49,815 research outputs found
Approximating Edit Distance Within Constant Factor in Truly Sub-Quadratic Time
Edit distance is a measure of similarity of two strings based on the minimum
number of character insertions, deletions, and substitutions required to
transform one string into the other. The edit distance can be computed exactly
using a dynamic programming algorithm that runs in quadratic time. Andoni,
Krauthgamer and Onak (2010) gave a nearly linear time algorithm that
approximates edit distance within approximation factor .
In this paper, we provide an algorithm with running time
that approximates the edit distance within a constant
factor
Ranking with Submodular Valuations
We study the problem of ranking with submodular valuations. An instance of
this problem consists of a ground set , and a collection of monotone
submodular set functions , where each .
An additional ingredient of the input is a weight vector . The
objective is to find a linear ordering of the ground set elements that
minimizes the weighted cover time of the functions. The cover time of a
function is the minimal number of elements in the prefix of the linear ordering
that form a set whose corresponding function value is greater than a unit
threshold value.
Our main contribution is an -approximation algorithm
for the problem, where is the smallest non-zero marginal value that
any function may gain from some element. Our algorithm orders the elements
using an adaptive residual updates scheme, which may be of independent
interest. We also prove that the problem is -hard to
approximate, unless P = NP. This implies that the outcome of our algorithm is
optimal up to constant factors.Comment: 16 pages, 3 figure
Sparse Covers for Sums of Indicators
For all , we show that the set of Poisson Binomial
distributions on variables admits a proper -cover in total
variation distance of size ,
which can also be computed in polynomial time. We discuss the implications of
our construction for approximation algorithms and the computation of
approximate Nash equilibria in anonymous games.Comment: PTRF, to appea
Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions
We investigate three related and important problems connected to machine
learning: approximating a submodular function everywhere, learning a submodular
function (in a PAC-like setting [53]), and constrained minimization of
submodular functions. We show that the complexity of all three problems depends
on the 'curvature' of the submodular function, and provide lower and upper
bounds that refine and improve previous results [3, 16, 18, 52]. Our proof
techniques are fairly generic. We either use a black-box transformation of the
function (for approximation and learning), or a transformation of algorithms to
use an appropriate surrogate function (for minimization). Curiously, curvature
has been known to influence approximations for submodular maximization [7, 55],
but its effect on minimization, approximation and learning has hitherto been
open. We complete this picture, and also support our theoretical claims by
empirical results.Comment: 21 pages. A shorter version appeared in Advances of NIPS-201
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