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ON A LINEAR PROORAMMING-COMBINA TORIAL APPROACH TO THE TRAVELING SALESMAN PROBLEM

G. B. Da.ntzig D. R. Fulke rson S. M. Johnson

P-1281 '

Revised April l o , 19S8

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PROCESSOR: j} j_ I

ON A LINEAR PROGRAMMING-COMBINATORIAL APPROACH TO THE TRAVELING SALESMAN PROBLEM

1. Introduc tion

G. B. Dantzig D. R. Fulke rson

S. M. Johnson

P-1281 2- 14-58

- 1-

In our paper published i n JORSA in November, 1954 , entit l ed

•solution of a Large Scale Trave ling Salesman Probl em," [1] , ·.ve

chose t he well known example of find i ng t he beat way to t ou r a

set or cities, one selected f r om each of t he 48 states, in orde r

to indicate t he power of a l inear pr ograrrnning approach to t he

traveling salesman problem. A way of using linear pro~rammin~

1n conjunction with com inatoria l methods was a lso mentioned

briefly 1n [1]. However, j udg ing from t he nurnt-er of queries 1·1e

have received from readers of our previous paper , t his met r ed

waa not elaborated suffic i ently to make t he proposal c lear .

Since it is our belief t hat a l inear programming-c omr natorin~

approach a~fords a prac tical way of solv ing travel in~;-salesli'an

problema, we shal l attempt, in t h i s note, to explain t hi s

approach in more detail.

To illustrate t he metr. od we have chosen t he example dis­

cussed by L. L. &rache t [2-J in t he De cember 19 -7 issue of J O'i.SA .

Ia that note he descri bes a procedur e of success i vely i mprovin

a aolution by us ing certa in necessary conditions for opt i ma li t y .

Be points out t hat t here is no guaran tee t hat t re fina l tout

ob~ined by his approach is opt i mal . In contrast , we s ra ll

atart with his initial t our , improve i t , and give a proof l a

hia t1nal solution i s indeed op imal .

P-1281 2-14-58

-2-

2. Earachet's Example and its Solution

We shall assume that the reader is familiar with our

earlier paper [\] . See also Qfl for a general discussion. The

linear programming approach suggested in 0Ü is to start with a

tour and a small number of linear equality and inequality con—

straints that are satisfied by all tours, then use the simplex

method to move to a new basic solution. If the new solution is

not a tour, impose an additional constraint on the problem that

cuts out this solution but no tours, and again, in the new

convex set thus defined, move to an adjacent solution. At a

suitable stage in the process, it is usually advantageous to

use the estimation procedure described in [l] in conjunction

with a combinatorial analysis of undominated tours.

In Darachet*s ten-city example, we shall find that in

addition to the starting conditions on the nonnegative variables

x^ (i < J),

(1) iiXlk * liA1'2 (k"0,1 9)'

only upper bounds on certain variables will take us to a stage

where the estimation procedure and combinatorial analysis suffice

to solve the problem. Thus we will not need the additional loop

constraints described in [l] , or any constraints of a more

complicated nature. Indeed, our experience indicates that (l)

and

(2) x1J < 1,

in conjunction with combinatorial arguments, often suffice for small

problems.

P-1281 2-14-58

-}-

fbr example, T. Robacker, while at RAND, waa able t o s olve a

aeries of ten 9-city problems, each ha ving a "random distance"

matrix, by the use of (1) and (2) only. More recently an e xpe r iment

waa run by Leola Cutler on the RAND electronic comput e r JOHHHIAC·

One hundred 10-<:ity problema, each having a "random distance" matri x ,

were aolved aaauming again only conditions (1) and {2 ). I n 56~ of

the caaea the optimal solution was a tour•.

Diatancea between the ten citie s 0,1, •.. , 9 f or Barachet 's

problem are given in Table 1. Pigure 1 is his map of the ci t i es

relative to each other, the heavy line being his starting t our. To

get a starting aet of conditions with respect t o ~hich this t our is

a baaic solution, we impose, in addition t o (1), an upper bound on

x12, and add the basic variable x68 (with value 0). The pre sence

ot an upper bound on x12 is depicted in Figure 1 by a bar on link

12. Alternatively, we may think of all upper bound r e lat i ons (2 )

aa being present in the probl em, and view x12 as a "non-basi c

Y&r1able" at ita upper bound value of unity (instead of ze r o ).

81nce we ahall uae only upper bounds t o s olve t his problem , the

latter point of view will simplify our de s cription of t he solution

proceaa, and we therefore advpt it.

the tirat atep ia to compute "po t entials" vi {s impl ex multi­

pliers or prices) aatiafying

(') yi + YJ • diJ

oorreaponding to basic variables xij" These are s hown adjacent

to tbe citiea in Wigure 1. Next comput e

~iJ -

(for ze r o non-basic variables-)

(for posit i ve non-basic variables) .

1or the 100 ca.es, 56 were t ours , 40 we r e l oops , 4 were f r actional .

.. P-1281 2-H1-58

If Ö-. < 0 for all non-basic variables, the tour Is established

as optimal. Otherwise, we select some non-basic variable corres-

ponding to 6«i > 0 for entry Into the basic set. Throughout

this problem we shall use the standard criterion used In the

simplex method for this selection, I.e. we take the largest 0--

and attempt to Increase the corresponding non-basic variable

x.. If It has value zero, or decrease It If Its value Is unity,

thereby obtaining a new basic set of variables.

Using the values of ir. as shown In Figure 1, we find that

5.7 ■ 33 is maximal and thus Introduce x.7 Into the basic set

(this Is symbolized In Figure 1 by the arrowheads on link 1,7).

Adjustments In the values of basic variables corresponding to

x17 * ^ 2l 0 are 8hown 1" brackets next to the appropriate links

In Figure 1. Since XQQ - 1 -f ^ > 1, we determine that ^ • 0

and drop XQQ from the basic set.

This brings the computation to Figure 2, where new potentials

are computed as shown there, and XQ,., with ö0- ■ 33, is to be

Introduced. This time the value of XQC can be Increased to

<? « 1 without violating any of the conditions (l) and (2), at

which point we obtain a new tour (l,2,3»^,5#0,9#8,6,7), which

Is therefore 33 units shorter than the original one. TVils

leads to Figure 3, where the solid lines now correspond to the

new tour, and link 0,1 has been dropped from the figure, i.e.

XQ. has become a zero non—basic variable (alternatively, we

could have dropped x^ or x««). The variable x-. Is then

selected to enter the basic set, and x^- becomes a non-laslc

variable havlnp value 1.

P- 1281 2-1 4-58

-:r-

Two more iterations pr oduce Figures 4 and 5. At t his s tage

or the computation (Figure 5) we note a fter calculat i ng potent ials

(aee Table 2) that t he maximum ~ i j is o46 • G, and t ha t t he

only othe r positive 5i j is o36 • 1. From the discuss i on of

the estimation procedure in [1] , it f ollows t ha t any zero non­

basic var: able xij whose 5iJ ie l es s t han - 7 • o46 + o3r

must stay at ze r o value in any optimal tour , while any positive

non-basic variable \ihose is less t han - 7 must

stay at value 1 in a ny optimal tour .

In Table 2 we have tabu l a ted all 5iJ ' and from t he table

we see that, 1n addition to t he Laa l e set , t he only links wh i c~

might be used to better t he t our of Figure 5 are t hose corres-

pond1ng to

Moreover, the ~ij corres ponding to positiv non-basic va riaLl cs

are

(6) 509 •- 2 , o12 a - 23 , 04 S • - 3 , O ~p c- 24 ,

...S hence it follows t hat l inks 1 , 2 and 6 , 8 mus be in ar1y

optlMl tour.

In eaaence, t he searc for a r,e tter tou r ha s l. een reduced

to the tollowing probl em. Add t he l inks corre s pon ding to t he

llJ ot (5) to t he network sho\vn in Fi gure 5 to oh t a in Fi gure

P-1281 2-14-58

Any better tour must use only linkt of Figure 6. To compare

the length of any tour of this network with our present tour,

use the $*.: a ^«i > 0 of (5) represents the decrease in

tour length by thst amount If link i,J Is used; a 6JJ < 0

of (3) represents the Increase In tour length (by the amount

- 64i) lf ^»J l8 used; on the other hand, a 0^, < 0 of (6)

represents the Increase In tour length (by — ^«i) if 1>J

Is not used.

It Is now a simple matter to deduce that the tour

(1,2,5,4,5#0,9#8,6,7) Is optimal. We proceed to give the

argument:

(a) Links ^,8 and 1,2 must be In (as asserted before)«

since, e.g., 668 " ~ 24 < " (556 * ö46) " " 7'

(b) Link 1,7 must be In, since there are only two links

at city 1. Hence also 2,3 Is In, since otherwise we would have

a subloop (1,2,7). Thus 3,7 Is out, since otherwise the sub*

loop (1,2,3,7) would result,

(c) Next look at cities 7 and 3. There are only two

more links from 7, namely 6,7 and 7,8, one of which must

be In. Thus link 3,6 Is out, since 6,7 and 3,6 In results

In a subloop (1,2,3,6,7), whereas 7,8 and 3*6 In yields

the subloop (1,2,3,6,8,7)* It follows that link 3,^ must be In.

(d) Similarly link 4,6 Is out, as otherwise we would have

either the subloop (1,2,3,4,6,7) or (1,2,3,^,6,8,7).

(e) There Is no better tour than (1,2,3,^,5,0,9,8,6,7),

since to better It, we must use at least one of the links 3,6

or 4,6, both of which have now been eliminated from consideration.

P-1281 2-14-58

-7-

One can go on to show that this tour is uniquely optimal,

and that a next best tour is (1,7,8,6,5,9,0,4,3,2), of length

3 units greater than the optimal tour (since, from the 6*0,

tht only loss incurred is in not using link 4,5 for which

öu- ■ — 5). The fact that this tour is second best can not be

deduced merely from Figure 6 and its corresponding Ö's, how-

ever, since, for example, the tour (1,7,8,6,4,5,9,0,3,2) is

also 3 units longer than the optimal one, but uses link 0,3,

which Is not in Figure 6 (6*, - - 9), The fxicrease in length

from optimal for this latter tour is given by - 6^, ^ 6^ -

9-6-3.

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5 7 28 ( 3 )

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81 54 30 10 ( 5)

85 57 2 8 20 22 ( 6 )

8 0 63 57 72 81 63 ( 7 )

113 85 57 45 4 1 28 80 ( 8)

8 9 63 40 20 !0 28 89 40 ( 9 )

80 63 57 45 4 1 63 113 eo 4 0 ( 0 )

( I ) ( 2) ( 3) ( 4 ) ( 5) ( 6 ) ( 7 ) ( 8) ( 9 )

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P-1281 2-1U-58

3. Practicality of the Method

We hope that our primary intent, to shed more llc^t on how

one can use the simplex multipliers in a combinatorial analysis

of undominateJ tourc at some otace of the linear programming

approach, has been fulfilled hy our discussion of the example.

It 1c, our feeling, based on our experience in solving some

thirty or more problems of various oizeo, that the susgested

method affords a practical meano of computing optimal tours for

problems that are not too huge. Certainly, we do not claim to

have proved that the impooition of simple conditions like (l)

and (2) will, in every problem, make the subsequent combinatorial

analysis sicnlficunt^y easier than a direct examination of all

tours, ^ut then, of course, no one has yet proved that the

simplex method, for example, cuts down the Job of computinc linear

programs significantly—compared to the crude method of examining

all basic solution.:, say. Nonetheless, people do use the simplex

method because of successful experience with many hundreds of

nractical problems.

REFERENCES

P- 1281 2-14-58

-1}-

1. Dantzig, G. B., D. R. Fulkerson, and S. M. Johnson, "Sol u­tion of a Large Scale Traveling Salesman Problem,"

. Operations Research , 2, 393-410 (1954).

2. Barachet, L. L., "Graphic Solution of t he Traveling Salesman Problem,• Operations Research , 5 , 841-845 (1957 ).

3. Dantzig, G. B., "Disc rete-var iable Ext remum Pr ot lems ," 9perationa Research , 5 , 266-277 (1957 ).