14,692 research outputs found
Distance-based exponential probability models on constrained combinatorial optimization problems
Estimation of distribution algorithms have already demonstrated their utility when solving a broad range of combinatorial problems. However, there is still room for methodological improvements when approaching constrained type problems. The great majority of works in the literature implement external repairing or penalty schemes, or use ad-hoc sampling methods in order to avoid unfeasible solutions. In this work, we present a new way to develop EDAs for this type of problems by implementing distance-based exponential probability models defined exclusively on the set of feasible solutions. In order to illustrate this procedure, we take the 2-partition balanced Graph Partitioning Problem as a case of study, and design efficient learning and sampling methods in order to use these distance-based probability models in EDAs
Approximation Algorithms for Distributionally Robust Stochastic Optimization with Black-Box Distributions
Two-stage stochastic optimization is a framework for modeling uncertainty,
where we have a probability distribution over possible realizations of the
data, called scenarios, and decisions are taken in two stages: we make
first-stage decisions knowing only the underlying distribution and before a
scenario is realized, and may take additional second-stage recourse actions
after a scenario is realized. The goal is typically to minimize the total
expected cost. A criticism of this model is that the underlying probability
distribution is itself often imprecise! To address this, a versatile approach
that has been proposed is the {\em distributionally robust 2-stage model}:
given a collection of probability distributions, our goal now is to minimize
the maximum expected total cost with respect to a distribution in this
collection.
We provide a framework for designing approximation algorithms in such
settings when the collection is a ball around a central distribution and the
central distribution is accessed {\em only via a sampling black box}.
We first show that one can utilize the {\em sample average approximation}
(SAA) method to reduce the problem to the case where the central distribution
has {\em polynomial-size} support. We then show how to approximately solve a
fractional relaxation of the SAA (i.e., polynomial-scenario
central-distribution) problem. By complementing this via LP-rounding algorithms
that provide {\em local} (i.e., per-scenario) approximation guarantees, we
obtain the {\em first} approximation algorithms for the distributionally robust
versions of a variety of discrete-optimization problems including set cover,
vertex cover, edge cover, facility location, and Steiner tree, with guarantees
that are, except for set cover, within -factors of the guarantees known
for the deterministic version of the problem
Convex lattice polygonal lines with a constrained number of vertices
A detailed combinatorial analysis of planar convex lattice polygonal lines is
presented. This makes it possible to answer an open question of Vershik
regarding the existence of a limit shape when the number of vertices is
constrained
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