtx dcrass by 5 4 and th wight of rtx dcrass by w 3. If rtx is an f3-rtx (rsp., u3, f4, u4, f5, or a u5-rtx), thn th wight dcras β of rtx would b w 3 (rsp., 3, w 4, 4, w 5, and 5 ). Thus, th total wight dcras in th branch of dlt( ) is at last (w 5 w 4 + w 3 + β). As a rsult, w gt th following six branching ctors: (w 5 + w 3 + 3m 1 + α, w 5 w 4 + w 3 + β) (20) for (α, β) {(w 3, w 3 ), (w 3, 3 ), ( 4 3, w 4 ), ( 4 3, 4 ), ( 5 4, w 5 ), ( 5 4, 5 )}. c-2. Thr xist rtics V f5 and N U (; V f3 ) such that N U () N U ( ) = { } (s Figur 5): W branch on dg. In th branch of forc( ), dg will b addd to F by th branching opration, and dgs,, and will b dltd from G by th rduction ruls. So, th wight of rtx dcrass by w 5, and th wight of rtx dcrass by w 3. Each of th rtics and can b ithr a typ f3, u3, f4, u4, f5, or u5-rtx, and ach of thir wights would dcras by at last m 1 = min { } w 3, w 3, 4 3, 4 3, 5 4, 5 4. Thr ar two possibl cass for th rtx typ of rtx. First, lt b an f3 or u3-rtx. Aftr prforming th branching opration, would bcom a rtx of dgr 1. By Lmma 1, cas (i), this is infasibl, and th algorithm will rturn a mssag of infasibility ISORA IET 51 Luoyang, China, Augus1 24, 2015

8 (a) forc( ) in c-2 (b) dlt( ) in c-2 : unforcd dgs : forcd dgs : nwly dltd dgs : nwly forcd dgs Figur 5: Illustration of branching rul c-2, whr rtx V f5 and rtx N U (; V f3 ) such that N U () N U ( ) = { }. t 6 t 6 (a) forc( ) in c-3 (b) dlt( ) in c-3 : unforcd dgs : forcd dgs : nwly dltd dgs : nwly forcd dgs Figur 6: Illustration of branching rul c-3, whr rtx V f5 and rtx N U (; V u3 ). Scond, lt b an f4, u4, f5, or u5-rtx. If is an f4-rtx (rsp., u4, f5, or a u5-rtx), thn th wight dcras α of rtx would b w 4 (rsp., w 4, 5 3, and 5 3 ). Thus, th total wight dcras in th branch of forc( ) is at last (w 5 +w 3 +2m 1 +α). In th branch of dlt( ), dg will b dltd from G by th branching opration, and dg will b addd to F by th rduction ruls. So, th wights of rtics and dcras by 5 4 and w 3, rspctily. If rtx is an f4-rtx (rsp., u4, f5, or a u5-rtx), thn th wight dcras β of rtx would b w 4 (rsp., 4, w 5, and 5 ). Thus, th total wight dcras in th branch of dlt( ) is at last (w 5 w 4 + w 3 + β). As a rsult, w gt th following four branching ctors: (w 5 + w 3 + 2m 1 + α, w 5 w 4 + w 3 + β) (21) for (α, β) {(w 4, w 4 ), (w 4, 4 ), ( 5 3, w 5 ), ( 5 3, 5 )}. c-3. Thr xist rtics V f5 and N U (; V u3 ) (s Figur 6): W branch on dg. Not that N U ( ) \ {} = {, t 6 }. In th branch of forc( ), dg will b addd to F by th branching opration, and dgs, and will b dltd from G by th rduction ruls. So, th wights of rtics and dcras by w 5 and 3, rspctily. Each of rtics, and can b ithr a typ u3, f4, u4, f5, or u5-rtx, and ach of thir wights would dcras by at last m 2 = min { w 3, 4 3, 4 3, 5 4, 5 4 }. Thus, th total wight dcras in th branch of forc( ) is at last (w 5 + w 3 w 3 + 3m 2 ). In th branch of dlt( ), dg will b dltd from G by th branching opration and dgs and t 6 will b addd to F by th rduction ruls. So, th wight of rtx dcrass by 5 4 and th wight of rtx dcrass by w 3. Each of th rtics and t 6 can b ithr a typ f3, u3, f4, u4, f5, or u5-rtx, and ach of thir wights would dcras by at last m 3 = min {w 3, 3, w 4, 4, w 5, 5 }. Thus, th total wight dcras in th branch of dlt( ) is at last (w 5 w 4 + w 3 + 2m 3 ). As a rsult, w gt th following branching ctor: (w 5 + w 3 w 3 + 3m 2, w 5 w 4 + w 3 + 2m 3 ). (22) c-4. Thr xist rtics V f5 and N U (; V f4 ) such that N U () N U ( ) = (s Figur 7): W branch on dg. Not that N U ( ) \ {} = {, t 6 }. t 6 (a) forc( ) in c-4 t 6 (b) dlt( ) in c-4 : unforcd dgs : forcd dgs : nwly dltd dgs : nwly forcd dgs Figur 7: Illustration of branching rul c-4, whr rtx V f5 and rtx N U (; V f4 ), such that N U () N U ( ) =. In th branch of forc( ), dg will b addd to F by th branching opration, and dgs,,, and t 6 will b dltd from G by th rduction ruls. So, th wight of rtx dcrass by w 5 and th wight of rtx dcrass by w 4. Each of th rtics, and can b ithr a typ f4, u4, f5, or u5-rtx, and ach of thir wights would dcras by at last m 4 = min { } 4 3, 4 3, 5 4, 5 4. Each of rtics t5 and t 6 can b ithr a typ f3, u3, f4, u4, f5, or u5- rtx, and ach of thir wights would dcras by 2015 ISORA IET 52 Luoyang, China, Augus1 24, 2015

9 at last m 1 = min { w 3, w 3, 4 3, 4 3, 5 4, 5 4 }. Thus, th total dcras in th branch of forc( ) is at last (w 5 + w 4 + 3m 4 + 2m 1 ). In th branch of dlt( ), dg will b dltd from G by th branching opration. So, th wight of rtx dcrass by 5 4 and th wight of rtx dcrass by 4 3. Thus, th total wight dcras in th branch of dlt( ) is at last (w 5 w 3 ). As a rsult, w gt th following branching ctor: (w 5 + w 4 + 3m 4 + 2m 1, w 5 w 3 ). (23) c-5. Thr xist rtics V f5 and N U (; V f4 ) such that N U () N U ( ). W distinguish two sub cass, according to th cardinality of th intrsction N U () N U ( ), (c-5(i)), N U () N U ( ) = 1, and (c-5(ii)), N U () N U ( ) = 2. c-5(i). Without loss of gnrality, assum that N U () N U ( ) = { } (s Figur 8): W branch on dg. Not that N U ( ) \ {} = { }. (a) forc(t1) in c-5(i) (b) dlt(t1) in c-5(i) : unforcd dgs : forcd dgs : nwly dltd dgs : nwly forcd dgs Figur 8: Illustration of branching rul c-5(i), whr rtx V f5 and rtx N U (; V f4 ), such that N U () N U ( ) = { }. In th branch of forc( ), dg will b addd to F by th branching opration, and dgs,,,, and will b dltd from G by th rduction ruls. So, th wight of rtx dcrass by w 5 and th wight of rtx dcrass by w 4. Vrtx can b ithr a typ f4, u4, f5, or u5-rtx, and its wight would dcras by at last m 5 = min { } w 4, w 4, 5 3, 5 3. Each of th rtics and can b ithr a typ f4, u4, f5, or u5- rtx, and ach of thir wights would dcras by at last m 4 = min { } 4 3, 4 3, 5 4, 5 4. Vrtx can b ithr a typ f3, u3, f4, u4, f5, or u5-rtx, and its wight would dcras by at last m 1 = min { } w 3, w 3, 4 3, 4 3, 5 4, 5 4. Thus, th total wight dcras in th branch of forc( ) is at last (w 5 + w 4 + m 5 + 2m 4 + m 1 ). In th branch of dlt( ), dg will b dltd from G by th branching opration. So, th wight of rtx dcrass by 5 4, and th wight of rtx dcrass by 4 3. Thus, th total wight dcras in th branch of dlt( ) is at last (w 5 w 3 ). As a rsult, w gt th following branching ctor: (w 5 + w 4 + m 5 + 2m 4 + m 1, w 5 w 3 ). (24) c-5(ii). Without loss of gnrality, assum that N U () N U ( ) = {, } (s Figur 9): W branch on dg. (a) forc( ) in c-5(ii) (b) dlt( ) in c-5(ii) : unforcd dgs : forcd dgs : nwly dltd dgs : nwly forcd dgs Figur 9: Illustration of branching rul c-5(ii), whr rtx V f5 and rtx N U (; V f4 ) such that N U () N U ( ) = {, }. In th branch of forc( ), dg will b addd to F by th branching opration, and dg,,, and will b dltd from G by th rduction ruls. So, th wight of rtx dcrass by w 5 and th wight of rtx dcrass by w 4. Each of rtics and can b ithr a typ f4, u4, f5, or u5-rtx, and ach of thir wights would dcras by at last m 5 = min { } w 4, w 4, 5 3, 5 3. Vrtx can b ithr a typ f3, u3, f4, u4, f5, or u5-rtx, and its wight would dcrass by at last m 4 = min { } 4 3, 4 3, 5 4, 5 4. Thus, th total wight dcras in th branch of forc( ) is at last (w 5 + w 4 + 2m 5 + m 4 ). In th branch of dlt( ), dg will b dltd from G by th branching opration. So, th wight of rtx dcrass by 5 4, and th wight of rtx dcrass by 4 3. Thus, th total wight dcras in th branch of dlt( ) is at last (w 5 w 3 ). As a rsult, w gt th following branching ctor: (w 5 + w 4 + 2m 5 + m 4, w 5 w 3 ). (25) c-6. Thr xist rtics V f5 and N U (; V u4 ) (s Figur 10): W branch on dg. Not that N U ( ) \ {} = {, t 6, t 7 }. In th branch of forc( ), dg will b addd to F by th branching opration, and dgs, and will b dltd from G by th rduction ruls. So, th wight of rtx dcrass by w 5 and th wight of rtx dcrass by 4. Each of th rtics, and can b ithr a typ u4, f5, or u5- rtx, and ach of thir wights would dcras by at 2015 ISORA IET 53 Luoyang, China, Augus1 24, 2015

10 t6 t 7 (a) forc( ) in c-6 (b) dlt( ) in c-6 : unforcd dgs : forcd dgs t6 t 7 : nwly dltd dgs : nwly forcd dgs Figur 10: Illustration of branching rul c-6, whr rtx V f5 and rtx N U (; V u4 ). last m 6 = min { 4 3, 5 4, 5 4 }. Thus, th total wight dcras in th branch of forc( ) is at last (w 5 + w 4 w 4 + 3m 6 ). In th branch of dlt( ), dg will b dltd from G by th branching opration. So, th wight of rtx dcrass by 5 4 and th wight of rtx dcrass by 4 3. Thus, th total wight dcras in th branch of dlt( ) is at last (w 5 w 4 + w 4 w 3 ). As a rsult, w gt th following branching ctor: (w 5 + w 4 w 4 + 3m 6, w 5 w 4 + w 4 w 3 ). (26) c-7. Thr xist rtics V f5 and N U (; V f5 ) such that N U () N U ( ) = (s Figur 11): W branch on dg. Not that N U ( ) \ {} = {, t 6, t 7 }. t6 t 7 (a) forc( ) in c-7 t6 t 7 (b) dlt( ) in c-7 : unforcd dgs : forcd dgs : nwly dltd dgs : nwly forcd dgs Figur 11: Illustration of branching rul c-7, whr rtx V f5 and rtx N U (; V f5 ), such that N U () N U ( ) =. In th branch of forc( ), dg will b addd to F by th branching opration, and dgs,,,, t 6 and t 7 will b dltd from G by th rduction ruls. So, both wights of rtx and rtx dcrass by w 5, ach. Each of rtics,,,, t 6 and t 7 can b ithr a typ f5, or u5- rtx, and ach of thir wights would dcras by at last m 7 = min { 5 4, 5 4 }. Thus, th total wight dcras in th branch of forc( ) is at last (2w 5 + 6m 7 ). In th branch of dlt( ), dg will b dltd from G by th branching opration. So, both wights of rtics and dcrass by 5 4, ach. Thus, th total wight dcras in th branch of dlt( ) is at last (2w 5 2w 4 ). As a rsult, w gt th following branching ctor: (2w 5 + 6m 7, 2w 5 2w 4 ). (27) c-8. Thr xist rtics V f5 and N U (; V f5 ) such that N U () N U ( ). W distinguish thr sub cass, according to th cardinality of th intrsction N U () N U ( ), (c-8(i)), N U () N U ( ) = 1, (c-8(ii)), N U () N U ( ) = 2, and (c-8(iii)), N U () N U ( ) = 3. c-8(i). Without loss of gnrality, assum that N U () N U ( ) = { } (s Figur 12): W branch on dg. Not that N U ( ) \ {} = {, t 6 }. t 6 (a) forc( ) in c-8(i) t 6 (b) dlt( ) in c-8(i) : unforcd dgs : forcd dgs : nwly dltd dgs : nwly forcd dgs Figur 12: Illustration of branching rul c-8(i), whr rtx V f5 and rtx N U (; V f5 ), such that N U () N U ( ) = { }. In th branch of forc( ), dg will b addd to F by th branching opration, and dgs,,,, and t 6 will b dltd from G by th rduction ruls. Both wights of rtx and rtx dcrass by w 5, ach. Vrtx can b ithr a typ f5 or u5-rtx, and its wight would dcrass by at last m 8 = min { 5 3, 5 3 }. Each of rtics t3,, and t 6 can b ithr a typ f5, or u5-rtx, and ach of thir wights would dcras by at last m 7 = min { 5 4, 5 4 }. Thus, th total wight dcras in th branch of forc( ) is at last (2w 5 + 4m 7 + m 8 ). In th branch of dlt( ), dg will b dltd from G by th branching opration. Both wights of rtics and dcrass by 5 4, ach. Thus, th total wight dcras in th branch of dlt( ) is at last (2w 5 2w 4 ). As a rsult, w gt th following branching ctor: (2w 5 + 4m 7 + m 8, 2w 5 2w 4 ). (28) 2015 ISORA IET 54 Luoyang, China, Augus1 24, 2015

11 c-8(ii). Without loss of gnrality, assum that N U () N U ( ) = {, } (s Figur 13): W branch on dg. Not that N U ( ) \ {} = { }. (a) forc( ) in c-8(ii) (b) dlt( ) in c-8(ii) : unforcd dgs : forcd dgs : nwly dltd dgs : nwly forcd dgs Figur 13: Illustration of branching rul c-8(ii), whr rtx V f5 and rtx N U (; V f5 ), such that N U () N U ( ) = {, }. In th branch of forc( ), dg will b addd to F by th branching opration, and dgs,,,, and will b dltd from G by th rduction ruls. So, both wights of rtx and rtx dcrass by w 5, ach. Each of rtics and can b ithr a typ f5, or u5-rtx, and ach of thir wights would dcras by at last m 8 = min { } 5 3, 5 3. Each of rtics t4 and can b ithr a typ f5, or u5-rtx, and ach of thir wights would dcras by at last m 7 = min { } 5 4, 5 4. Thus, th total wight dcras in th branch of forc( ) is at last (2w 5 +2m 8 +2m 7 ). In th branch of dlt( ), dg will b dltd from G by th branching opration. So, both wights of rtx and rtx dcrass by 5 4, ach. Th total wight dcras in th branch of dlt( ) is at last (2w 5 2w 4 ). As a rsult, w gt th following branching ctor: (2w 5 + 2m 8 + 2m 7, 2w 5 2w 4 ). (29) c-8(iii). Without loss of gnrality, assum that N U () N U ( ) = {,, } (s Figur 14): W branch on dg. In th branch of forc( ), dg will b addd to F by th branching opration, and dgs,,,, and will b dltd from G by th rduction ruls. So, both wights of rtx and rtx dcrass by w 5, ach. Each of rtics,, and can b ithr a typ f5, or u5- rtx, and ach of thir wights would dcras by at last m 8 = min { } 5 3, 5 3. Thus, th total wight dcras in th branch of forc( ) is at last (2w 5 + 3m 8 ). (a) forc( ) in c-8(iii) (b) dlt( ) in c-8(iii) : unforcd dgs : forcd dgs : nwly dltd dgs : nwly forcd dgs Figur 14: Illustration of branching rul c-8(iii), whr rtx V f5 and rtx N U (; V f5 ), such that N U () N U ( ) = {,, }. In th branch of dlt( ), dg will b dltd from G by th branching opration. Thus, both wights of rtx and rtx dcrass by 5 4, ach. Th total wight dcras in th branch of dlt( ) is at last (2w 5 2w 4 ). As a rsult, w gt th following branching ctor: (2w 5 + 3m 8, 2w 5 2w 4 ). (30) c-9. Thr xist rtics V f5 and N U (; V u5 ) (s Figur 15): W branch on dg. t 8 t6 t 7 (a) forc( ) in c-9 t 8 t6 t 7 (b) dlt( ) in c-9 : unforcd dgs : forcd dgs : nwly dltd dgs : nwly forcd dgs Figur 15: Illustration of branching rul c-9, whr rtx V f5 and rtx N U (; V u5 ). In th branch of forc( ), dg will b addd to F by th branching opration, and dgs, and will b dltd from G by th rduction ruls. So, th wight of rtx dcrass by w 5, and th wight of rtx dcrass by 5. Each of rtics, and can only b a typ u5-rtx, and ach of thir wights dcras by 5 4. Thus, th total wight dcras in th branch of forc( ) is at last (4w 5 3w 4 ). In th branch of dlt( ), dg will b dltd from G by th branching opration. Thus, th wight of rtx dcrass by 5 4, and th wight of rtx dcrass by 5 4. Th total wight dcras in 2015 ISORA IET 55 Luoyang, China, Augus1 24, 2015

12 th branch of dlt( ) is at last (w 5 + w 5 w 4 w 4 ). Thn, w gt th following branching ctor: N U (; V f4 ) (s Figur 17): W branch on dg. Not that N U ( ) \ {} = {t 6, t 7 }. (4 3w 4, 1 + w 5 w 4 w 4 ). (31) 3.2 Branching on Edgs Around u5- rtics (c-10 to c-14) If non of th first nin conditions can b xcutd, this mans that th graph has no f5-rtics. But this dos not man that th maximum dgr of th graph has bn rducd to 4, sinc thr might still b u5-rtics. This sction dris branching ctors for branchings on an optimal dg = incidnt to a u5-rtx, distinguishing th fi cass for conditions c-10 to c-14. c-10. Thr xist rtics V u5 and N U (; V f3 ) (s Figur 16): W branch on dg. Not that N U ( ) \ {} = {t 6 }. t 6 (a) forc( ) in c-10 (b) dlt( ) in c-10 t 6 : unforcd dgs : forcd dgs : nwly dltd dgs : nwly forcd dgs Figur 16: Illustration for c-10 whr rtics V u5 and N U (; V f3 ). In th branch of forc( ), dg will b addd to F by th branching opration, and dg t 6 will b dltd from G by th rduction ruls. So, th wight of rtx dcrass by 5, and th wight of rtx dcrass by w 3. If rtx t 6 is an f3-rtx (rsp., u3, f4, u4, or a u5-rtx), thn th wight dcras α of rtx t 6 would b w 3 (rsp., w 3, 4 3, 4 3, and 5 4 ). Thus, th total wight dcras in th branch of forc( ) is at last (w 5 w 5 + w 3 + α). In th branch of dlt( ), dg will b dltd from G by th branching opration, and dg t 6 will b addd to F by th rduction ruls. Th wight of rtx dcrass by 5 4, and th wight of rtx dcrass by w 3. If rtx t 6 is an f3-rtx (rsp., u3, f4, u4, or a u5-rtx), thn th wight dcras β of rtx t 6 would b w 3 (rsp., 3, w 4, 4, and 5 ). Thus, total wight dcras in th branch of dlt( ) is at last (w 5 w 4 + w 3 + β). As a rsult, w gt fi branching ctors: (1 w 5 + w 3 + α, 1 w 4 + w 3 + β) (32) for (α, β) {(w 3, w 3 ), (w 3, 3 ), ( 4 3, w 4 ), ( 4 3, 4 ), ( 5 4, 5 )}. c-11. Thr xist rtics V u5 and t 6 t 7 t 6 t 7 (a) forc( ) in c-11 (b) dlt( ) in c-11 : unforcd dgs : forcd dgs : nwly dltd dgs : nwly forcd dgs Figur 17: Illustration for c-11 whr rtics V u5 and N U (; V f4 ). In th branch of forc( ), dg will b addd to F by th branching opration, and dgs t 6 and t 7 will b dltd from G by th rduction ruls. So, th wight of rtx dcrass by 5, and th wight of rtx dcrass by w 4. Each of rtics t 6 and t 7 can b ithr a typ f3, u3, f4, u4, or u5- rtx, and ach of thir wights would dcras by at last m 9 = min { w 3, w 3, 4 3, 4 3, 5 4 }. Thus, th total wight dcras in th branch of forc( ) is at last (w 5 w 5 + w 4 + 2m 9 ). In th branch of dlt( ), dg will b dltd from G by th branching opration. So, th wight of rtx dcrass by 5 4, and th wight of rtx dcrass by 4 3. Th total wight dcras in th branch of dlt( ) is at last (w 5 w 4 + w 4 w 3 ). As a rsult, w gt th following branching ctor: (1 w 5 + w 4 + 2m 9, 1 w 4 + w 4 w 3 ). (33) c-12. Thr xist rtics V u5 and t N U (; V u3 ) (s Figur 18): W branch on dg. Not that N U ( ) \ {} = {t 6, t 7 }. t 6 t 7 (a) forc( ) in c-12 (b) dlt( ) in c-12 t 6 t 7 : unforcd dgs : forcd dgs : nwly dltd dgs : nwly forcd dgs Figur 18: Illustration of branching rul c-12, whr rtx V u5 and rtx t N U (; V u3 ). In th branch of forc( ), dg will b addd to F by th branching opration. So, th wight of rtx dcrass by 5, and th wight of rtx dcrass by 3. Th total wight dcras in th 2015 ISORA IET 56 Luoyang, China, Augus1 24, 2015

13 branch of forc( ) is at last (w 5 w 5 + w 3 w 3 ). In th branch of dlt( ), dg will b dltd from G by th branching opration, and dgs t 6 and t 7 will b addd to F by th rduction ruls. So, th wight of rtx dcrass by 5 4, and th wight of rtx dcrass by w 3. Each of rtics t 6 and t 7 can b ithr a typ f3, u3, f4, u4, or u5-rtx, and ach of thir wights would dcras by at last m 10 = min{w 3, 3, w 4, 4, 5 }. Thus, th total wight dcras in th branch of dlt( ) is at last (w 5 w 4 + w 3 + 2m 10 ). As a rsult, w gt th following branching ctor: (1 w 5 + w 3 w 3, 1 w 4 + w 3 + 2m 10 ). (34) c-13. Thr xist rtics V u5 and N U (; V u4 ) (s Figur 19): W branch on dg. t 6 t7 t 8 (a) forc( ) in c-13 (b) dlt( ) in c-13 : unforcd dgs : forcd dgs : nwly dltd dgs : nwly forcd dgs Figur 19: Illustration of branching rul c-13, whr rtx V u5 and rtx N U (; V u4 ). In th branch of forc( ), dg will b addd to F by th branching opration. So, th wight of rtx dcrass by 5, and th wight of rtx dcrass by 4. Thus, th total wight dcras in th branch of forc( ) is at last (w 5 w 5 + w 4 w 4 ). In th branch of dlt( ), dg will b dltd from G by th branching opration. So, th wight of rtx dcrass by 5 4, and th wight of rtx dcrass by 4 3. Thus, th total wight dcras in th branch of dlt( ) is at last (w 5 w 3 ). Thn, w gt th following branching ctor: t 6 (1 w 5 + w 4 w 4, 1 w 3 ). (35) c-14. Thr xist rtics V u5 and N U (; V u5 ) (s Figur 20): W branch on dg. In th branch of forc( ), dg will b addd to F by th branching opration. So, both wights of rtx and rtx dcrass by 5, ach. Thus, th total wight dcras in th branch of forc( ) is at last (2w 5 2w 5 ). In th branch of dlt( ), dg will b dltd from G by th branching opration. So, both wights of rtx and rtx dcrass by 5 4, ach. Thus, th total wight dcras in th branch of dlt( ) is at last (2w 5 2w 4 ). Thn, t7 t 8 t t 6 9 t 7 t 8 t t 6 9 t 7 t 8 (a) forc( ) in c-14 (b) dlt( ) in c-14 : unforcd dgs : forcd dgs : nwly dltd dgs : nwly forcd dgs Figur 20: Illustration of branching rul c-14, whr rtx V u5 and rtx N U (; V u5 ). w gt th following branching ctor: 3.3 Switching to TSP4 (2 2w 5, 2 2w 4 ). (36) If non of ths 14 cass can b xcutd, this mans that th graph has no mor dgr-5 rtics. In that cas, w can switch and us a fast algorithm for TSP in dgr-4 graphs (tsp4(g, F )) to sol th rmaining instancs. Xiao and Nagamochi [15, Lmma 3] ha shown how to lrag rsults obtaind by a masurand-conqur analysis, and that an algorithm can b usd as a subprocdur, gin that w know th rspcti wight stting mchanism. To gt a combination of total running tim bound of ths two algorithms, w can us th maximum branching factor for TSP in dgr-4 graphs algorithm and a masur µ is calculatd basd on th maximum ratio of rtx wights for TSP in dgr-4 graphs and TSP in dgr- 5 graphs [12]. Hr w us th O ( n )-tim algorithm by Xiao and Nagamochi [13], whr th wights of rtics in dgr-4 graphs ar st as follows; w 3 = , w 3 = , w 4 = , and w 4 = 1. For this stp, th running tim bound is { }) T (µ) O ( max w, w, w, 1 4 w Orall Analysis (37) Th branching factor of ach of th branching ctors from (20) to (37) dos not xcd Th tight constraints in th quasiconx program ar in conditions c-4, c-10, c-11, c-12, c-13 and th switching constraint. This complts a proof of Thorm 1. 4 Conclusion In this papr, w ha prsntd an xact algorithm for TSP in dgr-5 graphs. Our algorithm is a simpl 2015 ISORA IET 57 Luoyang, China, Augus1 24, 2015

14 branching algorithm, following th branch-and-rduc paradigm, and it oprats in spac which is polynomial of th siz of an input instanc. To th bst of our knowldg, this is th first polynomial spac xact algorithm dlopd spcifically for graphs of maximum dgr at mos, and xtnds prious algorithms for dgr 3 [11, 14], and dgr-4 graphs [13]. W usd th masur and conqur mthod for th analysis of th running tim of th proposd algorithm, and ha obtaind an uppr bound of O ( n ), whr n is th numbr of rtics in a gin instanc. This rsult compars faorably with th polynomialspac TSP algorithm for gnral graphs by Gurich and Shlah [8], which runs in O (4 n n log n )-tim. It rmains an opn qustion whthr this tim bound can b furthr improd by a modifid analysis tchniqu, or by a carful r-xamination of th branching ruls. Indd, it would b most intrsting to obtain a polynomial-spac algorithm with a running tim of O (2 n ) or lss, or simply show that this cannot b achid. Acknowldgmnts Th first author would lik to xprss gratitud to Tchnical Unirsity of Malaysia Malacca, Malaysia and Ministry of Highr Education (MOHE) Malaysia for th scholarship program. Rfrncs [1] Bllman, R. : Combinatorial Procsss and Dynamic Programming. In: Procding of th 10th Symposium in Applid Mathmatics, Amr. Math. Soc., Proidnc, RI, [2] Bodlandr, H. L., Cygan M., Kratsch S. and Ndrlof J. : Soling Wightd and Counting Variants of Connctiity Problms Paramtrizd by Trwidth Dtrministically in Singl Exponntial Tim. In: CoRR abs/ , [3] Eppstin, D. : Quasiconx Analysis of Multiariat Rcurrnc Equations for Backtracking Algorithms. In: ACM Transactions on Algorithms, Vol. 2, No. 4, pp , [4] Eppstin, D. : Th Traling Salsman Problm for Cubic Graphs. In: Journal of Graph Algorithms and Application, Vol. 11, No. 1, pp , [5] Fomin, F. V., Grandoni, F. and Kratsch, D. : A Masur and Conqur Approach for th Analysis of Exact Algorithms. In: J. ACM, Vol. 56, No. 5, Articl 25, [6] Fomin, F. V. and Kratsch, D. : Exact Exponntial Algorithms. In: Brlin Hidlbrg: Springr, [7] Gbaur, H. : Finding and Enumrating Hamilton Cycls in 4-rgular Graphs. In: Thortical Computr Scinc, Vol. 412, No. 35, pp , [8] Gurich, Y. and Shlah, S. : Expctd Computation Tim for Hamiltonian Path Problm. In: Siam Journal of Computation, Vol. 16, No. 3, pp , [9] Hld, M. and Karp, R. M. : A Dynamic Programming Approach to Squncing Problms. In: Journal of th Socity for Industrial and Applid Mathmatics, Vol. 10, No. 1, pp , [10] Iwama, K. and Nakashima, T. : An Improd Exact Algorithm for Cubic Graph TSP. In: CO- COON, LNCS 4598, pp , [11] Liskiwicz, M. and Schustr, M. R. : A Nw Uppr Bound for th Traling Salsman Problm in Cubic Graphs. In: CoRR abs/ , [12] Xiao, M. and Nagamochi, H. : Furthr Impromnt on Maximum Indpndnt St in Dgr-4 Graphs. In: COCOA 2011, LNCS 6831, pp , [13] Xiao, M. and Nagamochi, H. : An Improd Exact Algorithm for TSP in Graphs of Maximum Dgr-4. In: Thory Comput Syst, DOI /s x, [14] Xiao, M. and Nagamochi, H. : An Exact Algorithm for TSP in Dgr-3 Graphs ia Circuit Procdur and Amortization on Connctiity Structur. In: TAMC 2013, LNCS 7876, pp , [15] Xiao, M. and Nagamochi, H. : Exact Algorithms for Maximum Indpndnt St. In: ISAAC LNCS 8283, pp , ISORA IET 58 Luoyang, China, Augus1 24, 2015