Case Study: Solving a Classic TSP Instance

Transcription

1 Case Study: Solving a Classic TSP Instance Thomas Sauerwald Easter 20

2 Outline Introduction Solving via LPs and Branch & Bound Case Study: Solving a Classic TSP Instance Introduction 2

3 The Original Article () SOLUTION OF A LARGE-SCALE TRAVELING-SALESMAN PROBLEM* G. DANTZIG, R. FULKERSON, AND S. JOHNSON The Rand Corporation, Santa Monica, California (Received August, ) It is shown that a certain tour of cities, one in each of the 8 states and Washington, D. C., has the shortest road distance. THE TRAVELING-SALESMAN PROBLEM might be described as follows: Find the shortest route (tour) for a salesman starting from a given city, visiting each of a specified group of cities, and then returning to the original point of departure. More generally, given an n by n symmetric matrix D= (di), where doi represents the 'distance' from I to J, arrange the points in a cyclic order in such a way that the sum of the dj between consecutive points is minimal. Since there are only a finite number of possibilities (at most (n - )!) to consider, the problem is to devise a method of picking out the optimal arrangement which is reasonably efficient for fairly large values of n. Although algorithms have been devised for problems of similar nature, e.g., the optimal assignment problem,"8 little is known about the traveling-salesman problem. We do not claim that this note alters the situation very much; what we shall do is outline a way of approaching the problem that sometimes, at least, enables one to find an optimal path and prove it so. In particular, it will be shown that a certain arrangement of cities, one in each of the 8 states and Washington, D. C., is best, the djj used representing road distances as taken from an atlas. * HISTORICAL NOTE: The origin of this problem is somewhat obscure. It appears Case to Study: have been Solving discussed a Classic informally TSP Instance among mathematicians Introduction at mathematics

4 The 2 () Cities DANTZIG, FULKERSON, AND JOHNSON In order to try the method on a large problem, the following set of cities, one in each state and the District of Columbia was selected:. Manchester, N. HI. 8. Carson City, Nev.. Birmingham, Ala. 2. Montpelier, Vt.. Los Angeles, Calif.. Atlanta, Ga.. Detroit, Mich. 20. Phoenix, Ariz.. Jacksonville, Fla.. Cleveland, Ohio 2. Santa Fe, N. M.. Columbia, S. C.. Charleston, W. Va. 22. Denver, Colo.. Raleigh, N. C.. Louisville, Ky.. Cheyenne, Wyo.. Indianapolis, Ind.. Omaha, Neb.. Richmond, Va.. Washington, D. C. 8. Chicago, Ill.. Boston, Mass.. Milwaukee, Wis. 2. Kansas 2. Portland, Me. 0. Minneapolis, Minn. 2. Topeka, Kans. A. BaltimoreA Md.. Des Moines, Iowa City, Mo. 2. Bismark, N. D.. Oklahoma City, Okla. B. Wilmington, Del.. Helenar, MNt. 2. Dallas, Tex. C. Philadelphia, Penn.. Seattle, Wash. 0. Little Rock, Ark. D. Newark, N. J.. Portland, Ore.. Memphis, Tenn. E. New York, N. Y.. Boise, Idaho. Jackson, Miss. F. Hartford, Conn.. Salt Lake City, Utah. New Orleans, La. G. Providence, R. I. The reason for picking this particular set was that most of the road distances between them were easy to get from an atlas. The triangular table of distances between these cities (Table I) is part of the original one prepared by Bernice Brown of The Rand Corporation. It gives dj= K (di; - ) (IJ =, 2, *, 2), where dii is the road distance in miles between I and J. The di have been rounded to the nearest integer. Case Study: Solving a Classic TSP Instance Introduction

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6 i ci M) ci f- 'I.? 0 kf. C m Road Distances Hence this is an instance of the Metric TSP, but not Euclidean TSP. O Nm m 'i s) ooo OO QtN 0dt y 000 C~, i Z o o \O 8-Q-) C\ Mc S~~~~~~~diF~~~~~~~" r*to "_ Ge Vd iu(" N~h.- Neez F#0 ) -mmetm mbo mm C'Ocim -c -n --No 000 o 0 0w'0000. t-0 C",'- r- 00 'T0 CON 00 O sf-c-a f- ' iti c + ) Oa ' %,O 0 0O ' tt-\ 00\ )O c\ 0 0 ONC ON -\ C> Cf- O ooci C",00 '-ci'-o C-- ci 't'-ci f--f-- 0) ci' Y f ' 00 f-i 0 ' 0- i- i concon 0C - '-f 0 O Ci \ - 'I- i O \co O 00 0)0 cn ')ci '-'- i-i O- ON-C f--"-, '-) -'c ci (' ''c )0 0'-- '-i- -vi -l0'" 'CA~ici c 0c 0 Xci 0- ci 0 't0"0 0-'-800c C"Ci 0, 0 C')'t ', 00 C,) i'c cn tl ''' -i i'c f cici CAci I- \-.-0 -N#- CA -, c N.t '' C'. in-tc'c) ' ci ci ~~~ ~ 't'0 0-C o~~~~~~~~~~~~~~~~~~~~~~~c HO r- - r tn \,O m C CA -n+< c n t> f-o- 0 C, -'t O0 0 C0 ' t --C-) 0 o 00 0\, - OCA O 'ci00 t 00 0 t M ci C')'-ci f--- C''- ' t ' t o ci o) V '-cit ' t't 0'-t c')'-+w'c V ) t ci c O MOO - ', O O V 0\, C) ',) '.OONNC) 00 ", -- -C C) C'\00 0 O O' Cl Cn O t 0 - O -??? r 0M 'H'tc ci i Cn Ot CY'o -I ' ) CA Ci ci 000? W 0 Cc " ' O - -00N 0 M'- Mq0-~ \o 0 c 0N cc-\,on'.0 "C ci OON ci \,O 00 \000 t - 't~~~ci00cic')c00c~~ic0 '-~ \OCic CAic cn -,i,-t tj c 0 \O -CO cion rc ' o 't00 CA )o n ON O C' )\00 rt ci O ' rcic 00- o"- o '- c ci \i' C) "0f M ON CIA O 0Ci-' r0 00 C0 ci C 00 0-ON I- c i ' - 0 -c c 0 )'c\. to X o 0) Qr0 F- 0 C')'t t- C= 0 f\ 0 f- C% f 00 -* c')t C " 'cxio ) --\O t -f ON 0 ci c c cn t - 'to ) - I C, ) 0) F- H -00 0)0 ON ONOC= 0 - ci 'tci)'-ci00 00 f--i c o it N ci m c co n q- tn W 0e ci'- O~ci 0-?O ci ~i't"0 'ciono ci c I-,0 ON- '-cic'o -,o OC) CO C tn Cn \ d -- m0 00 ci mm00 't 'tci) ON i00 fm cn ' 0 '0 ' cm C\ m 0 rb t- -t r\. o moo CP\ 0 0 UN CP\00 r c 0 i 0 't ' 0 N, ' b0 ""O 0) ci ci '-'~0 '-ci'~0 'tf ONO c -o o -,, ci ~cif-~0 t r O 0 o 0 C C 0 c 0 F i ', 00 " - X~~~~~~~~~~~~~~~~~~~~- *0 C' - )+ mo v00 c tl HF CA C n,oo C t- o c c _I i c00 ron Co\ r' " 0 r \.O ci t- ',tr ~\C 0 wm r0 00 -\ "O 0-00X-= \O'-ci00 ~-A tit- ci 0N00\0 CA "C, 00 f c 0N0c'c ci 0 ~~ d )'t'ci0000 -~~~~~~~ci'c I0~0 cic -p\ - 0 ci 0 0 O~~~~(~~f--~~ '-cic~~ 0',tci0',00 O~~~~c-) ~~ t-0',tfl\,~~~~ t O 0 0)' 0 C, cd o-o coz\~ oo O -,o 0 0"0o ft+m? r- 00 s SA? ( Cit N', '' 0 c cn H ci00 0 Z t '.' t0 0 t"-'t00 a - H ci) 0 i tc Z-000 t ci 00 V-. ocall CS- tn W o o? >t 00 I M M f- r > 00 C O He % On m?i? 00 0tCI 0 S~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ r Xbt cn Xe\o \0 to ??o C, O 0 to cic~~ ci t es ) c 00m000-'-'M~ c 00c i t'thf?ci.00' 000 cn L"O st 0, i ~ ~-~0) 00 ci C) CS) j.0 t~ 'tci\o '-ci t0-~ ~, t 0- H 00 0C i Case Study: Solving a Classic TSP Instance Introduction 00) Q o an ~ b on H X? O H ct + tn a> a> 0 t? t (~~~~~~~n Itm-<. r -\o,o Co ~O roo e.-w H F,, E mn C- > C, + C > > t + \- + Qo? \0 +?t ac O s (0 ic it i0 t 00 I-, ~~~~~~~~~~~~~~~ e? 0 \o 0 o c - O t" 00 z >? ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C,> e?-\, roo +r" 0 t m+mo Q' > tw#)b 0 > 0 n cn t- 0 S~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i c Ct'I t n + + t-oo 0 N ]00 C~ t Q >~~~~~~~~~~~~~~~~~~~~~~~~~ O. 0 q O 00 ol o e cn C- I- tr\ cl cn cn -t 00 CNC o rn C CC, C-) f 0 00\,O n 0 n\ c,o tn 0 0 \ ' 00 n 00 e \0) co 0O

7 '- 0 The (Unique) Optimal Tour ( Units 2, miles) Ul)~~~~~ I X C) * ~~~A./H fcs CC 0~~~~~~~~~~0 A _~~~~~~~~~ V E M < * 0 ~ ~ ~ 0 Case Study: Solving a Classic TSP Instance Introduction

8 Modelling TSP as a Linear Program Idea: Indicator variable x(i, j), i > j, which is one if the tour includes edge {i, j} (in either direction) Case Study: Solving a Classic TSP Instance Introduction

9 Modelling TSP as a Linear Program Idea: Indicator variable x(i, j), i > j, which is one if the tour includes edge {i, j} (in either direction) minimize i= i j= c(i, j)x(i, j) subject to j<i x(i, j) + j>i x(j, i) = 2 for each i 0 x(i, j) for each j < i Case Study: Solving a Classic TSP Instance Introduction

10 Modelling TSP as a Linear Program Idea: Indicator variable x(i, j), i > j, which is one if the tour includes edge {i, j} (in either direction) minimize i= i j= c(i, j)x(i, j) subject to j<i x(i, j) + j>i x(j, i) = 2 for each i 0 x(i, j) for each j < i Constraints x(i, j) {0, } are not allowed in a LP! Case Study: Solving a Classic TSP Instance Introduction

11 CPLEX Case Study: Solving a Classic TSP Instance Introduction 8

12 Outline Introduction Solving via LPs and Branch & Bound Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

13 Iteration : Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 0

14 Iteration : Objective, Eliminate Subtour, 2,, Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 0

15 Iteration : Objective, Eliminate Subtour, 2,, Disallow subtour (, 2, 2, ) by adding this constraint to the LP: x(2, ) + x(, ) 2+ x(2, ) + x(, 2) + x(2, 2) + x(2, ) Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 0

16 Iteration : Objective, Eliminate Subtour, 2,, Disallow subtour (, 2, 2, ) by adding this constraint to the LP: x(2, ) + x(, ) 2+ x(2, ) + x(, 2) + x(2, 2) + x(2, ) Equivalent to: S = {, 2, 0, 2}, 0 x(max(i, j), min(i, j)) 2 i S,j V \S Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 0

17 Iteration 2: Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

18 Iteration 2: Objective, Eliminate Subtour Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

19 Iteration : Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 2

20 Iteration : Objective 8, Eliminate Subtour,, 2, Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 2

21 Iteration : Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

22 Iteration : Objective 82., Eliminate Small Cut by Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

23 Iteration : Objective 82., Eliminate Small Cut by Tour has to include at least two edges between S = {,,,, } and V \ S: x(max(i, 2 j), min(i, j)) 2. 2 i S,j V \S Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

24 Iteration : Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

25 Iteration : Objective 8, Eliminate Subtour 0,, Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

26 Iteration : Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

27 Iteration : Objective 8, Eliminate Subtour Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

28 Iteration : Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

29 Iteration : Objective 88, Eliminate Subtour Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

30 Iteration 8: Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

31 Iteration 8: Objective, Branch on x(, 2) Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

32 Iteration, Branch a x(, 2) = : Objective (Valid Tour) Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 8

33 Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

34 Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 20

35 Iteration 0, Branch b x(, 2) = 0: Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 2

36 Iteration 0, Branch b x(, 2) = 0: Objective 0 Branch & Bound procedure would stop here, since value of the best LP solution for x(, 2) = 0 is worse than a previously found tour Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 2

37 Iteration 0, Branch b x(, 2) = 0: Objective 0 Branch & Bound procedure would stop here, since value of the best LP solution for x(, 2) = 0 is worse than a previously found tour Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 2

38 Iteration, Branch b continued (just for fun...): Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 22

39 Solving Progress : LP solution Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

40 Solving Progress : LP solution Eliminate Subtour, 2,, 2 Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

41 Solving Progress : LP solution 2: LP solution Eliminate Subtour, 2,, 2 Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

42 Solving Progress : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

43 Solving Progress : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

44 Solving Progress : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

45 Solving Progress : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

46 Solving Progress : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

47 Solving Progress : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

48 Solving Progress : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

49 Solving Progress : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

50 Solving Progress : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

51 Solving Progress : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour : LP solution 88 Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

52 Solving Progress : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour : LP solution 88 Eliminate Subtour Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

53 Solving Progress : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour : LP solution 88 Eliminate Subtour 8: LP solution Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

54 Solving Progress : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour : LP solution 88 Eliminate Subtour 8: LP solution x(, 2) = Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

55 Solving Progress : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour : LP solution 88 Eliminate Subtour 8: LP solution x(, 2) = : Valid tour Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

56 Solving Progress : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour : LP solution 88 Eliminate Subtour 8: LP solution x(, 2) = x(, 2) = 0 : Valid tour Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

57 Solving Progress : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour : LP solution 88 Eliminate Subtour 8: LP solution x(, 2) = x(, 2) = 0 : Valid tour 0: LP solution 0 Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

58 Solving Progress : LP solution 2: LP solution : LP solution 8 : LP solution 82. : LP solution 8 : LP solution 8 : LP solution 88 Eliminate Subtour, 2,, 2 Eliminate Subtour Eliminate Subtour,, 2, 2 Eliminate Cut Eliminate Subtour 0,, 2 Eliminate Subtour Eliminate Subtour 8: LP solution x(, 2) = x(, 2) = 0 : Valid tour 0: LP solution 0 Cut branch, since LP solution worse than current best possible tour. Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

59 Solving Progress : LP solution 2: LP solution : LP solution 8 : LP solution 82. : LP solution 8 : LP solution 8 : LP solution 88 Eliminate Subtour, 2,, 2 Eliminate Subtour Eliminate Subtour,, 2, 2 Eliminate Cut Eliminate Subtour 0,, 2 Eliminate Subtour Eliminate Subtour 8: LP solution x(, 2) = x(, 2) = 0 : Valid Optimal tour 0: LP solution 0 Cut branch, since LP solution worse than current best possible tour. Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

60 Iteration 8: Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

61 Iteration 8: Objective What about choosing a different branching variable? Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

62 Solving Progress (Alternative Branch ) : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour : LP solution 88 Eliminate Subtour 8: LP solution Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

63 Solving Progress (Alternative Branch ) : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour : LP solution 88 Eliminate Subtour 8: LP solution x(8, ) = x(8, ) = 0 :??? 0:??? Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

64 Alternative Branch : x(8, ), Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 2

65 Alternative Branch : x(8, ), Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 2

66 Alternative Branch a: x(8, ) =, Objective 0 (Valid Tour) Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 2

67 Alternative Branch b: x(8, ) = 0, Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

68 Solving Progress (Alternative Branch ) : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour : LP solution 88 Eliminate Subtour 8: LP solution x(8, ) = x(8, ) = 0 : valid tour 0 0: LP solution 8 Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 2

69 Solving Progress (Alternative Branch 2) : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour : LP solution 88 Eliminate Subtour 8: LP solution Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 0

70 Solving Progress (Alternative Branch 2) : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour : LP solution 88 Eliminate Subtour 8: LP solution x(2, 22) = x(2, 22) = 0 :??? 0:??? Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound 0

71 Alternative Branch 2: x(2, 22), Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

72 Alternative Branch 2: x(2, 22), Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

73 Alternative Branch 2a: x(2, 22) =, Objective 08 (Valid tour) Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

74 Alternative Branch 2b: x(2, 22) = 0, Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

75 Solving Progress (Alternative Branch 2) : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour : LP solution 88 Eliminate Subtour 8: LP solution x(2, 22) = x(2, 22) = 0 : valid tour 08 0: LP solution. Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

76 Solving Progress (Alternative Branch ) : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour : LP solution 88 Eliminate Subtour 8: LP solution Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

77 Solving Progress (Alternative Branch ) : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour : LP solution 88 Eliminate Subtour 8: LP solution x(2, ) = x(2, ) = 0 :??? 0:??? Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

78 Alternative Branch : x(2, ), Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

79 Alternative Branch : x(2, ), Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

80 Alternative Branch a: x(2, ) =, Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

81 Alternative Branch b: x(2, ) = 0, Objective Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

82 Solving Progress (Alternative Branch ) : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut : LP solution 8 Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour : LP solution 88 Eliminate Subtour 8: LP solution x(2, ) = x(2, ) = 0 : LP solution. 0: LP solution 8 Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

83 Solving Progress (Alternative Branch ) : LP solution Eliminate Subtour, 2,, 2 2: LP solution Eliminate Subtour : LP solution 8 Eliminate Subtour,, 2, 2 : LP solution 82. Eliminate Cut Not only do we have to explore (and branch further in) both subtrees, : LP solution 8 but also the optimal tour is in the subtree with larger LP solution! Eliminate Subtour 0,, 2 : LP solution 8 Eliminate Subtour : LP solution 88 Eliminate Subtour 8: LP solution x(2, ) = x(2, ) = 0 : LP solution. 0: LP solution 8 Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

84 Conclusion (/2) How can one generate these constraints automatically? Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

85 Conclusion (/2) How can one generate these constraints automatically? Subtour Elimination: Finding Connected Components Small Cuts: Finding the Minimum Cut in Weighted Graphs Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

86 Conclusion (/2) How can one generate these constraints automatically? Subtour Elimination: Finding Connected Components Small Cuts: Finding the Minimum Cut in Weighted Graphs Why don t we add all possible Subtour Eliminiation constraints to the LP? Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

87 Conclusion (/2) How can one generate these constraints automatically? Subtour Elimination: Finding Connected Components Small Cuts: Finding the Minimum Cut in Weighted Graphs Why don t we add all possible Subtour Eliminiation constraints to the LP? There are exponentially many of them! Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

88 Conclusion (/2) How can one generate these constraints automatically? Subtour Elimination: Finding Connected Components Small Cuts: Finding the Minimum Cut in Weighted Graphs Why don t we add all possible Subtour Eliminiation constraints to the LP? There are exponentially many of them! Should the search tree be explored by BFS or DFS? Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

89 Conclusion (/2) How can one generate these constraints automatically? Subtour Elimination: Finding Connected Components Small Cuts: Finding the Minimum Cut in Weighted Graphs Why don t we add all possible Subtour Eliminiation constraints to the LP? There are exponentially many of them! Should the search tree be explored by BFS or DFS? BFS may be more attractive, even though it might need more memory. Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

90 Conclusion (/2) How can one generate these constraints automatically? Subtour Elimination: Finding Connected Components Small Cuts: Finding the Minimum Cut in Weighted Graphs Why don t we add all possible Subtour Eliminiation constraints to the LP? There are exponentially many of them! DANTZIG, FULKERSON, AND JOHNSON Should the It search can be shown tree be by explored introducing byall BFS links orfor DFS? which ai2 - A that x BFS may is the beunique moreminimum. attractive, There evenare though only itsuch might links need in addition more to memory. those shown in Fig., and consequently all possible tying tours were enumerated without too much trouble. None of them proved to be as good as x. CONCLUDING REMARK It is clear that we have left unanswered practically any question one might pose of a theoretical nature concerning the traveling-salesman problem; however, we hope that the feasibility of attacking problems involving a moderate number of points has been successfully demonstrated, and that perhaps some of the ideas can be used in problems of similar nature. REFERENCES. W. W. R. BALL, Mathematical Recreations and Essays, as rev. by H. S. M. Coxeter, th ed., Macmillan, New York,. Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

91 Conclusion (2/2) Eliminate Subtour, 2,, 2 Eliminate Subtour Eliminate Subtour 0,, 2 Eliminate Subtour Eliminate Subtour Eliminate Cut Eliminate Subtour,, 2, 2 Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound

92 Conclusion (2/2) Eliminate Subtour, 2,, 2 Eliminate Subtour Eliminate Subtour 0,, 2 Eliminate Subtour 8 DANTZIG FULKERSON, AND JOHNSON Eliminate Subtour use the remaining admissible links. By extending this type of combin- Eliminate atorial Cut argument to the range of values of the 'slack' variables yk, it is Eliminate often possible Subtour at, an earlier, 2, stage 2 of the iterative algorithm to rule out so many of the tours that direct examination of the remaining tours for minimum length is a feasible approach. THE -CITY PROBLEM* The optimal tour x is shown in Fig.. The proof that it is optimal is given in Fig.. To make the correspondence between the latter and its programming problem clear, we will write down in addition to 2 relations in non-negative variables (2), a set of relations which suffice to prove that D(x) is a minimum for L We distinguish the following subsets of the 2 cities: Si= {, 2,, 2 S=,,, S2i =,,, S=,,,, S ={, 2,,, 2, 0,..., 2} S{=,, 2, 2. S=, 2,...,} Except for two inequalities which we will discuss in a moment, the programming problem may now be written as the following relations:t Case Study: Solving a Classic TSP Instance Solving via LPs and Branch & Bound