Characterisation of the probabilistic travelling salesman problem

10 Pages·0.22 MB·English

by  Neill E. Bowler

Characterization of the probabilistic traveling salesman problem
Neill E. Bowler
∗
Department of Physics, University of Warwick, Coventry, CV4 7AL, England and
Met Office, Maclean Building, Crowmarsh-Gifford, Oxfordshire OX10 8BB, England
Thomas M. A. Fink
†
3 CNRS UMR 144, Institut Curie, 75005 Paris, France and
0 Theory of Condensed Matter, Cavendish Laboratory, Cambridge, CB3 0HE, England
0
2 Robin C. Ball
‡

n Department of Physics, University of Warwick, Coventry, CV4 7AL, England
a (Dated: February 2, 2008)
J
We show that Stochastic Annealing can be successfully applied to gain new results on the Prob-
0
abilistic Traveling Salesman Problem (PTSP). The probabilistic “traveling salesman” must decide
2
on an a priori order in which to visit n cities (randomly distributed over a unit square) before

learning that some cities can be omitted. We findthe optimized average length of the pruned tour
]
h follows E(L¯pruned) = √np(0.872 0.105p)f(np) where p is the probability of a city needing to
−
p be visited, and f(np) 1 as np . The average length of the a priori tour (before omitting
- → → ∞
p any cities) is found to follow E La priori = npβ(p) where β(p) = 1.25 0.182ln(p) is measured
m −
for 0.05 p 0.6. Scaling argum(cid:16)ents and(cid:17)indirect measurements suggest that β(p) tends towards
≤ ≤ p
o a constant for p < 0.03. Our stochastic annealing algorithm is based on limited sampling of the
c prunedtourlengths,exploitingthesamplingerrortoprovidetheanalogueofthermalfluctuationsin
. simulated(thermal)annealing. Themethodhasgeneralapplicationtotheoptimizationoffunctions
s
c whose cost to evaluate rises with theprecision required.
i
s
y PACSnumbers: 02.60.Pn,02.70.Lq
h
p
[ I. INTRODUCTION where G is some monotonic function of x and α is a
parameter. G is not known directly, but can only be
2
estimated. Theirtechniquetosolvethisproblemiscalled
Manyrealsystemspresentproblemsofstochasticopti-
v
stochastic approximation, and a number of variants of
3 mization. These include communications networks, pro-
this scheme have since been developed [4, 5, 6, 7].
2 tein design [1] and oil field models [2], in all of which
0 uncertainty plays a central role. We will consider the Stochastic optimization is a more general situation
1 case where the outcome g(x,ω) depends not only on pa- since the function to be minimized may have many lo-
1 rameters x to be chosen, but also on unknowns ω. We cal minima. We may classify the techniques to solve
0 can only average with respect to these unknowns, aim- stochasticoptimizationproblemsinto twoclasses- exact
0
/ ing to find the ‘solution’ x which optimises the average methods and heuristics. Heuristics are more appropriate
s outcome. Thus we seek to find x X which minimizes toNP-completeproblems,andthesearetheproblemson
c ∈ which we focus in this paper. A number of heuristics
i
s already exist to tackle stochastic optimization problems
y g¯(x)= g(x,ω)f(ω)dω (1)
[8, 9, 10, 11, 12]. Many of these are developments from
h Z
simulated annealing [13, 14, 15, 16], which has been
p
: where X is the solution space of the problem and f(ω) shown[17]tosolvestochasticoptimizationproblemswith
v
is the probability distribution of the uncertain variables. probability1,providedg¯(x)canbeestimatedwithpreci-
i
X Stochastic optimization was borne out of an idea by siongreaterthanO(t−γ)fortimestept,whereγ >1. A
Robbins&Monro[3]. They consideredsolvingtheprob- number of authors [14, 15, 16, 18] have used a modified
r
a lem of finding simulated annealing algorithm in which the acceptance
probability is modified to take some account of the pre-
G(x)=α (2) cision of the estimates of g¯(x), and in these cases there
are a number of convergence results [16, 18].
Stochasticannealing[1]isamodifiedsimulatedanneal-
ing algorithm which differs from the above approaches
∗Electronic address: [email protected]; in two key ways. Firstly the noise present in estimates
URL:http://uk.geocities.com/neill_bowler is positively exploited as mimicking thermal noise in a
Electronic address: [email protected];
† slowcooling, asopposedtobeingregardedassomething
URL:http://www.tcm.phy.cam.ac.uk/~tmf20/
whose influence should be minimised from the outset.
Electronic address: [email protected];
‡
URL:http://www.phys.warwick.ac.uk/theory/ Secondly, stochastic annealing can be modified to give
2
exact simulation of a thermal system. Although this is The original traveling salesman problem (TSP) is to
not specifically ruled out by the earlier approaches, no find the shortest tour around n cities, in which eachcity
attempt has been made with them to satisfy this condi- isvisitedonce. Forsmallnumbersofcitiesthisisaneasy
tion. task, but the problem is NP-complete: it is believed for
In stochastic annealing we estimate g¯(x) by taking large n that there is no algorithm which can solve the
r repeated, statistically independent, measurements of probleminatimepolynomialinn. Considerationofthe
g(x,ω) each of which we call an instance. For the im- travelingsalesmanproblembeganwithBeardwoodet al.
plementation of stochastic annealing used in this paper [23]. They showed that in the limit of large numbers of
we accept all moves for which one estimate (based on citieswhicharerandomlydistributedontheunitsquare,
r instances) for a new state is more favourable than an the optimal tour length (L ) follows [24]
TSP
equivalent estimate for the old. This simple procedure
E(L )=β √n+α (7)
does not exactly simulate a thermal system, where the TSP TSP
acceptance probabilities should obey
where β and α are constants. Here and below E(L)
TSP
denotesthequantityLaveraged,afteroptimization,with
P
A→B =e−β∆µ (3) respect to different city positions, randomly placed on
P
B→A the unit square. Numerical simulation [25] gives βTSP =
where β = 1 and ∆µ is the exact difference in g¯(x) 0.7211(3) and α = 0.604(5) as estimates when n 50.
kBT Significant divergence from this behaviour is foun≥d for
between states A and B. However if we assume that
n 10, but numerical estimates can be found quickly
our estimate change is Gaussiandistributed aroundg¯(x)
≤
with standard deviation σ , where r is the number of (see appendix).
√r The probabilistictravelingsalesmanproblem(PTSP),
instances used for each estimate, then it follows that the
introducedbyJaillet[26,27],is anextensionofthe trav-
acceptance probability is [1]
elingsalesmanproblemtooptimizationinthefaceofun-
known data. Whereas all of the cities in the TSP must
1 √r∆µ
PG = 1 erf . (4) be visited once, in the PTSP each city only needs to be
A→B 2(cid:20) − (cid:18) √2σ (cid:19)(cid:21) visited with some probability, p. One first decides upon
the order in which the cities are to be visited, the ‘a pri-
The approximation to a thermal acceptance rule is then
ori’ tour. Subsequently, it is revealed which cities need
quite good since
to be visited, and those which do not need to be visited
ln PPAB→BA =ln 11+−eerrff √√√rr2∆∆σµµ (5) athreesckitiipepseadretotolebaeveviasi‘tperdunisedprteosuerrv’.edThweheonrdperruinninwghsicuh-
(cid:16) → (cid:17) β(cid:18)∆µ (cid:0) √42σπ((cid:1)β(cid:19)∆µ)3 ... perfluouscities. Theobjectiveistochoseanaprioritour
≃− G −(cid:0) 4−8 (cid:1) G − which minimizes the average length of the pruned tour.
It is clear from figure 1 that near optimal a priori tours
where
may appear very different for different values of p.
√8r In our terminology, the average pruned tour length is
β = (6)
G √πσ averagedoverallpossibleinstancesofwhichcitiesrequire
to be visited. This was given by Jaillet as [26]
identifiestheequivalenteffectivetemperature. Thesmall
n 2
coefficient (≃0.02)ofthe cubic term in eq. 5 makes this L¯ = − p2(1 p)q L(q) (8)
a rather good approximationto true thermal selection. pruned − t
Increasingsamplesizer meansthatwearemorestrin- Xq=0
gent about not accepting moves that are unfavourable, where
equivalent to lowering the temperature, which is quan-
n
tified by eq. 6 for the Gaussian case. As with standard L(q) = d(j,1+(j+q) ) (9)
simulatedannealing[19,20,21],thequestionofprecisely t modn
j=1
whatcoolingscheduletouseremainssomethingofanart. X
is the sum of the distances between each city and its
(q+1)th following city on the a priori tour, and the fac-
II. PROBABILISTIC TRAVELING SALESMAN tors p2(1 p)q in the preceding equationsimply give the
PROBLEM (PTSP) probabilit−ythat any particular spanskippingq cities oc-
curs in the pruned tour. Jaillet’s closedform expression
We adopt the PTSP as a good test-bed amongst for the average pruned tour length renders the PTSP to
stochastic optimizationproblems,in muchthe same way some extent accessible as a standard (but still NP com-
as the TSP has been considered a standard amongst de- plete)optimizationproblem,andprovidessomecheckon
terministic optimization problems. The PTSP falls into the PTSP results by stochastic optimization methods.
the class of NP-complete problems [22], and the TSP is It has been conjectured [22] that, in the limit of large
a subset of the PTSP. n, the PTSP strategy is as good as constructing a TSP
3
tour on the cities requiring a visit, the re-optimization this to be false by comparisonto a space-filling curve al-
strategy. This would mean that gorithm which is generally superior as n . Such an
algorithmwasintroducedbyBartholdiet→al.∞[33]usinga
E L¯ technique based on a Sierpinski curve.
pruned
lim =β , (10)
n→∞ (cid:0)√np (cid:1)! TSP twFoocrittihese baneignuglanresaorretstwnitehighnbpo≫urs1o,nthteheprporbuanbeidlittyouorf
where E(L¯ ) is the pruned tour length further av- will be vanishingly small for cities which are separated
pruned from each other by a large angle on the a priori tour.
eraged over city positions after optimisation, which we
Thismeansthatonlycitiesthatareseparatedbyasmall
will refer to as the expected pruned tour length. Figure
anglecontributesignificantlytoeq. 8. Thusforanncity
2 shows the expected pruned tour length divided by the
tour chosen by angular sort, we may approximate
expectedre-optimizedtourlength. Since this quantity is
tending towards a value significantly greater than 1 for
L(q) L n (13)
p < 1 it demonstrates that the PTSP strategy can be t ≃ o
worsethanthe re-optimizationstrategy. Jaillet[26]and
where L is some fraction of the side of the unit square,
Bertsimaset al. [28]havealsoshownthatthereisalimit o
sincecitieswhicharesortedwithrespecttoanglewillbe
to how much worse it can be, with
unsorted with respect to radial distance. This leads to
E L¯pruned n 2
nl→im∞ (cid:0)√np (cid:1)!=βpruned(p) (11) E(L¯ang)≃Lonp2 − (1−p)q. (14)
q=0
X
where
For np 1 and p 1, we then find that the angular
≫ ≪
β sort yields
TSP
β β (p) Min(0.9212, ). (12)
TSP pruned
≤ ≤ √p
E(L¯ ) L np. (15)
ang o
→
OneattempttosolvethePTSPusinganexactmethod
was taken by Laporte et al. [29] who introduced the use By contrast it has been shown[28] that
of integer linear stochastic programming. Although use
E(L¯ )
ofalgorithmswhichmayexactlysolvethePTSPareuse- τsf =C (16)
ful, they are always very limited in the size of problem E(L¯Reopt)
which may be attempted. Furthermore, the stochastic
programming algorithm even fails to solve the PTSP on with probability 1, where E(L¯τsf) is the expected length
certainoccasions,thustheaccuracyofanystatisticsthat ofatourgeneratedbyaheuristicbasedontheSierpinski
would be generated using this method is dubious. curve and E(L¯Reopt) is the expected length for the re-
Three studies have used heuristics to solve the PTSP optimization strategy. Using previous computational re-
[30, 31, 32]. None of these studies used global search sults [25, 28], we estimate C 1.33,which is worse than
heuristics, and all were very restricted in the problem we achieve using stochastic a≃nnealing. Hence, E(L¯τsf) is
size attempted due to computational cost. The evalua- given by
tionofamoveforthePTSPusingequation8involvesthe
computationofO(n2)termscomparedtoO(1)computa- E(L¯τsf)=O(√np) (17)
tionstoevaluateamoveintheTSP.Thus,tosolvea100
which leads to
city problem for the PTSP would take O(10,000) times
longer than it would to solve a 100 city problem for the E(L¯ ) 1
TSP.Itshouldbenoted,however,thatitisonlypossible E(L¯τsf) =O(√np). (18)
ang
to make this comparisondue to the relativesimplicity of
the PTSP. For many more stochastic optimization prob- So for large enough np, the angular sort is not optimal.
lems, standard optimization techniques are simply not From inspection of near-optimal PTSP tours such as
applicable. fig. 1, we propose that the tour behaves differently on
different length scales; the tour being TSP-like at larger
length scales, but resembling a locally directed sort at
III. FORM OF THE OPTIMAL TOUR & smallerlengthscales. We mayconstructsucha tourand
SCALING ARGUMENTS
use scaling arguments to analyse both the pruned and a
priorilengthsoftheoptimaltour. Considerdividingthe
Optimal a priori PTSP tours for small p, as exempli- unit square into a series of ‘blobs’, each blob containing
fied in figure 1 for p = 0.1, resemble an “angular sort” 1/p cities so that of order one city requires a visit. The
- where cities are ordered by their angle with respect to number of such blobs is given by
the centre of the square. Bertsimas [30] proposed that
an angular sort be optimal as p 0, but we can show N np (19)
→ ≃
4
andforthesetoapproximatelycovertheunitsquaretheir approach is difficult, since we need a large number of
typical linear dimension ξ must obey citiestoproducereliabledataforthisregime. Extraction
of this behaviour can howeverbe achievedby comparing
Nξ2 1. (20) simulations for different values of n, but fixed np. We
∼
accomplishthisbyinsistingthateachinstancehas4cities
Sinceeachblobisvisitedoforderoncebyaprunedtour,
ontheprunedtour. 4citytoursarechosensincetheyare
we can estimate the expected pruned tour length to be
thesmallestforwhichitmattersinwhichorderthecities
E(L¯ ) Nξ √np (21) are visited. This can be viewed as an efficient way to
pruned
∼ ∼ simulate(approximately)thePTSPstrategywithp= 4.
n
which we will see below is verified numerically. We can Since we are considering the PTSP at fixed np, if
similarly estimate the a priori tour length to be n times β (p) tends towardsa (non-zero)constant as p 0
apriori
→
the distance between two cities in the same blob. Thus, then we expect E L4city /n to tend towards a con-
the expected a priori tour length is apriori
stant as p 0. S(cid:16)imulatio(cid:17)ns in this regime were per-
→
n formed for N = 12 210, with 100 different ran-
E(L ) nξ (22) −
apriori ∼ ∼ p dom city configurations used for N < 30, 20 configu-
r rations for N 90 and 10 configurations for N 120.
which is more difficult to confirm numerically. Figure 5 show≤s a linear-log plot of n a≥gainst
2E(L4city )
apriori
ln(n/4) = ln(1/p). For small n these results reasonably
∼
match the direct measurements of β (p), shown for
IV. COMPUTATIONAL RESULTS FOR THE apriori
comparison. However, for surprisingly large n 100
PTSP
∼
which is beyond the range of our β (p) data, our
apriori
earlier proposal of scaling behaviour is vindicated by
We have investigated near optimal PTSP tours for a
rangeofdifferentnumbersofcities,andvariousvaluesof E L4acpirtiyori /n approaching a constant value. In sum-
p. We used stochastic annealing with effective temper- ma(cid:16)ry we ha(cid:17)ve
atures in the range kT = 0.07 0.01, corresponding to
samplesizesintheranger =2 −500. Between10and80 n
− E(Lapriori)= βapriori(p) (25)
different random city configurations were optimized (80 p
r
configurations of 30 cities, 40 configurations of 60 cities,
20 configurations of 90 cities and 10 configurations for where
n 120 cities).
≥ = 1 p>0.03
Figure3showsamastercurvefortheexpectedpruned β (p) 1.25 0.82ln(p) (26)
apriori =β − p<0.03.
tour length divided by √np. The shift factors have a (cid:26) 0
linear fit and the data are consistent with
E(L¯pruned) VI. NOTES ON ALGORITHM
=f(np) (23)
√np(a bp) IMPLEMENTATION
−
for n 1, where a = 0.872 0.002, b = 0.105 0.005
We applied stochastic annealing to the PTSP using a
≫ ± ±
and f(np) 1 for large np. The shift factors indicate
combinationofthe2-optand1-shiftmove-sets[34]estab-
thatthe PT→SPstrategycanbe no morethan 0.872 1=
0.767− lishedfortheTSP.Bothmove-setsworksimilarlytothat
14( 1) worse than the re-optimization strategy.
whichwouldbe expected forthe deterministic case. The
±
The master curve for the a prioritour length is shown
expectedprunedtourlengthchangeforthemovewases-
in fig. 4. Our scaling arguments predict that the shift
timated by averagingthe change in the tour length for a
factorsβ (p)shouldtendtowardsaconstantforp
apriori number of instances. For a given instance it is not nec-
→
0. However, data are fit very well by the relation
essary to decide whether every city is present, but only
1 the set of cities closest to the move which determine the
βapriori(p)= (24) change in the pruned tour length (see figure 6). For the
1.25 0.82ln(p)
− PTSP, the location of the nearest cities on the pruned
which would tend to zero as p 0 in conflict with our tour to the move is determined from a simple Poisson
→
scaling arguments. To resolve this dilemma we need to distribution.
probe very small p. When using stochastic optimization, the only variable
over which we have control is the sample size (the num-
berofinstances)r,whereastheeffectivetemperature σ
√r
V. THE LIMITING CASE p 0 alsoentailsthestandarddeviationσoftheprunedlength
→
change over instances. As shown in figure 7, annealing
We are interested in finding whether β (p) tends by controlling r alone exhibits a relatively sharp tran-
apriori
towardsa constant as p 0. To do this using the above sition in the expected pruned tour length. The rapid
→
5
transition appears to ‘freeze in’ limitations in the tours troduced 4-city tours to probe the behaviour of a priori
found (analogous to defects in a physical low tempera- tour length downto very smallp. As summarisedby eq.
ture phase). By comparisonwe obtain a much smoother 25, we find a wide pre-asymptoticregime until recover-
change when σ is controlled. ingtheexpectedcrossoverscalingonlyforp<0.03. Un-
√r
The sharpness of the transition under control by r is derstanding these anomalies in the a priori tour length,
caused by the fact that σ may vary from move to move, andconfirmingthemanalytically,isleft asafuture chal-
and is on average lower when the expected pruned tour lenge.
length is less. The jump in the pruned tour length is Wehaveshownstochasticannealingtobearobustand
accompaniedbyajump inσ andhencethe temperature. effective stochastic optimization technique, taking the
We suggest that quite generally controlling σ gives a PTSPasarepresentativedifficultstochasticoptimization
√r
problem. In this case it enabled us to obtain represen-
better cooling schedule than focussing on r alone.
tativeresultsouttounprecedentedproblemsizes, which
in turn supported a whole new view of how the tours
VII. CONCLUSION behave. Of relevance to wider applications of stochas-
tic optimmisation, we have seen that smoother anneal-
ing can be obtained by directly controlling the effective
We have shown that earlier incompatible ideas about
temperature σ [1] rather than simply the bare depth of
the form of PTSP tours especially at small p[22, 30, 33] √r
sampling r alone.
are resolved by a new crossover scaling interpretation.
The crossoverscale corresponds to a group of cities such
thatoforderone willtypicallyhaveto be visited; below NEBwouldliketothankBPAmoco&EPSRCforthe
this scale the (optimsed) a priori PTSP tours resemble support of a CASE award during this research.
a local sort whereas they are TSP-like on scales larger
than the crossover. Our computational results for the
prunedtourlengtharesummarisedbyeq. 23andclearly THE LENGTH OF A TSP TOUR FOR SMALL
support the crossover scaling. NUMBERS OF CITIES
Computationallytheaprioritourlengthismoresubtle
thantheprunedtourlength, althoughitdoesultimately Numerical estimates of the length of a TSP tour for
conform to expectations from crossover scaling. We in- n 10 are given below
≤
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7
FIG. 1: Typical near optimal a priori PTSP tourswith n=300 for p=0.5(left) and p=0.1(right), respectively.
1.12
p=0.05
p=0.1
p=0.2
1.1 p=0.4
p=0.5
p=0.6
1.08
L )pruned) LtpoeR 1.06
E((E
1.04
1.02
1
0 50 100 150 200
np
FIG. 2: The expected pruned tour length divided by the ex-
pected re-optimized tourlength. This indicates the improve-
ment one would expect from re-optimization.
Numberof cities n Numberof instances I Average tour length σ/√I 1
−
2 100000 1.043 0.002
3 100000 1.564 0.002
4 5000 1.889 0.006
5 5000 2.123 0.006
6 5000 2.311 0.005
7 5000 2.472 0.005
8 5000 2.616 0.005
9 5000 2.740 0.005
10 5000 2.862 0.005
TABLE I: The length of theoptimal TSP tourfor n cities.
8
1.2
p=0.05 0.88
p=0.1 p)
1.15 ppp===000...245 (pruned0.86
p=0.6 bor 0.84
E(L)pruned pn )p(denurp 1.1 Shift fact0.08.280 0.1 0.2 0.3p0.4 0.5 0.6 0.7
b
1.05
1
0 50 100 150 200
np
FIG.3: Themastercurvefor theprunedtourlength divided
byβpruned(p)√np. Thedatafollowsasmoothcurveforn>
30,andtheshiftfactorsfollowalinearrelationship,suggesting
E L¯pruned
that = f(np). Three points with n = 30
√np(cid:16)(0.872 0.10(cid:17)5p)
−
can be seen to fit less well (here and also in fig. 4), showing
breakdown of themaster curveat small n.
1.2
p=0.05 0.7
p=0.1 p)
1.15 ppp===000...245 (a-priori00..56
E(L)a-priori p/n )p(iroirp-a11.0.15 p=0.6 bShift factor 000...2340 0.1 0.2 0.3p0.4 0.5 0.6 0.7
b
1
0.95
0 20 40 60 80 100 120 140 160 180
np
FIG.4: Themastercurvefortheaprioritourlengthdivided
by npβa priori(p). The shift factors, inset, are expected to
tendtowardsaconstantforp 0. Theslight,butsignificant,
p →
deviationfromlinearsuggeststhatthismightnotbethecase.
9
4.5
4
3.5
1b (p)a-priori2.53 pp==04./0n5-0.6
2
1.5
1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
ln(1/p)
FIG. 5: Reciprocal shift factors for a-priori tours (dia-
monds) compared to estimates from 4 city tours (crosses),
1 n . The4citytourdataisoptimized
βa−priori(p) ≃ 2E L4acpirtiyori
when each of the instances have 4 cities on the pruned tour.
(cid:0) (cid:1)
Thedirectmeasurementsdonotappeartosaturatewithinthe
accessible range of p. The crosses show matching behaviour,
with saturation at larger n corresponding to inaccessible p,
suggesting that E(La priori)=β0 np for small p.
p
We take one paticular instance and
An a-priori tour, and proposed move
estimate length change from that
FIG. 6: When estimating the expected length change due to
amove,werandomlygenerateinstances. Onlythecitiesthat
arenearesttothemoveareneededtocalculatethechangein
thepruned tourlength.
10
16
s r-1/2
14 r-1/2
12
)d 10
e
n
u
Lpr 8
(E
6
4
2
0
0 0.02 0.04 0.06 0.08 0.1
s r-1/2, 1/2r-1/2
FIG.7: Theexpectedprunedtourlengthforannealingswhen
rand1/T = √r areincreasedmonotonically. Thesharpdrop
σ
in the pruned tour length is seen when only r is controlled,
demonstratingthatthis“freezesin”imperfectionsinthetour.
The system was annealed at each value of the temperature
andvalueofrfor50,000MonteCarlostepswithn=300and
p=0.1.