A travelling salesman problem in the k-dimensional unit cube

Operations Research Letters 11 (1992) 19-21 North-Holland

February 1992

A travelling salesman problem in the k-dimensional unit cube B. Bollobfis Department of Pure Mathematics and Mathematical Statistics, Unit,ersity of Cambridge, 16 Mill Lane, Cambridge CB2 ISB. UK

A. Meir Department of Mathematics, Louisiana State Uniuersity, Baton Rouge, LA, USA, and Department of Mathematics, Unit,ersity of Alberta, Edmonton, Alberta, Canada Received July 1990 Revised March 1991

Answering a question of the second author in Operations Research Letters 6 (1987) 289-291, we show that for every k _> 1 there is a constant c k with the following property. Let x 1,..., xn be points in a k-dimensional cube. Then there is a tour x, x,: ... xi, ' of these points such that, with x , n + = xit, we have (YTj'~=I [xi,- x,,+~ ]k)l/k _ 3, it is not known what is the best constant cff one can take, but we show that it satisfies 21/k~/k <_c~ <_( 3 / 2 ) ~k -I)/k6V~.

1. Introduction

In what follows, for a vector (point) x ~ [~k we shall d e n o t e by I xl the Euclidean length of x. T h u s if x(~: 1, 5¢2. . . . . ~k ), t h e n I xl = ( Z ki = l l ~ : i l Z ) l / Z ; i f x, y ~ R k , t h e n I x - Y l is the Euclidean distance b e t w e e n the points x and y. Let S = { x 3. . . . ,xn} be an n-point subset of the k-dimensional unit cube [0, 1] k. For a Hamilton cycle H =XiXi2"''Xin through these points, et

j=l

w h e r e Xs.., is defined to be x i . Thus if H has edge-lengths e~, e 2 . . . . , e~ (i.e.

ej= lxij-xi,+,l, j = 1, 2 . . . . . n), then Sk(H)= Ilellk = (ET=le~) l/k. Define S k ( S ) = m i n { s k ( H ) : H is a H a m i l t o n cycle through the points of S} and Sk(n)=sup{Sk(S): S is an n-point subset of [0, 1]k}. H o w large is Sk(n)?

This variant of the usual travelling salesman p r o b l e m has been solved for k = 2: N e w m a n [4, p. 9, P r o b l e m 57] proved that Sz(n) = 2 for n > 2. Results related t o s 2 ( n ) have been proved by Bentley and Saxe [1] and Meir [3]. In the same note, Meir [3] asked w h e t h e r Sk(n) is b o u n d e d from above by a constant Ck, and the same question was raised by Y a t c h e w (see [5] for some related problems). O u r aim is to answer this question in the affirmative. Note that k is the smallest value of p for which sp(S) is b o u n d e d as S ranges over the finite subsets of [0, 1] k. Indeed, let n = ( m + 1) k and

S m = { ( i , / m , i2/m . . . . . i k / m ) : ij = O, 1 . . . . . m}. T h e n the distance b e t w e e n any two points of S,n is at least l / m , so if a H a m i l t o n cycle H joining the points in S,, has edge-lengths e 1, e e . . . . . e,, then e i > 1/rn for every i, so

Sp(H) =

ep

> ( n ( 1 / m ) P ) I/p

i

=(m+l)k/P/m>m

0167-6377/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

k/p t. 19

Volume I 1, Number 1

OPERATIONS RESEARCH LETTERS

Hence Sp( S m ) >_~m k / I ' - I

and so

s u p ( s p ( S ) : S is a finite subset o f [ 0 , 1] k} = ~ , as claimed. T h e p r o o f of our t h e o r e m is based on two simple lemmas, the first of which concerns minimal trees on a point set in ~k. T h e length of a tree T = (V, E ) with vertex set V = {x 1. . . . . x~} c R k is I(T) = I2{I x~ - xj I: i < j and xix j ~ E}. W e say that T is a minimal length tree on V if I(T) < I(T') w h e n e v e r T ' = (V, E ' ) is a tree on V.

February 1992

distance [ z - w[ is decreasing and p, q, r and s stay constant. Let us stop this rotation only if t decreases to r. Thus we may assume that either the points x, z, y and w are collinear and (1) holds, or else t = r and (1) holds. In the first case we are done: x and y are outside the o p e n ball of radius r and centre v and w is inside so z is outside this ball, implying 1 I z - w l >_5r> ¼(p+r).

In the second case the assertion is almost immediate. Since [ u - v l = [ y - v l and I x - y [ < I x - u [ , we h a v e ( y - u , x - v ) > O s o

1½(x-v) I _< 1½(x- v)

+ ½(y-u)[

= 1 ½ ( x + y ) - ½(u + v ) [ = I z - w l .

Let T = ( V, E ) be a minimal length tree on a finite set V c ~k. For each edge xy ~ E let Bxv be the open ball of centre ½(x + y ) and radius ¼1 x - y I. Then B xy n B,, = 0 whenever xy and uv are distinct edges of T. L e m m a 1.

But, as s > max{p, r}, 15' ( x - v ) l

= ' gs>_ ¼ ( P + r ) .

Hence

I z - w l >_¼(p+r). Proof. Construct the edge set E by succesively joining vertices at minimal distance from each other, provided the new edge does not create a cycle (see [2]). Suppose that x is joined to y before u is joined to v. Let us consider the forest F we have constructed just before we join u to v. T h e n u and v belong to different c o m p o n e n t s of F while x and y belong to the same c o m p o n e n t . T h e r e f o r e we may assume that v is neither in the c o m p o n e n t of x and y, nor in the c o m p o n e n t of u. As uv is the next edge to be a d d e d to our forest, lu-vl

_< min{ I v - x l ,

Iv-yl}.

Also, since xy is an edge of F, Ix-yl

_< max{ l u - x l ,

Note that for k > 2 the factor ~1 in the lemma is as large as possible: if V = {x, y, z} with Ix yl = l y - z l = ]z-xl and E = { x y , yz}, say, then (aBxy) n (aBy z) 4:0 for a > 1. As usual, we write V(G) for the vertex set of a graph G and E ( G ) for its edge set. T h e h-th power G h of a graph G = (V, E ) is the graph with vertex set V and edge set E ( G h)={xy: x , y V , l < d ( x , y)_
lu-yl},

say I x - y l < l u - x l . F u r t h e r m o r e , as x was joined to y before u was joined to v, I x - Y l < l u - v [ . Thus the quantities p = I x - y I, q = Ix -u[, r= lu-vl, s = Is-vl, and t = l y - v l satisfy the inequalities p < min{q, r} and r < min{s, t}.

(1)

Set z = l ( x + y ) and w = ½(u + v). T h e l e m m a 1 claims that I z - w l > ~(p + r ) / 4 , so in proving the lemma we may decrease [ z - w [ if we keep p + r constant. R o t a t i n g y towards w in the (affine) plane d e t e r m i n e d by x, y and w, the 20

[]

L e m m a 2. Let x be a vertex of a tree T of order at least 3. Then T 3, the cube of T, contains a Hamilton cycle H such that every edge of T is used exactly twice by H, and one of the edges of H incident with x is an edge of T.

Proof. For the purpose of this proof, trees of o r d e r 1 and 2 will be d e e m e d to be Hamiltonian: if V ( T ) = {x}, then x is a ' H a m i l t o n cycle' of T, and if V ( T ) = {x, y}, then xy is a ' H a m i l t o n cycle' of T, with the edge xy used twice by this 'cycle'.

Volume 11, Number l

OPERATIONS RESEARCH LETTERS

Let us prove the assertion by induction on the order n = l T ] of T. For n = l and 2 there is nothing to prove. Suppose that n >_ 3 and the assertion holds for trees of order at most n - 1. Let y~ . . . . Yk be the neighbours of x in T and let Tl . . . . . T k be the b r a n c h e s of T, rooted at x, with roots yl . . . . Yk" Thus V ( T ) is the disjoint union of V ( T l) . . . . . V ( T k) and {x}. By the induction hypothesis, each Ti3 contains a H a m i l t o n cycle H i = Yi ... zi (in the wider sense) such that the first edge is an edge of T~ and every edge of T, is used twice by H i. Let us write yiH~z~ for the sequence of vertices of H i from Yi to z i, i.e. for the path obtained from H i by deleting the edge YiZi . In particular, if H i is the 'cycle' Yi then yiHizi is just yi, and if H i is the 'cycle' YiZi then yiHizi is just the edge yizi. It is easily checked that the paths y~Hiz i can be used to construct a p r o p e r H a m i l t o n cycle in T 3. Namely, H = x y ~ H ~ z ~ Y z H 2 z 2 . . . Y k H k Z k is a H a m i l t o n cycle in T 3 such that the first edge, xy, is an edge of T, and every edge of T is used twice by H. [] We are ready to prove our main result.

February 1992

so n--1

E

_<

k.

(2)

i-I

Let e 1. . . . . e,, be the edge lengths of a Hamilton cycle H in T 3 g u a r a n t e e d by L e m m a 2. Suppose that the edges of T used by the i-th edge of H have lengths di~ . . . . . di~ (l < 3). Set fi = 2~= idij. T h e n e i <--fi for every i, each f,. is a sum of at most three dss and each dj occurs in the representations of two fis. O u r aim is to give an u p p e r bound for Y~7=l fi k in terms of E /n= 1 Ida'l " As it is easy to obtain the best u p p e r bound implied by the conditions above, we shall do so. Each fi is the sum of at most three dis, so we can form vectors t,~, v 2, ~,,. ~ ~n such that f = I23= lvi, every coordinate of v i is a dy or 0, and every dj occurs exactly twice as a coordinate in the three cis. Thus E{ IN ci Ill = 2Y~sn -=I i dj.k H e n c e tlfll-< ~

Ilviltk--<3

i=1

~ d -Xj=I

T h e r e f o r e , by (2),

sk(g ) =

e~

< II f IIk < 9 ( ~ ) l / k ~ / k

ki=l

Theorem 3. Let x l . . . . . x , ~ [0, 1] k c [~k where k

> 2. Then there is a tour H = x 6 x i 2 . . . x i , that

) Ilk

s/,(H)=

~

xi-xi,+,l k

--

,

E (J,) i=1

It is trivial that Sk(2) = 21/k~/k so

for every n _> 2. It is likely that, in fact, Sk(n) = 21/kv~- for all k >_ 1 and n >_ 2.

Xi,,.

Proof. We may assume that n > 3 since if n = 1 then S k ( H ) = 0 and if n = 2 then sk(H)<_ 2 ~ / k ~ -. Let d t . . . . , d~ ~ be the edge-lengths of a minimal tree T on {Xl . . . . . Xn}. For the i-th edge xtxm, let B~ be the open ball of radius ~d i and 1 centre ~(x~ +Xm). By L e m m a 1, the balls B 1. . . . . B~_~ are disjoint. Also, ¼di/4 < ¼vCk/4 so U T , l l B i c B , where B is the ball of radius 1 1 ½v/k + ~v~= ] f k - and centre (½, ~1 . . . . . ~-) [0, 1] k. Hence, writing Vk for the volume of the k-dimensional unit ball, n-1

[]

2 ' / k v ' k <_Sk(n ) <_ 9 ( } ) ' / k f k _< 9(2)1/k f k -

j=l

where xi, ' + ,

as claimed.

such

References [1] J.L. Bentley and J.B. Saxe, "An analysis of two heuristics for the Eulerian travelling salesman problem", in: 18th Annual Allerton Conference on Communications, Control and Computing, Proc. Conf. Monticello, 1980, 41-49. [2] Graham, R.L. and Hell, P., "On the history of the minimum spanning tree problem", Technical Report, Computing Science, Simon Fraser University, Burnaby, B.C., 1982. [3] A. Meir, "A geometric problem involving the nearest neighbour algorithm", Oper. Res. Lett. 6, 289-291 (1987). [4] D.J. Newman, A Problem Seminar, Springer-Verlag, Berlin, 1982. [5] A.D. Yatchew, "Some tests of non-parametric regression models", in: W.A. Barnett, E.R. Berndt and H. White (eds.), Dynamic Econometric Modelling: Proceedings of the Third International Symposium in Economic Theory and Econometrics, Cambridge University Press, 1989.

21