Restrictions on sums over

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

• p prime
• d dimension of the vectors
• point-wise multiplication mod p
• p 3, d3

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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)2d1 Elsholtz
• f(n,2) ? conjectured equal to 4n-3

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dimensionality restrictions

• is an arbitrary subset of size
  p(d2)
• numbers of subsets of A summing up to
• what is the probability that the number of
  zero-sum subsets of A is odd, when p2?
• how smaller can the number of zero-sum subsets
  be, than then number of v-sum subsets?

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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 p2? 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
  

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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

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motivation

• Let
• If there are k disjoint terms, there is a
  multilinear term. If not, fk is in the ideal
  ltx12,x2,2,..gt

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example

• Define the sets
• Then

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motivation

• Let
• If there are k disjoint terms, there is a
  multilinear term. If not, fk is in the ideal
  ltx12,x2,2,..gt
• Basic idea evaluate fk over a small
  commutative ring with a polynomial number of
  operations and exploit the squares

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example

• Assign distinct to element i and
  substitute v0xvi in xi
• Then for every i
• If there is no set packing of size k, then fk is
  a multiple of (1x)2
• How large must d be so that the multilinear term
  is not a multiple of (1x)2 ? must be
  linear, unfortunately

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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

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representation theory for

• The coefficient of xi in H(1,j) is the number of
  vj-sum sets of cardinality i in A.
• For x1, H(1,1) zero-sum1
• H(1,j) vj-sum

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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

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parity of zero sum subsets

• For x1, H(1,1) zero-sum1, 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 2d1. All entries of H are even.
• zero-sum1 even , vj-sumeven

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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

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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 tgt2log n can be
  determined completely .

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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

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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 tgt2log n can be
  completely determined.
• Question What are the admissible values for
  the 2log n free numbers, over selections
  ?

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generalizations to
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conclusions back to motivation

• Assign distinct to element i and
  substitute v0xvi in xi
• Then for every i
• If there is no set packing of size k, then fk is
  a multiple of (1x)2
• How large must d be so that the multilinear term
  is not a multiple of (1x)2 ? must be
  linear, unfortunately

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conclusions back to motivation

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