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 functionals. 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 (∀i0, (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

Separating subadditive Euclidean functionals

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