8,364 research outputs found
An Atypical Survey of Typical-Case Heuristic Algorithms
Heuristic approaches often do so well that they seem to pretty much always
give the right answer. How close can heuristic algorithms get to always giving
the right answer, without inducing seismic complexity-theoretic consequences?
This article first discusses how a series of results by Berman, Buhrman,
Hartmanis, Homer, Longpr\'{e}, Ogiwara, Sch\"{o}ening, and Watanabe, from the
early 1970s through the early 1990s, explicitly or implicitly limited how well
heuristic algorithms can do on NP-hard problems. In particular, many desirable
levels of heuristic success cannot be obtained unless severe, highly unlikely
complexity class collapses occur. Second, we survey work initiated by Goldreich
and Wigderson, who showed how under plausible assumptions deterministic
heuristics for randomized computation can achieve a very high frequency of
correctness. Finally, we consider formal ways in which theory can help explain
the effectiveness of heuristics that solve NP-hard problems in practice.Comment: This article is currently scheduled to appear in the December 2012
issue of SIGACT New
Asymptotic Sum-Capacity of Random Gaussian Interference Networks Using Interference Alignment
We consider a dense n-user Gaussian interference network formed by paired
transmitters and receivers placed independently at random in Euclidean space.
Under natural conditions on the node position distributions and signal
attenuation, we prove convergence in probability of the average per-user
capacity C_Sigma/n to 1/2 E log(1 + 2SNR).
The achievability result follows directly from results based on an
interference alignment scheme presented in recent work of Nazer et al. Our main
contribution comes through the converse result, motivated by ideas of
`bottleneck links' developed in recent work of Jafar. An information theoretic
argument gives a capacity bound on such bottleneck links, and probabilistic
counting arguments show there are sufficiently many such links to tightly bound
the sum-capacity of the whole network.Comment: 5 pages; to appear at ISIT 201
Shallow Circuits with High-Powered Inputs
A polynomial identity testing algorithm must determine whether an input
polynomial (given for instance by an arithmetic circuit) is identically equal
to 0. In this paper, we show that a deterministic black-box identity testing
algorithm for (high-degree) univariate polynomials would imply a lower bound on
the arithmetic complexity of the permanent. The lower bounds that are known to
follow from derandomization of (low-degree) multivariate identity testing are
weaker. To obtain our lower bound it would be sufficient to derandomize
identity testing for polynomials of a very specific norm: sums of products of
sparse polynomials with sparse coefficients. This observation leads to new
versions of the Shub-Smale tau-conjecture on integer roots of univariate
polynomials. In particular, we show that a lower bound for the permanent would
follow if one could give a good enough bound on the number of real roots of
sums of products of sparse polynomials (Descartes' rule of signs gives such a
bound for sparse polynomials and products thereof). In this third version of
our paper we show that the same lower bound would follow even if one could only
prove a slightly superpolynomial upper bound on the number of real roots. This
is a consequence of a new result on reduction to depth 4 for arithmetic
circuits which we establish in a companion paper. We also show that an even
weaker bound on the number of real roots would suffice to obtain a lower bound
on the size of depth 4 circuits computing the permanent.Comment: A few typos correcte
Polynomial time quantum computation with advice
Advice is supplementary information that enhances the computational power of
an underlying computation. This paper focuses on advice that is given in the
form of a pure quantum state and examines the influence of such advice on the
behaviors of an underlying polynomial-time quantum computation with
bounded-error probability.Comment: 9 page
- …