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Restrictions on sums over
Yiannis Koutis
Computer Science Department
Carnegie Mellon University
Carnegie Mellon
School of Computer Science
03/17/2004
Vectors over
•
•
•
•
p: prime
d: dimension of the vectors
point-wise multiplication mod p
p = 3, d=3
Carnegie Mellon
School of Computer Science
03/17/2004
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]
Carnegie Mellon
School of Computer Science
03/17/2004
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?
Carnegie Mellon
School of Computer Science
03/17/2004
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 ]
Carnegie Mellon
School of Computer Science
03/17/2004
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
Carnegie Mellon
School of Computer Science
03/17/2004
motivation
• Let
• If there are k disjoint terms, there is a multilinear
term. If not, fk is in the ideal <x12,x2,2,..>
Carnegie Mellon
School of Computer Science
03/17/2004
example
• Define the sets
• Then :
Carnegie Mellon
School of Computer Science
03/17/2004
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 fk over a ‘small’ commutative
ring with a polynomial number of operations and
exploit the squares
Carnegie Mellon
School of Computer Science
03/17/2004
example
• Assign distinct
v0+xvi in xi
to element i and substitute
• 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]
Carnegie Mellon
School of Computer Science
03/17/2004
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
Carnegie Mellon
School of Computer Science
03/17/2004
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
Carnegie Mellon
School of Computer Science
03/17/2004
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
Carnegie Mellon
School of Computer Science
03/17/2004
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
Carnegie Mellon
School of Computer Science
03/17/2004
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]
Carnegie Mellon
School of Computer Science
03/17/2004
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 .
Carnegie Mellon
School of Computer Science
03/17/2004
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
Carnegie Mellon
School of Computer Science
03/17/2004
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
?
Carnegie Mellon
School of Computer Science
03/17/2004
generalizations to
Carnegie Mellon
School of Computer Science
03/17/2004
conclusions – back to motivation
• Assign distinct
v0+xvi in xi
to element i and substitute
• 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]
Carnegie Mellon
School of Computer Science
03/17/2004
conclusions – back to motivation
• If there is no set packing of size k, then f k 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?
Carnegie Mellon
School of Computer Science
03/17/2004