163 research outputs found
A Proof of the Cameron-Ku conjecture
A family of permutations A \subset S_n is said to be intersecting if any two
permutations in A agree at some point, i.e. for any \sigma, \pi \in A, there is
some i such that \sigma(i)=\pi(i). Deza and Frankl showed that for such a
family, |A| <= (n-1)!. Cameron and Ku showed that if equality holds then A =
{\sigma \in S_{n}: \sigma(i)=j} for some i and j. They conjectured a
`stability' version of this result, namely that there exists a constant c < 1
such that if A \subset S_{n} is an intersecting family of size at least
c(n-1)!, then there exist i and j such that every permutation in A maps i to j
(we call such a family `centred'). They also made the stronger `Hilton-Milner'
type conjecture that for n \geq 6, if A \subset S_{n} is a non-centred
intersecting family, then A cannot be larger than the family C = {\sigma \in
S_{n}: \sigma(1)=1, \sigma(i)=i \textrm{for some} i > 2} \cup {(12)}, which has
size (1-1/e+o(1))(n-1)!.
We prove the stability conjecture, and also the Hilton-Milner type conjecture
for n sufficiently large. Our proof makes use of the classical representation
theory of S_{n}. One of our key tools will be an extremal result on
cross-intersecting families of permutations, namely that for n \geq 4, if A,B
\subset S_{n} are cross-intersecting, then |A||B| \leq ((n-1)!)^{2}. This was a
conjecture of Leader; it was recently proved for n sufficiently large by
Friedgut, Pilpel and the author.Comment: Updated version with an expanded open problems sectio
Sparse Sums of Positive Semidefinite Matrices
Recently there has been much interest in "sparsifying" sums of rank one
matrices: modifying the coefficients such that only a few are nonzero, while
approximately preserving the matrix that results from the sum. Results of this
sort have found applications in many different areas, including sparsifying
graphs. In this paper we consider the more general problem of sparsifying sums
of positive semidefinite matrices that have arbitrary rank.
We give several algorithms for solving this problem. The first algorithm is
based on the method of Batson, Spielman and Srivastava (2009). The second
algorithm is based on the matrix multiplicative weights update method of Arora
and Kale (2007). We also highlight an interesting connection between these two
algorithms.
Our algorithms have numerous applications. We show how they can be used to
construct graph sparsifiers with auxiliary constraints, sparsifiers of
hypergraphs, and sparse solutions to semidefinite programs
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