23,769 research outputs found
Robust randomized matchings
The following game is played on a weighted graph: Alice selects a matching
and Bob selects a number . Alice's payoff is the ratio of the weight of
the heaviest edges of to the maximum weight of a matching of size at
most . If guarantees a payoff of at least then it is called
-robust. In 2002, Hassin and Rubinstein gave an algorithm that returns
a -robust matching, which is best possible.
We show that Alice can improve her payoff to by playing a
randomized strategy. This result extends to a very general class of
independence systems that includes matroid intersection, b-matchings, and
strong 2-exchange systems. It also implies an improved approximation factor for
a stochastic optimization variant known as the maximum priority matching
problem and translates to an asymptotic robustness guarantee for deterministic
matchings, in which Bob can only select numbers larger than a given constant.
Moreover, we give a new LP-based proof of Hassin and Rubinstein's bound
New algorithms for the dual of the convex cost network flow problem with application to computer vision
Motivated by various applications to computer vision, we consider an integer convex optimization problem which is the dual of the convex cost network flow problem. In this
paper, we first propose a new primal algorithm for computing an optimal solution of the problem. Our primal algorithm iteratively updates primal variables by solving associated
minimum cut problems. The main contribution in this paper is to provide a tight bound for the number of the iterations. We show that the time complexity of the primal algorithm is K ¢ T(n;m) where K is the range of primal variables and T(n;m) is the time needed to compute a minimum cut in a graph with n nodes and m edges.
We then propose a primal-dual algorithm for the dual of the convex cost network flow problem. The primal-dual algorithm can be seen as a refined version of the primal algorithm by maintaining dual variables (flow) in addition to primal variables. Although its time complexity is the same as that for the primal algorithm, we can expect a better performance
practically.
We finally consider an application to a computer vision problem called the panoramic stitching problem. We apply several implementations of our primal-dual algorithm to some
instances of the panoramic stitching problem and test their practical performance.
We also show that our primal algorithm as well as the proofs can be applied to the L\-convex function minimization problem which is a more general problem than the dual of the convex cost network flow problem
Oracle-Based Robust Optimization via Online Learning
Robust optimization is a common framework in optimization under uncertainty
when the problem parameters are not known, but it is rather known that the
parameters belong to some given uncertainty set. In the robust optimization
framework the problem solved is a min-max problem where a solution is judged
according to its performance on the worst possible realization of the
parameters. In many cases, a straightforward solution of the robust
optimization problem of a certain type requires solving an optimization problem
of a more complicated type, and in some cases even NP-hard. For example,
solving a robust conic quadratic program, such as those arising in robust SVM,
ellipsoidal uncertainty leads in general to a semidefinite program. In this
paper we develop a method for approximately solving a robust optimization
problem using tools from online convex optimization, where in every stage a
standard (non-robust) optimization program is solved. Our algorithms find an
approximate robust solution using a number of calls to an oracle that solves
the original (non-robust) problem that is inversely proportional to the square
of the target accuracy
Efficient Algorithms for Envy-Free Stick Division With Fewest Cuts
Given a set of n sticks of various (not necessarily different) lengths, what
is the largest length so that we can cut k equally long pieces of this length
from the given set of sticks? We analyze the structure of this problem and show
that it essentially reduces to a single call of a selection algorithm; we thus
obtain an optimal linear-time algorithm.
This algorithm also solves the related envy-free stick-division problem,
which Segal-Halevi, Hassidim, and Aumann (AAMAS, 2015) recently used as their
central primitive operation for the first discrete and bounded envy-free cake
cutting protocol with a proportionality guarantee when pieces can be put to
waste.Comment: v3 adds more context about the proble
Continuous Multiclass Labeling Approaches and Algorithms
We study convex relaxations of the image labeling problem on a continuous
domain with regularizers based on metric interaction potentials. The generic
framework ensures existence of minimizers and covers a wide range of
relaxations of the originally combinatorial problem. We focus on two specific
relaxations that differ in flexibility and simplicity -- one can be used to
tightly relax any metric interaction potential, while the other one only covers
Euclidean metrics but requires less computational effort. For solving the
nonsmooth discretized problem, we propose a globally convergent
Douglas-Rachford scheme, and show that a sequence of dual iterates can be
recovered in order to provide a posteriori optimality bounds. In a quantitative
comparison to two other first-order methods, the approach shows competitive
performance on synthetical and real-world images. By combining the method with
an improved binarization technique for nonstandard potentials, we were able to
routinely recover discrete solutions within 1%--5% of the global optimum for
the combinatorial image labeling problem
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