30,203 research outputs found
Exponential Time Complexity of the Permanent and the Tutte Polynomial
We show conditional lower bounds for well-studied #P-hard problems:
(a) The number of satisfying assignments of a 2-CNF formula with n variables
cannot be counted in time exp(o(n)), and the same is true for computing the
number of all independent sets in an n-vertex graph.
(b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed
in time exp(o(n)).
(c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time
exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it
cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs.
Our lower bounds are relative to (variants of) the Exponential Time
Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF
formulas cannot be decided in time exp(o(n)). We relax this hypothesis by
introducing its counting version #ETH, namely that the satisfying assignments
cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds,
we transfer the sparsification lemma for d-CNF formulas to the counting
setting
Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process
Consider the zero set of the random power series f(z)=sum a_n z^n with i.i.d.
complex Gaussian coefficients a_n. We show that these zeros form a
determinantal process: more precisely, their joint intensity can be written as
a minor of the Bergman kernel. We show that the number of zeros of f in a disk
of radius r about the origin has the same distribution as the sum of
independent {0,1}-valued random variables X_k, where P(X_k=1)=r^{2k}. Moreover,
the set of absolute values of the zeros of f has the same distribution as the
set {U_k^{1/2k}} where the U_k are i.i.d. random variables uniform in [0,1].
The repulsion between zeros can be studied via a dynamic version where the
coefficients perform Brownian motion; we show that this dynamics is conformally
invariant.Comment: 37 pages, 2 figures, updated proof
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