Covering maxima. Frank Vallentin (TU Delft, CWI Amsterdam) Mathieu Dutour Sikiric (RBI Zagreb) Achill Schurmann (TU Delft)

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1 Covering maxima Frank Vallentin (TU Delft, CWI Amsterdam) Mathieu Dutour Sikiric (RBI Zagreb) Achill Schurmann (TU Delft)

2 1999 Develop a (nice) theory for lattice coverings! Rudolf Scharlau

3 b 1,...,b n basis of Euclidean space E L = Zb Zb n lattice b 2 V (L) = Voronoi cell V (L) b 1 vol L = volume of V (L) λ(l) = inradius of V (L) µ(l) = circumradius of V (L)

4 lattice packing problem = max L δ(l) = volume of ball w/ radius λ(l) vol L δ(l) = packing density

5 δ(l) = lattice packing problem = max L δ(l) = volume of ball w/ radius λ(l) vol L δ(l) = packing density

6 lattice covering problem = min L Θ(L) Θ(L) = volume of ball w/ radius µ(l) vol L = covering density

7 Θ(L) = lattice covering problem = min L Θ(L) Θ(L) = volume of ball w/ radius µ(l) vol L = covering density

8 Far easier than general sphere packing / covering... but already difficult enough

9 * every dimension gives a new surprise * many local optima * computing packing density is NP-hard (Ajtai 1998) * computing covering density conjectured to be NP-hard, probably not in NP (DSV 2009: Counting vertices of a Voronoi cell is #P-hard.)

10 Known results n lattice Θ 1 Z 1 2 A A A A Solution in dimension 5 by Ryshkov, Baranovski (1975) in a 140-pages paper A n = { x Z n+1 : n+1 i=1 x i =0 } L = {y E : x y Z for all x L}

11 1967 What is the first n for which A n is not optimal? Sergey Ryshkov

12 n 114 Ryshkov 1967 n 24 Conway, Parker, Sloane 1982 n 9 Baranovski 1994 n =6 Ph.D. thesis the candidate for optimal lattice covering

13 Algorithm which finds all local minima * classify all combinatorial types of Voronoi cells subdivide space of all lattices in polyhedral cones * for every type solve a convex optimization problem local lattice covering problem is SDP representable * issues - far too many types in dimension 6, only 222 in dimension 5 - solution generally not nice (rational)

14 The space of lattices lattices { }} { GL n (Z) \ GL n (R) / O(R n ) }{{} semidefinite matrices S n 0 lattice basis b 1,...,b n semidefinite matrix (b i b j )

15 lattices { }} { GL n (Z) \ GL n (R) / O(R n ) }{{} semidefinite matrices S n 0 fundamental domain in S n 0 for GL n(z)-action Decompose S n 0 into polyhedral cones. Voronoi I vs. Voronoi II good for packing good for covering

16 Voronoi II polyhedral cone = domain where type of Voronoi cell is fixed S 2 0

17 Running SDP gives with covering density covering density of A 6 is

18 1999 Develop a (nice) theory for lattice coverings! Rudolf Scharlau

19 What happens to the nice lattices? lattice packing covering Z A 2 D 4 E 6 E 7 E 8 K 12 BW 16 Λ 24 global opt. global opt. global opt. global opt. global opt. not loc. opt. global opt.? global opt.? global opt.? conj. global opt.? conj. global opt.? global opt.?

20 What happens to the nice lattices? lattice packing covering Z global opt. global opt. A 2 D 4 E 6 E 7 E 8 K 12 BW 16 Λ 24 global opt. global opt. global opt. not loc. opt. global opt.? global opt.? global opt. not loc. opt.? conj. global opt.? (SV 2005) conj. global opt.? global opt. loc. opt.? (SV 2005)

21 SECOND TRY * Characterize local maxima for Θ δ * Local minima for don t exist if n > 1 * New theory links Voronoi I and Voronoi II

22 Voronoi I Q S n 0 Q[v] =v Qv homogeneous minimum λ(q) = min Q[v] v Z n \0 minimal vectors Min Q = {v Z n : Q[v] =λ(q)} Hermite invariant λ(q) γ(q) = (det Q) 1/n maximizing Hermite = maximizing packing

23 Voronoi I Theorem. Voronoi 1907 Q local maximum for γ Q perfect and eutactic. perfectness: lin{vv ( ) : v Min Q} has maximal dimension n+1 2 eutaxy: Q 1 = v Min Q α v vv with α v > 0 Consequence: Local maxima are rational

24 Voronoi II inhomogeneous minimum µ(q) = max deep hole vectors min x R n v Z n Q[x v] c R n s.t. µ(q) = min Q[c v] v Zn Min c Q = {v Z n : Q[c v] =µ(q)} inhomogeneous Hermite invariant H(Q) = µ(q) (det Q) 1/n maximizing i-hermite = maximizing covering

25 Voronoi II Theorem. DSV 2009 Q local maximum for H Q i-perfect and i-eutactic. inhomogeneous perfectness: { (1 ) 1 lin : v Min v)( v c Q} has max. dim. ( ) n for all deep holes c inhomogeneous eutaxy: ( 1 c c cc + µ(q) n Q 1 ) = v Min c Q α v ( 1 v )( 1 v ) with α v > 0 for all deep holes c Consequence: Local maxima are rational

26 Venkov theory Nice lattices are strongly perfect. Q strongly perfect: Min Q is spherical 4-design Theorem. Venkov 2001 Q strongly perfect = Q perfect and eutactic.

27 Spherical designs X S n 1 spherical t-design f(x)dω(x) = 1 S n 1 X x X f(x) for all polynomials of degree t. point conf. strength n-gon n - 1 simplex 2 cross polytope 3 icosahedron

28 Venkov i-theory Q strongly i-perfect: Min c Q is spherical 4-design for all deep holes c Theorem. DSV 2009 Q strongly i-perfect = Q i-perfect and i-eutactic.

29 This happens to the nice lattices: lattice packing covering Z A 2 D 4 E 6 E 7 E 8 K 12 BW 16 Λ 24 global opt. global min. global opt. global min. global opt. i-eutactic global opt. strongly i-perfect global opt. strongly i-perfect global opt. i-eutactic conj. global opt. i-eutactic conj. global opt. strongly i-perfect global opt. local min.