Separating subadditive Euclidean functionals

0.45 MB·English

by  Alan Frieze

SEPARATING SUBADDITIVE EUCLIDEAN FUNCTIONALS
ALANFRIEZEANDWESLEYPEGDEN
5
1
0
Abstract. Ifwearegivennrandompointsinthehypercube[0,1]d,thenthe
2
minimumlengthofaTravelingSalespersonTourthroughthepoints,themin-

y imumlengthofaspanningtree,andtheminimumlengthofamatching,etc.,
a d−1
are known to be asymptotically βn d a.s., where β is an absolute constant
M
in each case. We prove separation results for these constants. In particular,
concerning the constants βd , βd , βd , and βd from the asymptotic
TSP MST MM TF
1 formulasfortheminimumlengthTSP,spanningtree,matching,and2-factor,
2 respectively, we prove that βd < βd , 2βd < βd , and βd < βd
MST TSP MM TSP TF TSP
for all d ≥ 2. We also asymptotically separate the TSP from its linear pro-

] gramming relaxation in this setting. Our results have some computational
R
relevance, showing that a certain natural class of simple algorithms cannot
P solvetherandomEuclideanTSPefficiently.
.
h
t
a
m
[ 1. Introduction

4
v Beardwood,Halton,andHammersley[3]studiedthelengthofaTravelingSalesper-
4
sonTourthroughrandompointsinEuclideanspace. Inparticular,ifx ,x ,... isa
4 1 2
randomsequenceofpointsin[0,1]d andX ={x ,...,x },theirresultsimplythat
9 n 1 n
1 there is an absolute constant βTdSP such that the length TSP(Xn) of a minimum
0 length tour through X satisfies
n
.
01 (1) TSP(Xn)∼βTdSPnd−d1 a.s.
5
1 This result has many extensions; for example, we know that identical asymp-
: totic formulas hold for the the cases of the minimum length of a spanning tree
v
i MST(Xn)[3], and the minimum length of a matching MM(Xn) [23]. Steele [25]
X
provided a general framework which enables fast assertion of identical asymptotic
r formulas for these and other suitable problems. For example, we will see in Sec-
a
tion 2 that his results imply that the length TF(X ) of a minimum length 2-factor
n
admits the same asymptotic characterization.
A major remaining problem in this area is to obtain analytic results regarding
the constants β in such formulas. In particular, the best rigorous bounds on such
constants are generally very weak, with known results for d = 2 given in Table 1.
In particular, the bounds on β2 were not improved since 1959, until the paper
TSP
[29] of Steinerberger improved the lower bound by 19 ≈.0036... and the upper
5184
Date:May22,2015.
ResearchsupportedinpartbyNSFgrantDMS-1362785.
ResearchsupportedinpartbyNSFgrantDMS-1363136.
1
2 ALANFRIEZEANDWESLEYPEGDEN
lower upper
β2 .62866 [29] .92116...[3, 29]
TSP
β2 .60082 [2] √1 ≈.707 [12]
MST 2
2β2 .5 [5] .92116
MM
Table 1. Bounds on constants for d=2.
bound by ≈10−6 . On the other hand, there was some success as d grows large, as
Bertsimas and Van Ryzen [5] showed that, asymptotically in d,
(cid:114)
d
(2) βd ∼2βd ∼ ,
MST MM 2πe
(cid:113)
and conjectured that βd ∼ d as well.
TSP 2πe
It seems that it has been overlooked that local geometric arguments are sufficient
to prove the separation of constants for many natural examples of Euclidean func-
tionals. In particular, in the present paper, we will show that βd < βd ,
MST TSP
βd < βd , and 2βd < βd for all d. These are the first asymptotic separa-
TF TSP MM TSP
tions for Euclidean functionals where the Euclidean metric is playing an essential
role: theonlypreviousseparationwasshown(byBern[4];seealso[18])forthemin-
imum length rectilinear Steiner tree vs. the minimum rectilinear length spanning
tree, which is equivalent to asymptotically distinguishing Steiner trees from trees
intheL norm. (TherectilinearSteinertreeisalsotheonlycasewheretheasymp-
1
totic worst-case length is known exactly [7].) Finally, we will also asymptotically
separate the TSP from its linear programming relaxation.
We begin by considering the degrees of vertices in the minimum spanning trees
among n random points. Steele, Shepp, and Eddy [28] showed that the number
Λ (X ) of vertices of degree k satisfies
k n
Λ (X )∼α n
k n k,d
for constants α , and proved that α(1,d) > 0. Note that we must have α = 0
k,d k,d
when k > τ(d), where τ(d) is the kissing number of d dimensional space (6 in the
case d = 2). Indeed, we must have α = 0 whenever k > τ(cid:48)(d), where τ(cid:48)(d)
k,d
denotes a strict kissing number of d, which we define as the maximum K such
that there exists ε > 0 such that there is, in d dimensions, a configuration of K
disjoint spheres of radius 1+ε each tangent to a common unit sphere. (Note that
τ(cid:48)(d)≤τ(d), and in particular, τ(cid:48)(2)=5.) We prove:
Theorem 1.1. α(k,d)>0 if k ≤τ(cid:48)(d).
ConsideringEuclideanfunctionalsMST (X)(withcorrespondingconstantsβd )
k MSTk
definedastheminimumlengthofaspanningtreeofX whoseverticesallhavedegree
≤k, we will then get separation as follows:
Theorem 1.2. We have that
(3) βd =βd >βd >···>βd =βd
TSP MST2 MST3 MSTτ(cid:48)(d) MST
SEPARATING SUBADDITIVE EUCLIDEAN FUNCTIONALS 3
for all d.
Thus, the MST constants are as diverse as are allowed by the simple geometric
k
constraint of τ(cid:48)(d).
Still, there are only finitely many constants βd for each d; while we can draw
MSTk
treeswithverylargedegrees,largedegrees(relativetod)arenotusefulforminimum
spanning trees in Euclidean space. In contrast to this scenario, let us recall that
a 2-factor is a disjoint set of cycles covering a given set of points. We will see in
Section2thatthelengthoftheminimum2-factorisindeedasubadditiveEuclidean
functional,andthusthislengthsatisfiesTF(Xn)∼βTdFnd−d1 forsomeconstantβTdF.
Moreover,ifTF (X)istheminimumlengthofa2-factorthroughX whosecyclesall
g
have length ≥ g, then we will see that TF is also a subadditive linear functional,
g
so that we have TFg(Xn) ∼ βTdFgnd−d1. Naturally, we must have βTdF = βTdF3 ≤
βd ≤ βd ≤ ··· . In analogy to the high-degree vertices in a tree, we can of
TF4 TF5
course draw 2-factors with small cycles, but it is not clear a priori whether small
cycles will be asymptotically essential to optimum 2-factors in random point sets.
The following theorem shows that they are:
Theorem 1.3. β is a monotone increasing sequence βd < βd < βd <
TFg TF3 TF4 TF5
···.
On the other hand, we prove that 2-factors with long (but constant) girth require-
ments produce close approximations to the TSP:
Theorem 1.4. lim βd =βd .
g→∞ TFg TSP
With a bit more work, our method for proving Theorem 1.3 will also allow us to
deduce the following:
Theorem 1.5. 2βd <βd .
MM TSP
We note in contrast to Theorem 1.3 that in the independent case where the edge
lengths X ,e ∈ (cid:0)[n](cid:1) are independent uniform [0,1] random variables, Frieze [11]
e 2
showedthatwithprobability1−o(1), theweightoftheminimal2-factorisasymp-
totically equal to the minimum length of a tour.
We continue by mentioning a natural generalization of MM(X ). Given a fixed
n
graph H on k vertices, an H-factor of a set of points S is a set of edges isomorphic
to (cid:98)|X|/k(cid:99) vertex disjoint copies of H. As a subadditive Euclidean functional, the
minimum length HF(X ) of an H factor of X satisfies
n n
HF(Xn)∼βHdnd−d1.
We pose the following conjecture:
Conjecture 1.6. Given H ,H and d ≥ 2, we have that βd (cid:54)= βd unless H
1 2 H1 H2 1
and H are each isomorphic to a disjoint union of copies of some graph H . In
2 3
particular, βd (cid:54)=βd if H ,H are connected and non-isomorphic.
H1 H2 1 2
4 ALANFRIEZEANDWESLEYPEGDEN
We prove at least the following, showing diversity in the constants even for fixed
edge density:
Theorem 1.7. For any fixed d ≥ 2 and rational r ≥ 1, there are infinitely many
distinct constants βd over connected graphs H with edge density |E(G)| =r.
H |V(G)|
OurfinalseparationresultconcernsthelinearprogrammingrelaxationoftheTSP.
The TSP through a set of points can be given as the following integer program, on
variables x (i,j ∈ V,i < j) indicating the presence of an edge between vertices
{ij}
i and j in the tour, where c gives the cost of the edge {i,j}:
{ij}
(cid:88)
min c x
{ij} {ij}
{i,j}⊆V
subject to
(cid:88)
(∀i) x =2
{ij}
j(cid:54)=i
(cid:88)
(∀∅(cid:54)=S (cid:40)V) x ≤|S|−1
{ij}
{i,j}⊆S
(∀i<j ∈V) x ∈{0,1}
{ij}
The linear programming relaxation of the above integer program replaces the final
constraint with the requirements that 0 ≤ x ≤ 1 for each x . It is often
{ij} {ij}
referred to as the Held-Karp relaxation. of the TSP, but its origins go back to the
thepaperofDantzig,FulkersonandJohnson[9]. Itshouldbenotedthat,although
this LP has an exponential number of constraints, it can be solved in polynomial
time, e.g., using the ellipsoid algorithm [14].
We denote by HK(X ) the value of a solution to the Held-Karp relaxation on X .
n n
Of course, HK(X ) ≤ TSP(X ). The Held-Karp bound on the TSP is generally
n n
consideredtobeagoodboundwhichisalgorithmicallyusefulon‘typicalinstances’
(see [6, 15, 19, 24, 31]), and, with this motivation, Goemans and Bertsimas [13]
showedthatHK(Xn)∼βHKnd−d1 forsomeconstantβHK,andaskedwhetherβHK =
β . Experimental evidence has suggested that any gap between the constants
TSP
would be less than 1%. [20]
Theorem 1.8. For all d≥2, βd <βd .
HK TSP
Our separation results have implications for the practical problem of solving the
Euclidean TSP. Branch and bound algorithms are a standard approach to solving
NP-hard problems, in which a bounding estimate is used to prune an exhaustive
search of the solution space. There has been a great deal of success solving real-
world instances of the TSP with branch-and-bound augmented with sophisticated
SEPARATING SUBADDITIVE EUCLIDEAN FUNCTIONALS 5
techniques based on cutting planes for the TSP polytope (see, for example Apple-
gate, Bixby, Chv´atal and Cook [1]).
Ourresultsshow,however,showthatseveralnaturalapproximators(2-factor,twice
a matching, the Held-Karp bound, etc) are asymptotically distinct from the TSP.
This will be algorithmically relevant as follows:
Theorem 1.9. Suppose that we use branch and bound to solve the TSP on X ,
n
using TF or HK as a lower bound. Then, w.h.p, the algorithm runs in time
g
eΩ(n/log6n).
In particular, this gives a rigorous explanation for the observation (see [22], for
example) that branch-and-bound heuristics using the Assignment Problem as a
bounding estimate (even weaker than the 2-factor) perform poorly on random Eu-
clidean instances, and indicatesthat the success of theHeld-Karp bound inbranch
and bound algorithms will be limited for sufficiently large Euclidean point sets.
Remark 1.10. Asymptotic formulas are available for subadditive Euclidean func-
tionals in more general settings. If x ,x ,··· ∈ Rd are independent identically
1 2
distributed random variables with bounded support, then the length L(x ,...,x )
1 n
of the functional on the points X satisfies
n
(cid:90)
L(x1,...,xn)/nd−d1 →βLd f(x)d−d1dx,
Rd
where f isthe absolutely continuouspart ofthedistributionofthe x ’sand βd isa
i L
constantdependingonlyondandL(see[3,25]). Notethatthisgivesanasymptotic
formula for L(x ,...,x ) unless the right hand side is zero. The latter case will
1 n
happen if the x ’s lie exclusively on some m-dimensional manifold embedded in Rd
i
where m < d, but the BHH theorem also has a suitable extension to this setting
[8], allowing asymptotic formulas involving nmm−1. Our results are all immediately
valid in these more general settings, however: as the constants βd depend only on
L
L and d or m (in particular, not on the distribution or, in the second case, the
particular manifold), it is enough study the constants in the case of points which
are uniformly distributed in the hypercube.
2. Subadditive Euclidean Functionals
Steele defined a Euclidean functional as a real valued function L on finite subsets
of Rd which is invariant under translation, and scales as L(αX) = αL(X). It is
nearly monotone with respect to addition of points if
d−1
(4) L(X∪Y)≥L(X)−o(n d ) for n=|X|.
It has finite variance if, fixing n, we have
(5) Var(L(X ))<∞
n
(in particular, if it is bounded for fixed n) and it is subadditive if, for Y a random
n
set of n points from [0,t]d, it satisfies
(cid:88)
L(Y )≤ L(S ∩Y )+Ctmd−1
n α n
α∈[m]d
6 ALANFRIEZEANDWESLEYPEGDEN
for some absolute constant C, where here {S } (α ∈ [m]d) is a decomposition of
α
[0,t]d into md subcubes of side length u=t/m.
Steele proved:
Theorem 2.1 (Steele [25]). If L is a subadditive Euclidean functional on Rd of
finite variance, x ,x ,... is a random sequence of points from [0,1]d, and X =
1 2 n
{x ,x ,...,x }, then there is an absolute constant βd s.t.
1 2 n L
(6) L(Xn)/nd−d1 →βLd a.s.
In particular, we have a.s. that either L(Xn)=o(nd−d1) (if βLd =0) or that
(7) L(Xn)∼βLdnd−d1.
This can thus be used to easily give the existence of the simple asymptotic for-
mulas for the functionals TF (X), MST (X), and HF(X) by showing that these
g k
functionals are subadditive. We note that the constant βd cannot be zero in any
L
of these cases and thus that (6) implies (7), since for any 1≤i≤n,
E(min dist(xi,x))=Ω(n−d1),
x∈Xn
x(cid:54)=xi
giving a lower bound of Ω(nd−d1) for each functional by Linearity of Expectation.
Proposition2.2. TF (X),MST (X),andHF(X)aresubadditiveEuclideanfunc-
g k
tionals.
Before writing a proof, we note that for the definition of the 2-factor functionals
TF (X), we can only require that the 2-factors whose length we minimize cover all
g
thepointswhenthereareatleastmax(g,3)points. Similarly,theHF(X)functional
is required just to cover at least n−|H|+1 points.
Proof. Webeginbynotingthatforeachofthesefunctionals,wecanassertanupper
bound Cnd−d1 for some constant C, even over worst-case arrangements of n points
in [0,1]d. The analogous statement for the TSP was proved by Toth [30] and by
Few [10], and implies these bounds for the functionals considered here. Indeed, a
tour through n points itself gives a tree of max-degree 2 (after deleting one edge),
andisa2-factorsubjecttoanyconstantgirthrestriction. ForH factors,atourcan
bedividedintopathsoflength|H|(exceptfor<|H|remainingvertices)whichcan
thenbecompletedtoinstancesofH(cid:48) ⊇H byaddingedges. Eachaddededgehasa
cost bounded by the length of the path it lies in and so this construction increases
the total cost by at most a factor equal to the number of edges in H.
SubadditivityofTF (X),andHF(X)isnowaconsequenceofthefactthataunion
g
of 2-factors (subject to restrictions on the cycle length, perhaps) or H-factors is
again a 2-factor (subject to the same restrictions) or an H factor, respectively.
In particular, the subadditive error term for these functions comes just from the
fact that points may be uncovered in some of the subcubes S , for the exceptional
α
reasons noted above. Since there are at most (g−1)md or |H|md such uncovered
SEPARATING SUBADDITIVE EUCLIDEAN FUNCTIONALS 7
points, however, the error is suitably bounded by the minimum cost factor on a
worst-case arrangement of the remaining points.
Subadditivity of MST (X) (k ≥2) is similar: after finding minimal spanning trees
k
ofmax-degreek ineachsubcubeS , wemustjointogetherthesetreesintoasingle
α
tree. We choose 2 leaves of each subcube’s tree and denote one red and the other
blue. We let α ,α ,...,α denote a path through the decomposition {S }, so
1 2 md α
that the subcubes S and S are adjacent. For each i < md, we join the red
αi αi+1
leaf of the tree in S to the blue leaf in the tree of S . The result is a spanning
αi αi+1
tree of the whole set of points with the same maximum degree and with extra cost
√ √
at most 2 dumd =2 dtmd−1. (cid:3)
3. Separating asymptotic constants
InthefollowingwewillusethesimplestapplicationoftheAzuma-Hoeffdingmartin-
gale tail inequality: It is often referred to as McDiarmid’s inequality [21]. Suppose
thatwehavearandomvariableZ =Z(X ,X ,...,X )whereX ,X ,...,X are
1 2 N 1 2 N
independent. Further,supposethatchangingoneX canonlychangeZ byatmost
i
c in absolute value. Then for any t>0,
(cid:26) t2 (cid:27)
(8) Pr(|Z−EZ|≥t)≤2exp − .
c2N
We will also use the following inequality, applicable under the same conditions,
when EZ is not large enough.
(cid:16)e(cid:17)αEZ/c
(9) Pr(Z ≥αEZ)≤ .
α
Ourmethodtodistinguishconstantsisbasedonachievingconstantfactorimprove-
ments to the values of functions via local changes. Given ε,D ∈R and a finite set
of points S ⊆ Rd and a universe X, we say that T ⊆ X is an (ε,D)-copy of S
if there is a bijection f between T and a point set S(cid:48) congruent to S such that
||x−f(x)||<ε for all x∈T, and such that T is at distance >D from X\T. Here
we will further assume that ||x−y||>ε for x(cid:54)=y ∈S.
For our purposes, it will be convenient notationally to work with n random points
Y from [0,t]d where t = n1/d, in place of n random points X from [0,1]d. At
n n
the end, we will scale our results by a factor n−1/d in order to get what is claimed
above.
Observation 3.1. Given any finite point set S, any ε > 0, and any D, Y a.s
n
contains at least CS n (ε,D)-copies of S, for some constant CS >0.
ε,D ε,D
Proof of Observation 3.1. Let Z denote the number of (ε,D)-copies of S in Y .
n
We divide [0,t]d into n/(3D)d subcubes C ,C ,..., of side 3D. Then let C(cid:48) ⊆ C
1 2 i i
be a centrally placed subcube of side D. Now choose a set S(cid:48) congruent to S
somewhere inside C(cid:48) and let B ,B ,...,B , s = |S| be the collection of balls of
1 1 2 s
radius ε, centered at each point of S(cid:48). The with probability at least α=α >0,
ε,D
each B contains exactly one point of Y and there are no other points of Y in
i n n
C . Thus EZ ≥ βn where β = α/(3D)d. Now changing the position of one point
1
8 ALANFRIEZEANDWESLEYPEGDEN
in Y changes the number of (ε,D)-copies of S by at most two and so we can use
n
McDiarmid’s inequality [21] to show that Z ≥ 1EZ a.s. (cid:3)
2
To use this to prove Theorem 1.1, we will need just a bit more.
Observation 3.2. If Y ⊆ Rd and x lies in the interior of the convex hull of Y,
then when D is sufficiently large, any point at distance >D is closer to some point
of Y than to x. (cid:3)
Ifv ,v ,...,v arevectorsinRdwithpairwisenegativedot-product,thenv ,...,v
0 1 k 1 k
lieinthehalf-spacev ·x<0,andtheprojectionsofv ,...,v ontothehyperplane
0 1 k
v ·x=0havepairwisenegativedot-products. Thisgivesthefollowing,byinduction
0
on d:
Observation 3.3. If v ,...,v ∈ Rd are vectors with negative pairwise dot-
1 d+1
products, then 0 is a positive linear combination of the v ’s. (cid:3)
i
This allows us to prove:
Lemma 3.4. If d+1 ≤ k ≤ τ(cid:48)(d), then there exists a set of points S¯(k) ⊆ Rd
consisting of a single point at the origin, surrounded by a set S(k) of k points
on the unit sphere centered at the origin and separated pairwise by at least some
ε(cid:48) >0 more than unit distance, such that S(k) does not lie in open half-space whose
boundary passes through the origin.
Proof. We first observe that the definition of τ(cid:48) already gives us a set S(k) with
the desired properties, except that it may all lie in some open half-space through
the origin. In this case, however, we can delete a point and replace it with the
point x onthe unitsphere opposite thehalf-space H, andfurthest away fromthe
H
halfspace. Wedothisrepeatedlyandnotethatbecausetheaboveexchangeofpoints
only happens when all points are on one side of a half-space H(cid:48), x remains as the
H
unique point which is in the open half-space opposite to H. Furthermore, doing
this repeatedly, we can achieve either a set S(k) with all the desired properties, or
can find after at most k steps a set S(k) of points on the sphere separated pairwise
by at least ε(cid:48) > 0 more than unit distance, and whose pairwise dot products as
vectorsinRd areallnegative. ButthenObservation3.3andk ≥d+1impliesthat
the points cannot all lie in the interior of some half-space whose boundary passes
through the origin. (cid:3)
We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1. Given k ≥ 2, we choose any d(cid:48) ≤ d such that d(cid:48)+1 ≤
k ≤τ(cid:48)(d(cid:48)).
WeapplyLemma3.4withk,d(cid:48) togetasetS(cid:48)(k) ⊆Rd. Observefirstthattheorigin
must lie in the convex hull X of the set S(cid:48)(k) given by Lemma 3.4; otherwise, there
wouldbeasupportinghalf-spaceH ofX notcontainingtheorigin,andS(cid:48)(k) would
SEPARATING SUBADDITIVE EUCLIDEAN FUNCTIONALS 9
bb b b b b bb b
Figure 1. A configuration for forcing degree 2 in 2-dimensions.
lieintheopenhalf-spacethroughtheoriginwhichisparalleltoH,acontradiction.
Now we take S(k) = S(cid:48)(k) ×{0}d−d(cid:48), and the origin is still in the convex hull of
S(k).
Now, letting ∆d denote a unit simplex centered at the origin (with d+1 points),
we let
U =S¯(k)∪ (cid:91) {(1.5)p+.1·∆d}.
p∈S(k)
SoU isasetof1+k+(d+1)k points. (Figure1showsU forthecased=2,k =2;
note that in this case, d(cid:48) =1.)
We now let U denote an (ε,D) copy of U, for sufficiently small ε > 0 and
ε,D
sufficiently large D. Observe that since the origin is in the convex hull of S(k), the
k small copies of the d-simplex in U ensure that the origin is in the interior of the
convex hall of U, and thus also in the interior of U for sufficiently small ε.
ε,D
Observe that (for large D) the distance between any pair of points in an U is
ε,D
less than the minimum distance between U and Y \U . In particular, if T
ε,D n ε,D
denotes the minimum length spanning tree on Y , the subgraph T[U ] induced
n ε,D
by the points in U must be connected (and so a tree), or we could exchange
ε,D
a long edge for a short edge. Moreover, the minimum length spanning tree on T
mustrestricttoaminimumlengthspanningtreeonU ,andbyconstruction,the
ε,D
point of U corresponding to the origin point in U has degree k in the MST on
ε,D
U. Finally, nopointsinY \U canbeadjacenttothecenterofthestarwhenD
n ε,D
is sufficiently large, by Observation 3.2. Thus Observation 3.1 gives that α > 0
k,d
for d+1≤k ≤τ(cid:48)(d).
Finally, α >0 is an immediate consequence of α >0. (cid:3)
1,d 3,d
Indeed, Theorem 1.2 follows immediately as well:
ProofofTheorem1.2. Suppose2≤k <τ(cid:48)(d),andT isaminimumspanningtree
of Y subject to the restriction that the maximum degree is ≤ k. By Observation
n
3.1wehavethatthereareCn(ε,D)copiesofthesetU fromthepreviousproof,for
some constant C, and from the argument above we see that each such copy S will
i
induce a (connected subtree) T[S ], which will have maximum degree at most k in
i
an instance of MST . Replacing each T[S ] by the optimum (k+1)-star produces
k i
a spanning tree of maximum degree k +1, whose length is less by at least some
constantC(cid:48)n. RescalingbytgivesthatthelengthdifferenceisatleastC(cid:48)nd−d1. (cid:3)
Remark 3.5. Thesameargumentallowsustoseparateβd fromβd where
MST Steiner
the latter corresponds to the minimum length Steiner tree. We just need to use
(ε,D) copies of an equilateral triangle. We remark that adding the Steiner points
10 ALANFRIEZEANDWESLEYPEGDEN
corresponding to the Fermat points of the copies will reduce the tree length. The
details can be left to the reader.
We turn our attention now to 2-factors. We begin with two very simple geometric
lemmas:
Lemma 3.6. Suppose that points p,q,r,s satisfy
||p−q||,||r−s||≥∆ and ||q−r||≤δ,
where ∆(cid:29)δ i.e ∆ is sufficiently large with respect to δ.
Let θ(x;y,z) denote the angle between the line segments xy and xz. If
max{θ(p;q,s),θ(s;p,r)}≥∆−1/3
then
∆1/3
||p−s||≤||p−q||+||r−s||+δ− .
4
Proof. We have
||p−s||≤||p−q||cosθ(p;q,s)+δ+||r−s||cosθ(s;p,r).
Now use cosx≤1−x2/3 for x≤1. (cid:3)
Lemma 3.7. Suppose that points p ,q ,r ,s ,i=1,2 satisfy
i i i i
(10) ||p −q ||,||r −s ||≥∆ for i=1,2
i i i i
and also that q ,r ,q ,r are contained in a ball of radius δ. Then there is a
1 1 2 2
matching on {p ,p ,s ,s } whose total length is at most
1 2 1 2
(11) ||p −q ||+||r −s ||+||p −q ||+||r −s ||+4δ− 1∆.
1 1 1 1 2 2 2 2 2
Proof. Withoutlossofgeneralitywelettheq ,r bewithindistanceδ oftheorigin,
i i
andthenletθ(x,y)denotetheanglebetweenxandy viatheoriginthatislessthan
or equal to π. There are three possible pairings of the points P = {p ,p ,s ,s },
1 2 1 2
and for at least one such pairing, θ(x,y)< 1π for one of the pairs.
2
Let us take {x,y} and {w,z} to be the pairs in such a pairing of P, with θ(x,y)≤
1π. We let T denote the triangle with vertices x,y,0, let a,b,c denote the side-
2
lengths, where a is length of the side opposite 0, and s denote the semi-perimeter
(a+b+c)/2. Now a≤(b2+c2)1/2 and in fact
(cid:18) (cid:18) (cid:19)(cid:19)
2bc
b+c−a≥b+c−(b2+c2)1/2 =(b+c) 1− 1−
(b+c)2
bc 1 1
≥ ≥ min{b,c}≥ ∆.
b+c 2 2
Thus we find a pairing of P for which the total length is at most ||p ||+||p ||+
1 2
||s ||+||s ||− 1∆, and we will be done after applying the triangle inequality four
1 2 2
times and using the fact that ||q ||,||r ||≤δ for i=1,2. (cid:3)
i i