12,351 research outputs found
Algorithm Engineering in Robust Optimization
Robust optimization is a young and emerging field of research having received
a considerable increase of interest over the last decade. In this paper, we
argue that the the algorithm engineering methodology fits very well to the
field of robust optimization and yields a rewarding new perspective on both the
current state of research and open research directions.
To this end we go through the algorithm engineering cycle of design and
analysis of concepts, development and implementation of algorithms, and
theoretical and experimental evaluation. We show that many ideas of algorithm
engineering have already been applied in publications on robust optimization.
Most work on robust optimization is devoted to analysis of the concepts and the
development of algorithms, some papers deal with the evaluation of a particular
concept in case studies, and work on comparison of concepts just starts. What
is still a drawback in many papers on robustness is the missing link to include
the results of the experiments again in the design
Partial resampling to approximate covering integer programs
We consider column-sparse covering integer programs, a generalization of set
cover, which have a long line of research of (randomized) approximation
algorithms. We develop a new rounding scheme based on the Partial Resampling
variant of the Lov\'{a}sz Local Lemma developed by Harris & Srinivasan (2019).
This achieves an approximation ratio of , where is the minimum covering
constraint and is the maximum -norm of any column of the
covering matrix (whose entries are scaled to lie in ). When there are
additional constraints on the variable sizes, we show an approximation ratio of
(where is the maximum number
of non-zero entries in any column of the covering matrix). These results
improve asymptotically, in several different ways, over results of Srinivasan
(2006) and Kolliopoulos & Young (2005).
We show nearly-matching inapproximability and integrality-gap lower bounds.
We also show that the rounding process leads to negative correlation among the
variables, which allows us to handle multi-criteria programs
Approximation Algorithms for Covering/Packing Integer Programs
Given matrices A and B and vectors a, b, c and d, all with non-negative
entries, we consider the problem of computing min {c.x: x in Z^n_+, Ax > a, Bx
< b, x < d}. We give a bicriteria-approximation algorithm that, given epsilon
in (0, 1], finds a solution of cost O(ln(m)/epsilon^2) times optimal, meeting
the covering constraints (Ax > a) and multiplicity constraints (x < d), and
satisfying Bx < (1 + epsilon)b + beta, where beta is the vector of row sums
beta_i = sum_j B_ij. Here m denotes the number of rows of A.
This gives an O(ln m)-approximation algorithm for CIP -- minimum-cost
covering integer programs with multiplicity constraints, i.e., the special case
when there are no packing constraints Bx < b. The previous best approximation
ratio has been O(ln(max_j sum_i A_ij)) since 1982. CIP contains the set cover
problem as a special case, so O(ln m)-approximation is the best possible unless
P=NP.Comment: Preliminary version appeared in IEEE Symposium on Foundations of
Computer Science (2001). To appear in Journal of Computer and System Science
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
Curse of dimensionality reduction in max-plus based approximation methods: theoretical estimates and improved pruning algorithms
Max-plus based methods have been recently developed to approximate the value
function of possibly high dimensional optimal control problems. A critical step
of these methods consists in approximating a function by a supremum of a small
number of functions (max-plus "basis functions") taken from a prescribed
dictionary. We study several variants of this approximation problem, which we
show to be continuous versions of the facility location and -center
combinatorial optimization problems, in which the connection costs arise from a
Bregman distance. We give theoretical error estimates, quantifying the number
of basis functions needed to reach a prescribed accuracy. We derive from our
approach a refinement of the curse of dimensionality free method introduced
previously by McEneaney, with a higher accuracy for a comparable computational
cost.Comment: 8pages 5 figure
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