nealing [1, 4, 5] and on the study of finite-temperature properties.

(*) Laboratoire de Physique Statistique.

(**) Laboratoire _Propre du Centre National de la Recherche _{Scientifique,} associé à l’Ecole Normale _Supérieure et à l’Université de Paris-Sud.

214 EUROPHYSICS LETRERS

Our _{analytic approach} does not use

_replicas,

but the cavity method [2]. The _cavity

equations have been written down in _[6], within the _hypothesis that there exists only one

pure state. _Adding a _{new point} i = _{0 to a}

system of N _points i _{= 1,..., N,} the

_{magnetization}

of the new _point is m0= 2A1/((A1)2 -

A 2

), where _Ak

_{= Tk0i}

_ikmand miis the _{magnetization} on

site i before the addition of the new _point. The _coupling constant is _T0i=

exp

[-03B2N03B4l0i],

with

03B2 the inverse _temperature. For a _length distribution

03C1(l)

scaling as _03C1(l) ~ lr/r! _{(l ~ 0),} 03B4 = 1/(r + 1) must be chosen to have a

good

thermodynamic limit [7].

As we are interested in the _{zero-temperature} limit _{03B2 ~} ~, it is natural to _parametrize

m i=

exp

[03B2~i],

and the _{cavity equation} for m0 simplifies at zero _temperature to

In eq. (1) we reordered the _{points i = 1, .... N} in such a _way that

N03B4l01 - ~1

N

03B4

l

02

- ~

2

... N

03B4

l

0N

- ~

N

.

Denoting

by

( )

the _average over the distribution of links, _eq. (1) implies a self-consistent

equation for the

_probability

distribution of _{the ~i:}

To write this _{self-consistency equation explicitly,} we first deduce from eq. (2) the

distribution 03A0_(~)

_{of ~}

=

N03B4l - ~

(For N finite the _integral has a _cut-off, which

_diverges

with N.) In the cavity method the N

random variables

_{0i~-~il03B4=Ni}

are independent, so that the distribution of the second

smallest of all the _{~i’s (see eq.} (1)) is _easily derived. _{Using eqs. (2),} (3), this leads to

which for

_large

N is _equal to _{P(~) =}

_(dG/d~)G(~)

_{exp [-}

_G(~)],

with

From this we can deduce the closed integral equation for G, the order _parameter function of

the TSP:

G can be

_computed

_{precisely by} iteration. Its relation with the order _parameter function defined in the replica

approach

is _complicated.

From G _(x), we now _compute the

length

of the

optimal

tour. Let us define the distribution

W. KRAUTH et al.: THE CAVITY METHOD AND THE TRAVELLING-SALESMAN PROBLEM 215

where n

ij

is the thermal average of the occupation number of link ij. At zero temperature,

the cavity equation for _n0l, written down in _[6], tells us that _n01=

n

02= _1, and _n0i= 0 (i 3).

(The points are ordered as introduced after _eq. (1).) _Using this result and _{eq. (7),} we find

after some work the following expression for L at zero _temperature:

where G is the solution _{of eq.} (6). The

_length

of the _optimal tour _{(as N ~ ~)} is then

L~N1-1/(r+1)

r

with

We summarize our theoretical results: for _any r we can solve for G (x) in eq. (6) and then

get from eqs. (8), (9) the _length of the _optimal tour and the distribution of

_lengths

of _occupied links in this tour.

In the case r = 0 (flat distances) we find

L=r=0

= 2.0415... and for r = 1 we have

r=1

= 1.8175.... This random link case with r = 1 is _an

approximation for the Euclidean TSP

in 2 dimensions.

_Following

[9], we estimate the _length of a TSP for N _{points uniformely}

distributed in the unit _square as

r=1/~203C0

= _0.7251 which is _{close to the known bounds}

[10].

A _{commonly accepted} numerical result is 0.749.

We now turn to the numerical checks of these results (restricted to the case of flat

distances r = 0, i.e.,

lij = lji

are uniform random numbers on [0,1]). The TSP

being

NP-

complete,

there are no tractable _algorithms for _solving

_large

instances of it. The _long-

standing interest into the TSP _{has, however,} led to the _following situation: there do exist

(very involved) algorithms using linear

_{programming [11]}

or branch and bound

strategies [12], the best of which are _{presently capable} of solving instances with several

hundreds of _points.

On the other _hand, a number of methods are available which

provide good

sub-optimal

tours and thus _upper bounds on the _optimal tour

_length.

_Furthermore,

_algorithms

are

known which solve «relaxed» TSP _problems,

_leading

to lower bounds on the _length of the shortest tour. It has been known for some time [12,13] that these bounds _might be _quite

tight, even for the random link TSP.

We used two well-known heuristics to determine upper and lower bounds for the TSP,

following

[13]. For the lower bound we used the

_Lagrangian

one-tree relaxation of Held and

Karp

[14] (our

implementation

follows ref. _[12]). This relaxation solves the _problem

where the min (for fixed 03BB) is taken on certain _spanning

_graphs

_containing

_exactly

one cycle

(cf. [12]). The upper bound on the optimal TSP-tour was determined with the

Lin-Kernighan

algorithm, which was

_implemented

_using the

_original

ref. [15]. Both methods are _quite fast:

for an instance of 400 _points we can calculate both the _{upper and} the lower bound in less than

1 minute on a Convex C1 _computer.

Our numerical results for the bounds are _given in _fig. 1. In both cases we find _good

agreement with a _dependence linear in 1/N. As N ~ ~ we _get 2.039 ± 3·10-3

Fig. 1. - _Upper and lower bounds for the _optimal tour _length of the random link TSP (flat distances

l

ij~ _[0,1]). The _upper curve was _{produced using} the Lin-Kernighan algorithm, averaged over, e.g., 360

samples for N = _{800, 5000 samples} for _N = _{25. The} lower _curve results from the _{Langrangian one-tree}

relaxation _{with, e.g.,} 3000 _samples (N = _{400), 20000 samples} (N = _25).

in fact it coincides within our 2o/oo level of _precision with the results of the one-tree

relaxation.

We _compared also the _probability distribution of links both of the _{Lin-Kernighan} solution and of the one-tree relaxation with the theoretical _prediction. To eliminate _{sample-to-sample}

fluctuations, we rescalded the _lengths of _{occupied links,} in each _{solution, by} the average

length of the _occupied links. The distribution of these rescaled _lengths, for N = 800, _was

then _{averaged over, e.g., 100 Lin-Kernighan} tours and _compared to the theoretical

prediction

The _agreement is excellent, see _fig. 2. We found _equally

good

_agreement with the one-tree

relaxation. We do not know _why a _{simple length rescaling} makes _agree the _{Lin-Kernighan}

solution (which we _suspect to be 10% above _optimal) the one-tree relaxation (which does not

produce a _tour) and the theoretical result for the _optimal solution.

The simulations

_give

the numerical value of the _optimal TSP tour _length, albeit on a

rather crude level. _They also illustrate the _power of the _{Lin-Kernighan algorithm, compared}

to _simpler heuristics which up to now have all _given bounds on the

optimal

tour _length

Fig. 2. - _Integrated distribution function I(x) for the _occupied links in TSP solutions vs. reduced

length variable x (x = 1 _{corresponds to} the _{mean length} of _occupied links in each _sample). The _curve

gives the theoretical _result, the dotted line the numerical results for _{Lin-Kernighan} solutions (100

samples at N = _800).

tree relaxation and the _proposed value for the _optimal tour _suggests to us the _exciting

conjecture that the two values _may indeed be identical. It would be _interesting to _apply the

cavity method _directly to the one-tree relaxation.

We found some _support for the _conjecture in additional simulations using the Lin-

Kernighan

heuristics _repeatedly on each _sample and

_keeping

the lowest result. In this way

we can lower the upper bound _considerably for finite _N, but not for N~~. (Preliminary

results with an exact _{algorithm, by}

_Kirkpatrick

_{[16], agree} with an optimal TSP length

around 2.04.)

In conclusion we remark that the _{recently developed cavity} method of statistical _physics

in combination with rather traditional tools of _Operations Research have led to a better

understanding of the random link TSP.

***

Stimulating discussions with G. PARISI and with S. KIRKPATRICK are

_gratefully

acknowledged. The simulations on the Convex C1 were _{supported by} GRECO 70

«Expérimentation Numérique». WK _acknowledges financial _{support by Studienstiftung} des

deutschen Volkes.

REFERENCES

[1] KIRKPATRICK _S., GELATT C. D. and VECCHI M. P., Science, 220 (1983) 671.

[2] MÉZARD _M., PARISI G. and VIRASORO M. _{A., Spin} Glass Theory and _Beyond (World Scientific, Singapore) 1987.

[3] MÉZARD M. and PARISI _G., J. _Phys. (Parts), 47 (1986) 1285.

[4] KIRKPATRICK S. and TOULOUSE G., J. _{Phys. (Paris),} 46 (1985) 1277.

[5] SOURLAS _{N., Europhys. Lett.,} 2 (1986) 919.

218 EUROPHYSICS LETTERS

[7] VANNIMENUS J. and MÉZARD _M., J. _{Phys. (Paris) Lett.,} 45 (1984) L-1145.

[8] BASKARAN _G., Fu Y. and ANDERSON P. _W., J. Stat. Phys., 45 (1986) 1.

[9] MÉZARD M. and PARISI G., LPTENS _{preprint 88/21,} to _appear in J. _{Phys. (Paris).}

[10] BEARDWOOD J., HALTON J. H. and HAMMERSLEY J. _M., Proc. _Cambridge Philos. _Soc., 55 (1959)

299.

[11] PADBERG M. W. and GRÖTSCHEL M., in The _Traveling Salesman Problem, edited _by E. L.

LAWLER, J. K. LENSTRA, A. H. G. RINNOOY KAN and D. B. SHMOYS (Wiley, Chichester) 1985, p. 307.

[12] BALAS E. and TOTH _P., in The _Traveling Salesman Problem, op. cit. [11].

[13] JOHNSON _D., Nature _(London), 330 (1987) 525.

[14] HELD R. M. and KARP R. _{M., Oper. Res.,} 18 (1970) 1138.

[15] LIN S. and KERNIGHAN B. _{W., Oper. Res.,} 21 (1973) 498.

[16] KIRKPATRICK _S., Talk at the _Cargèse school «Common trends in _particle and condensed matter

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