1 / 21

Restrictions on sums over

Restrictions on sums over. Yiannis Koutis Computer Science Department Carnegie Mellon University. Vectors over. p: prime d: dimension of the vectors point-wise multiplication mod p p = 3, d=3. Work on sum of vectors.

shereea
Download Presentation

Restrictions on sums over

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Restrictions on sums over Yiannis Koutis Computer Science Department Carnegie Mellon University

  2. Vectors over • p: prime • d: dimension of the vectors • point-wise multiplication mod p • p = 3, d=3

  3. Work on sum of vectors • f(n,d) : the minimum number such that every subset of has a sub-subset that sums up to • f(n,1) = 2n-1 [ Erdos, Ginzburg, Ziv] • f(n,d) · cd n [Alon, Dubiner] • f(n,d) ¸ (9/8)d/3 (n-1)2d+1 [Elsholtz] • f(n,2) = ? [conjectured equal to 4n-3]

  4. dimensionality restrictions • is an arbitrary subset of size p(d+2) • numbers of subsets of A summing up to • what is the probability that the number of zero-sum subsets of A is odd, when p=2? • how smaller can the number of zero-sum subsets be, than then number of v-sum subsets?

  5. dimensionality restrictions • is an arbitrary subset • numbers of subsets of A summing up to • what is the probability that the number of 0-sum subsets of A is odd, when p=2? [ prob = 1 ] • how smaller can the number of zero-sum subsets be, than then number of v-sum subsets? [zero is attractive : worst case one less ]

  6. motivation • The Set Packing problem: Given a collection C of sets on a universe U of n elements, is there a sub-collection of k mutually disjoint sets ? • Algebraization: • Assign variables xi to the elements of U • for each set S, let

  7. motivation • Let • If there are k disjoint terms, there is a multilinear term. If not, fk is in the ideal <x12,x2,2,..>

  8. example • Define the sets • Then :

  9. motivation • Let • If there are k disjoint terms, there is a multilinear term. If not, fk is in the ideal <x12,x2,2,..> • Basic idea: evaluate fkover a ‘small’ commutative ring with a polynomial number of operations and exploit the squares

  10. example • Assign distinct to element i and substitute v0+xviin xi • Then for every i • If there is no set packing of size k, then fk is a multiple of (1+x)2 • How large must d be so that the multilinear term is not a multiple of (1+x)2 ? [must be linear, unfortunately]

  11. representation theory for • each element is represented by a matrix • addition is isomorphic to matrix multiplication • 1-1: elements with entries in the first row

  12. representation theory for • The coefficient of xi in H(1,j) is the number of vj-sum sets of cardinality i in A. • For x=1, H(1,1) = #zero-sum+1 H(1,j) = #vj-sum

  13. representation theory for • All matrices  are simultaneously diagonalizable • V is a Hadamard matrix, every entry is 1 or -1 • () is diagonal, containing the eigenvalues which are all 1 and -1

  14. parity of zero sum subsets For x=1, H(1,1) = #zero-sum+1, H(1,j) = #vj-sum Each matrix (I+() ) has eigenvalues 0 and 2 For d +1 terms in the product, the eigenvalues are either 0 or 2d+1. All entries of H are even. #zero-sum+1 = even , #vj-sum=even

  15. number of zero-sum subsets 2dH(1,1) = trace(H) =sum of eigenvalues 2dH(1,j) = weight eigenvalues by 1 and -1 H(1,1)¸ H(1,j) [zero is attractive : #zero sum +1 ¸ #vj-sum]

  16. restrictions on sums • Let N(v,k) be the number of v-sum subsets of cardinality k Theorem: Given N(v,2t) mod 2,for 1· t · 2log n, the numbers N(v,2t) mod 2, for t>2log n can be determined completely .

  17. restrictions on sums - outline • Form • Also • v’ has only a 1 in the extra dimension • The coefficient aj of xj in H’, is zero mod 2 when j¸ 4d • aj is a linear combination of the coefficients of xj for j\leq 4d in H

  18. restrictions on sums • Let N(v,k) be the number of v-sum subsets of cardinality k Theorem: Given N(v,2t) mod 2,for 1· t · 2log n, the numbers N(v,2t) mod 2, for t>2log n can be completely determined. Question:What are the ‘admissible’ values for the 2log n free numbers, over selections ?

  19. generalizations to

  20. conclusions – back to motivation • Assign distinct to element i and substitute v0+xviin xi • Then for every i • If there is no set packing of size k, then fk is a multiple of (1+x)2 • How large must d be so that the multilinear term is not a multiple of (1+x)2 ? [must be linear, unfortunately]

  21. conclusions – back to motivation • If there is no set packing of size k, then fk is a multiple of (1+x)2 • But now, we know that fk must also satisfy many linear restrictions • Question: Can we exploit this algorithmically?

More Related