Applications of semidefinite programming in Algebraic Combinatorics

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1 Applications of semidefinite programming in Algebraic Combinatorics Tohoku University The 23rd RAMP Symposium October 24, 2011

2 We often want to 1 Bound the value of a numerical parameter of certain combinatorial configurations. parameter = size, index, configurations = codes, designs, spreads, ovoids, 2 Show that optimal (or nearly optimal) configurations have certain additional regularity. 3 Classify the optimal (or nearly optimal) configurations.

3 A chart ( AS stands for association schemes ) Delsarte (1973) Used AS as framework Applied LP (duality) Lovász (1979) -number Terwilliger (1992) Representation theory of AS Lovász-Schrijver (1991) Matrix-cuts Schrijver (2005) SDP bound for codes

4 Codes Q = {0, 1,..., q 1} : an alphabet of size q 2 C Q n : an (unrestricted) code of length n H (, ) : the Hamming distance on Q n : H (x, y) := {i = 1,..., n : x i y i } for x = (x 1,..., x n ), y = (y 1,..., y n ) Q n. d(c) := min{ H (x, y) : x, y C, x y} : the minimum distance of C

5 A classical problem Remark Set e := d(c) 1 2. Then e x y (C x, y, z ) z In other words, C is e-error-correcting. A q (n, d) := max{ C : d(c) d} Problem Determine A q (n, d). (Hard in general)

6 More modest problem Problem* Find a good upper bound on A q (n, d). Example B e (x) := {y Q n : H (x, y) e} : the ball with radius e and center x, where e := d 1 2 : e x A q (n, d) B e (x) = A q (n, d) e ( n ) i=0 i (q 1) i q n A code attaining equality in this sphere-packing bound is called a perfect code.

7 Association schemes X : a finite set C X : the X -dimensional column vector space over C C X X : the set of X X matrices over C R = {R 0,..., R n } : a set of non-empty subsets of X X A 0,..., A n C X X : the adjacency matrices : { 1, if (x, y) R i, (A i ) xy := 0, if (x, y) R i.

8 R = {R 0,..., R n } : a set of non-empty subsets of X X Definition The pair (X, R) is a (symmetric) association scheme if (AS1) A 0 = I (the identity matrix), (AS2) A A n = J (the all 1 s matrix), (AS3) A T i = A i (i = 0,..., n), (AS4) A i A j A := span{a 0,..., A n } (i, j = 0,..., n).

9 (AS4) A i A j = n h=0 ph ij A h A := span{a 0,..., A n } x h y i j z h p ij Remark A is a commutative matrix -algebra, so has a basis of primitive idempotents, i.e., E i E j = δ ij E i, E E n = I. A : the Bose Mesner algebra of (X, R)

10 Things to keep in mind The Bose Mesner algebra A of (X, R) is commutative and has a basis of 0-1 (so nonnegative) matrices A 0,..., A n, a basis of idempotent (so positive semidefinite) matrices E 0,..., E n. Each of the bases is an orthogonal basis with respect to M, N = trace(m N) (M, N C X X ).

11 The Hamming schemes Remark Q = {0, 1,..., q 1} X = Q n def (x, y) R i H (x, y) = i (i = 0,..., n) R = {R 0,..., R n } H(n, q) = (X, R) : the Hamming scheme H(n, q) admits G := S q S n as the group of automorphisms. A coincides with the commutant of G in C X X

12 The LP bound (Delsarte, 1973) For x X, set ^x = (0,..., 0, 1, 0,..., 0) T C X (a 1 in position x) C X : a code with minimum distance d(c) d χ C = x C ^x : the characteristic vector of C M := 1 C χ Cχ T C CX X : nonnegative & positive semidefinite M, I = 1, M, J = C M, A i = 0 for i = 1,..., d 1

13 The LP bound (Delsarte, 1973), continued Consider the following SDP problem: l LP = l LP (n, q, d) = max M, J subject to 1 M, I = 1, 2 M, A i = 0 (i = 1,..., d 1), 3 M : nonnegative & positive semidefinite. Then A q (n, d) l LP. Remark l LP is the strengthening of Lovász s ϑ-number due to Schrijver (1979).

14 max M, J ; M, I = 1, M, A i = 0 (i = 1,..., d 1), Example By projecting M to A, l LP turns to an LP: subject to 1 M, I = 1, (A q (n, d) ) l LP = l LP (n, q, d) = max M, J 2 M, A i = 0 (i = 1,..., d 1), 3 n M, A i i=0 A i, A i A i = n M, E i i=0 E i, E i E i 0 & 0, i.e., M, A i 0 (i = d,..., n), M, E i 0 (i = 1,..., n). l LP (16, 2, 6) = 256. In fact: A 2 (16, 6) = 256 (attained by the Nordstrom Robinson code).

15 Delsarte exploited duality of LP Remark Many of the known universal bounds on A q (n, d), including the sphere packing bound, are obtained by constructing nice feasible solutions to the dual problem of l LP. e := d 1 2 Ψ e (z) := e i=0 ( 1)i( )( z 1 n z i e i) (q 1) e i : Lloyd polynomial Theorem (Lloyd) If a perfect e-error-correcting code exists, then Ψ e (z) has e distinct zeros among the integers 1, 2,..., n.

16 More generally, In fact, this reduction to LP works for any SDP problem max M, B 0 subject to 1 M, B i = b i (i = 1,..., m), 2 M is positive semidefinite, whenever B 0,..., B m A for an association scheme (X, R). This was worked out in detail by Goemans and Rendl (1999) for the MAX-CUT problem.

17 Yet one more application Wilson (1984) used Delsarte s method to show the following Erdős Ko Rado theorem: Theorem (Erdős Ko Rado, 1961) Let v > (t + 1)(n t + 1) and let C be a collection of n-element subsets of Ω := {1,..., v} with the property x y t for all x, y C. Then C ( v t n t), with equality if and only if C = {x Ω : x = n, w x} for some t-element subset w Ω. This theorem has been extended to many other association schemes; cf. T. (2006, 2010).

18 The SDP bound (Schrijver, 2005) Remark For simplicity, we only consider binary codes, i.e., codes in H(n, 2). Q = {0, 1}, X = Q n A 0, A 1,..., A n : the adjacency matrices The Bose Mesner algebra A = span{a 0,..., A n } coincides with the commutant of G := S 2 S n in C X X. Below we shall consider the commutant of S n in C X X.

19 The Terwilliger algebra of H(n, 2) 0 := (0,..., 0) X E 0,..., E n C X X : the dual idempotents : { 1, if x = y, (0, x) R (E i, i ) xy := 0, otherwise. T := C[A 0,..., A n, E 0,..., E n ] : the Terwilliger algebra T = span{e i A j E h : i, j, h = 0,..., n}

20 T = span{e i A j E h : i, j, h = 0,..., n} Remark (E i A j E h ) xy = { 1, if (0, x) R i, (x, y) R j, (0, y) R h, 0, otherwise. 0 i h x j y E i A j E h = 0 unless i, j, h satisfy the triangle inequality.

21 Two matrices from a code C X : an (unrestricted) code Lemma (Schrijver, 2005) The matrices with M 1 SDP = i,j,h λ ijh E i A j E h, M 2 SDP = i,j,h(λ 0jj λ ijh )E i A j E h λ ijh := X C {(x, y, z) C3 : (x, y, z) satisfies ( )} {(x, y, z) X 3 : (x, y, z) satisfies ( )} are nonnegative & positive semidefinite, where x i h ( ) y j z It follows that λ 000 = 1 and n i=0 ( n i) λ0ii = C.

22 λ 000 = 1, n i=0 ( n i) λ0ii = C Consider the following SDP problem: subject to 1 λ 000 = 1, l SDP = l SDP (n, 2, d) = max n i=0 ( n i) λ0ii 2 λ ijh = λ i j h if (i, j, h ) is a permutation of (i, j, h), 3 i,j,h λ ijhe i A j E h : nonnegative & positive semidefinite, 4 i,j,h (λ 0jj λ ijh )E i A j E h : nonnegative & positive semidefinite, 5 λ ijh = 0 if {i, j, h} {1, 2,..., d 1}. Then A 2 (n, d) l SDP.

23 Computational results (Schrijver, 2005) Bounds on A 2 (n, d) best best upper lower bound bound previously n d known l SDP known l LP

24 2 λ ijh = λ i j h if (i, j, h ) is a permutation of (i, j, h). Remark If we omit 2, then lsdp essentially coincides with the application of matrix cuts (Lovász Schrijver, 1991) to l LP, followed by projecting to T. Gijswijt (2005) observed that 2 in the resulting bound. makes a huge difference Currently, several hierarchies of SDP bounds on A 2 (n, d) of the form l LP l (1) SDP l(2) SDP l(k) SDP... ( A 2(n, d)) have been proposed, and some numerical computations have also been carried out (e.g., Lasserre (2001), Laurent 2007), Gvozdenović Laurent Vallentin (2009), Gijswijt Mittelmann Schrijver (2010)).

25 l SDP is huge E i A j E h T CX X, X = 2 n. We reduce the size of l SDP by describing the Wedderburn decomposition (or block-diagonalization) of the matrix -algebra T (in a form convenient for the computation). T is the commutant of S n in C X X, and its Wedderburn decomposition was also found by Dunkl (1976) in the study of Krawtchouk polynomials.

26 Structure of irreducible T-modules T = span{e i A j E h : i, j, h = 0,..., n} n i=0 E i = I, E i E j = δ ij E i W C X : an irreducible T-module r := min{i = 0,..., n : E i W 0} : the endpoint of W Theorem (Go, 2002) We have More precisely, W = E r W E r+1w E n rw. dim E i W = { 1, if i = r, r + 1,..., n r, 0, otherwise. The isomorphism class of W is determined by the endpoint r.

27 Structure of irreducible T-modules, continued With respect to the decomposition W = E r W E r+1w E n rw, the matrix representing E i E i W = on W is 1 r. i. n r Remark (E i A j E h ) W is a sparse matrix. (In fact, its (i, h)-entry is written by a dual Hahn polynomial.) l SDP (n, q, d) (q 3) was studied by Gijswijt Schrijver T. (2006).

28 The quadratic assignment problem (QAP) X : a finite set S(X) : the symmetric group on X π(g) R X X : the permutation matrix of g S(X): (π(g)) xy = δ x,gy (x, y X). W, A R X X : the distance and flow matrices Consider the following QAP (without linear term): 1 min 2 W, g S(X) π(g)t Aπ(g).

29 min g S(X) 1 2 W, π(g)t Aπ(g) Suppose A = A 1 A for some association scheme (X, R). Write E i = 1 X n j=0 Q jia j (i = 0,..., n). Then (A 0,..., A n ) (C X X ) n+1 satisfies 1 A 0 = I, and A 1,..., A n are nonnegative, 2 A A n = J, 3 n j=0 Q jia j is positive semidefinite (i = 0,..., n). Conditions 1, 2, 3 hold for (π(g) T A 0 π(g),..., π(g) T A n π(g)) for any g S(X), as well as their convex combinations.

30 SDP relaxation of QAP (de Klerk et al., to appear) Thus, the SDP problem min 1 2 W, M 1 subject to 1 M 0 = I, and M 1,..., M n are nonnegative, 2 M M n = J, 3 n j=0 Q jim j is positive semidefinite (i = 0,..., n), gives a lower bound on QAP.

31 Minimum bisection problem X = 2m W : nonnegative ( ) 0m J A = m J m 0 m ( ) Jm I A 1 := A, A 2 := m 0 m 0 m J m I m A = span{i, A 1, A 2 }

32 Traveling salesman problem (de Klerk et al., 2008) W : nonnegative A = A = span{i, A, A 2,..., A X /2 }

33 Recent progress Remark When A is the commutant of a finite group G in C X X, then this SDP relaxation of QAP can be strengthened further (de Klerk Sotirov, to appear). For H(n, 2) (so G = S 2 S n ), the Terwilliger algebra T plays a role again in this case.

34 Related topics 1 Spherical codes and spherical designs LP bound Kissing number problem (k(4) = 24 (Musin, 2008)) SDP bound for spherical codes (Bachoc Vallentin, 2008) 2 Designs Dual concept to codes LP bound