41,409 research outputs found
On the complexity of nonlinear mixed-integer optimization
This is a survey on the computational complexity of nonlinear mixed-integer
optimization. It highlights a selection of important topics, ranging from
incomputability results that arise from number theory and logic, to recently
obtained fully polynomial time approximation schemes in fixed dimension, and to
strongly polynomial-time algorithms for special cases.Comment: 26 pages, 5 figures; to appear in: Mixed-Integer Nonlinear
Optimization, IMA Volumes, Springer-Verla
Complexity of Bradley-Manna-Sipma Lexicographic Ranking Functions
In this paper we turn the spotlight on a class of lexicographic ranking
functions introduced by Bradley, Manna and Sipma in a seminal CAV 2005 paper,
and establish for the first time the complexity of some problems involving the
inference of such functions for linear-constraint loops (without precondition).
We show that finding such a function, if one exists, can be done in polynomial
time in a way which is sound and complete when the variables range over the
rationals (or reals). We show that when variables range over the integers, the
problem is harder -- deciding the existence of a ranking function is
coNP-complete. Next, we study the problem of minimizing the number of
components in the ranking function (a.k.a. the dimension). This number is
interesting in contexts like computing iteration bounds and loop
parallelization. Surprisingly, and unlike the situation for some other classes
of lexicographic ranking functions, we find that even deciding whether a
two-component ranking function exists is harder than the unrestricted problem:
NP-complete over the rationals and -complete over the integers.Comment: Technical report for a corresponding CAV'15 pape
Computing a k-sparse n-length Discrete Fourier Transform using at most 4k samples and O(k log k) complexity
Given an -length input signal \mbf{x}, it is well known that its
Discrete Fourier Transform (DFT), \mbf{X}, can be computed in
complexity using a Fast Fourier Transform (FFT). If the spectrum \mbf{X} is
exactly -sparse (where ), can we do better? We show that
asymptotically in and , when is sub-linear in (precisely, where ), and the support of the non-zero DFT
coefficients is uniformly random, we can exploit this sparsity in two
fundamental ways (i) {\bf {sample complexity}}: we need only
deterministically chosen samples of the input signal \mbf{x} (where
when ); and (ii) {\bf {computational complexity}}: we can
reliably compute the DFT \mbf{X} using operations, where the
constants in the big Oh are small and are related to the constants involved in
computing a small number of DFTs of length approximately equal to the sparsity
parameter . Our algorithm succeeds with high probability, with the
probability of failure vanishing to zero asymptotically in the number of
samples acquired, .Comment: 36 pages, 15 figures. To be presented at ISIT-2013, Istanbul Turke
Superpolynomial lower bounds for general homogeneous depth 4 arithmetic circuits
In this paper, we prove superpolynomial lower bounds for the class of
homogeneous depth 4 arithmetic circuits. We give an explicit polynomial in VNP
of degree in variables such that any homogeneous depth 4 arithmetic
circuit computing it must have size .
Our results extend the works of Nisan-Wigderson [NW95] (which showed
superpolynomial lower bounds for homogeneous depth 3 circuits),
Gupta-Kamath-Kayal-Saptharishi and Kayal-Saha-Saptharishi [GKKS13, KSS13]
(which showed superpolynomial lower bounds for homogeneous depth 4 circuits
with bounded bottom fan-in), Kumar-Saraf [KS13a] (which showed superpolynomial
lower bounds for homogeneous depth 4 circuits with bounded top fan-in) and
Raz-Yehudayoff and Fournier-Limaye-Malod-Srinivasan [RY08, FLMS13] (which
showed superpolynomial lower bounds for multilinear depth 4 circuits). Several
of these results in fact showed exponential lower bounds.
The main ingredient in our proof is a new complexity measure of {\it bounded
support} shifted partial derivatives. This measure allows us to prove
exponential lower bounds for homogeneous depth 4 circuits where all the
monomials computed at the bottom layer have {\it bounded support} (but possibly
unbounded degree/fan-in), strengthening the results of Gupta et al and Kayal et
al [GKKS13, KSS13]. This new lower bound combined with a careful "random
restriction" procedure (that transforms general depth 4 homogeneous circuits to
depth 4 circuits with bounded support) gives us our final result
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