Exercises

11.4.5 Construction of a family of expander graphs

Explicit constructions of a family of expander graphs are very much non-trivial.

An easy construction of a family of3-regular expander graphs which neverthe-less relies on a deep result in number theory (Selberg’s 3{16 theorem) is as follows: Letpbe a prime. The vertex set ofG_pisZpand a vertexxis connected tox`1,x´1andx^´1where all operations are performed modulopand where the inverse of0is defined to be0.

11.4 Computer exercise: Compute the optimal distortion embedding of the semimetric in Table 11.1 and draw a random projection onto the two-dimensional Euclidean plane.

BIBLIOGRAPHY

[1] R. Diestel,Graph Theory, Springer, 2010.

http://diestel-graph-theory.com/

[2] S. Hoory, N. Linial, A. Wigderson,Expander graphs and their applications, Bull. Amer. Math. Soc.43(2006), 439–561.

www.ams.org/bull/2006-43-04/S0273-0979-06-01126-8/

[3] M.X. Goemans,Semidefinite programming in combinatorial optimization, Math. Program.79(1997), 143–161.

http://math.mit.edu/~goemans/PAPERS/semidef-survey.ps [4] A. Lubotzky Expander graphs in pure and applied mathematics, Bull.

Amer. Math. Soc.49(2012), 113–162.

http://www.ams.org/journals/bull/2012-49-01/

S0273-0979-2011-01359-3/home.html

[5] J. Matouˇsek,Lectures on discrete geometry, Springer 2002.

http://kam.mff.cuni.cz/~matousek/dg-nmetr.ps.gz

[6] P. Sarnak,What is . . . an Expander?, Notices of the AMS51(2004), 762–

763.

http://www.ams.org/notices/200407/what-is.pdf

CHAPTER 12

PACKINGS ON THE SPHERE

Packing problems are fundamental in geometric optimization and coding the-ory: How densely can one pack given objects into a given container?

In this lecture the container will be the unit sphere S^n´1“ txPRⁿ:x¨x“1u

and the objects we want to pack are spherical caps of angleγ. Thespherical cap with angleγP r0, πsand centerxPS^n´1is given by

Cpx, γq “ tyPS^n´1:x¨yěcosγu.

Its normalized volume equals (by integration with spherical coordinates) wpγq “ωn´1pS^n´2q

ω_npS^n´1q ż1

cosγ

p1´u²q^pn´3q{2du,

whereω_npS^n´1q “ p2π^n{2q{Γpn{2qis the surface area of the unit sphere. Two spherical capsCpx₁, γqandCpx₂, γqintersect in their topological interior if and only if the inner product ofx1andx2lies in the half-open intervalpcosp2γq,1s.

Conversely we have

Cpx₁, γq^˝XCpx₂, γq^˝“ H ðñ ´1ďx₁¨x₂ďcosp2γq.

Apackingof spherical caps with angleγ, is a collection of any number of spher-ical caps with this angle and pairwise-disjoint topologspher-ical interiors. Given the dimensionnand the angleγwe define¹

Apn,2γq “maxtN :Cpx₁, γq, . . . , Cpx_N, γqis a packing inS^n´1u.

1Note here that we use2γin the definition ofApn,2γqbecause we want to make the notation consistent with the common literature. There one emphasizes that2γis the angle between the centers of the spherical caps.

One particular case of packings of spherical caps has received a lot of atten-tion over the last centuries.

In geometry, thekissing numberτnis the maximum number of non-overlapping equally-sized spheres that can simultaneously touch a central sphere. It is easy to see thatτn “Apn, π{3qbecause the points where the spheres touch the cen-tral sphere form the centers of a packing of spherical caps with angleπ{6.

Today, the kissing number is only known for dimensions1,2,3,4,8and24.

It is easy to see that the kissing number in dimension1is2, and in dimension 2it is6. The kissing number problem has a rich history. In 1694 Isaac Newton and David Gregory had a famous discussion about the kissing number in three dimensions. The story is that Gregory thought thirteen spheres could fit while Newton believed the limit was twelve. Note that the easy area argument, which provesτ2“6, only gives that

τ₃ď Z 1

wpπ{3q

^

“

Z 4π 2πp1´cospπ{6qq

^

“t14.92. . .u“14.

It took many years, until 1953, when Sch¨utte and van der Waerden proved Newton right.

Figure 12.1: Construction of 12 kissing spheres. Image credit: Anja Traffas In the 1970s advanced methods to determine upper bounds for the kiss-ing number based on linear programmkiss-ing were introduced. Uskiss-ing these new techniques, the kissing number problem in dimension8and24was solved by Odlyzko, Sloane, and Levensthein. For four dimensions, however, the optimiza-tion bound is25, while the exact kissing number is24. In a celebrated work Oleg Musin proved this in 2003, see [3].

The goal of this lecture is to provide a proof ofτ8“240.

12.1 α and ϑ for packing graphs

Many, often notoriously difficult, problems in combinatorics and geometry can be modeled as packing problems of graphsG “ pV, Eq where the vertex set V can be an infinite or even a continuous set. All possible positions of the objects which we can use for the packing are vertices of a graph and we draw edges between two vertices whenever the two corresponding objects cannot be

simultaneously present in the packing because they overlap in their interior.

Now every independent set in this conflict graph gives a valid packing.

For the problem of determining the optimal packing of spherical caps with angleγ,Apn,2γq, we define the packing graphGpn,2γqwith vertex set

V “S^n´1“ txPRⁿ:x¨x“1u, and edge set

x„yðñx¨yP pcosp2γq,1q.

Then,

Apn,2γq “αpGpn,2γqq, and τn “Apn, π{3q “αpGpn, π{3qq

Now it is an “obvious” strategy to compute the theta number for this graph in order to find upper bounds for the independence numberαpGpn,2γqq.

To generalize the theta number for infinite graphs, we will need a notion of positive semidefiniteinfinite matricesbecause in the definition of the theta number we need matrices whose rows and columns are indexed by the vertex set of the graph.

This leads to positive semidefinite, continuousHilbert-Schmidt kernels.

Definition 12.1.1. A continuous function (called continuous Hilbert-Schmidt ker-nel)

K:S^n´1ˆS^n´1ÑR

is calledsymmetric if Kpx, yq “ Kpy, xqholds for allx, y P S^n´1. It is called positive semidefiniteif for allNand allx₁, . . . , x_N PS^n´1the symmetricNˆN matrix

`Kpxi, xjq˘

1ďi,jďN ľ0

is positive semidefinite. We denote the cone of positive semidefinite continuous Hilbert-Schmidt kernels byCpS^n´1ˆS^n´1qľ0

We use this coneCpS^n´1ˆS^n´1qľ0to define the theta prime number of the packing graphGpn,2γq:

ϑ¹pGpn,2γqq “inf λ

KPCpS^n´1ˆS^n´1qľ0

Kpx, xq “λ´1for allxPS^n´1 Kpx, yq ď ´1for alltx, yu RE.

We have tx, yu R E whenever the spherical caps Cpx, γq and Cpy, γq do not intersect in their topological interior, i.e. wheneverx¨y P r´1,cosp2γqs.

The definitionϑ¹ is similar to the dual formulations in Lemma 6.4.1. We use a prime to indicate that we replace the equalityKpx, yq “ ´1 by the inequality Kpx, yq ď ´1.

Similar to the finite case,ϑ¹ provides an upper bound for the independence number:

Theorem 12.1.2.

αpGpn,2γqq ďϑ¹pGpn,2γqq

Proof. Let C Ď S^n´1 be an independent set. Let K be a feasible solution of ϑ¹pGpn,2γqq. BecauseKis positive semidefinite we have

0 ď ÿ

xPC

ÿ

yPC

Kpx, yq

“ ÿ

xPC

Kpx, xq looooomooooon

“|C|pλ´1q

` ÿ

x‰y

Kpx, yq looooomooooon

ďp´1qp|C|²´|C|q

ď |C|pλ´1q ´ p|C|²´ |C|q This implies|C| ďλ, yielding the theorem.

Note that if we are in the lucky case thatαpGpn,2γqq “ ϑ¹pGpn,2γqq, the inequalities in the proof of the theorem are tight. This can only happen when Kpx, yq “ ´1fortx, yu RE. We will use this observation later when we deter-mineτ8.