501,215 research outputs found
Strongly Polynomial Primal-Dual Algorithms for Concave Cost Combinatorial Optimization Problems
We introduce an algorithm design technique for a class of combinatorial
optimization problems with concave costs. This technique yields a strongly
polynomial primal-dual algorithm for a concave cost problem whenever such an
algorithm exists for the fixed-charge counterpart of the problem. For many
practical concave cost problems, the fixed-charge counterpart is a well-studied
combinatorial optimization problem. Our technique preserves constant factor
approximation ratios, as well as ratios that depend only on certain problem
parameters, and exact algorithms yield exact algorithms.
Using our technique, we obtain a new 1.61-approximation algorithm for the
concave cost facility location problem. For inventory problems, we obtain a new
exact algorithm for the economic lot-sizing problem with general concave
ordering costs, and a 4-approximation algorithm for the joint replenishment
problem with general concave individual ordering costs
Online and quasi-online colorings of wedges and intervals
We consider proper online colorings of hypergraphs defined by geometric
regions. We prove that there is an online coloring algorithm that colors
intervals of the real line using colors such that for every
point , contained in at least intervals, not all the intervals
containing have the same color. We also prove the corresponding result
about online coloring a family of wedges (quadrants) in the plane that are the
translates of a given fixed wedge. These results contrast the results of the
first and third author showing that in the quasi-online setting 12 colors are
enough to color wedges (independent of and ). We also consider
quasi-online coloring of intervals. In all cases we present efficient coloring
algorithms
Compositions and Averages of Two Resolvents: Relative Geometry of Fixed Points Sets and a Partial Answer to a Question by C. Byrne
We show that the set of fixed points of the average of two resolvents can be
found from the set of fixed points for compositions of two resolvents
associated with scaled monotone operators. Recently, the proximal average has
attracted considerable attention in convex analysis. Our results imply that the
minimizers of proximal-average functions can be found from the set of fixed
points for compositions of two proximal mappings associated with scaled convex
functions. When both convex functions in the proximal average are indicator
functions of convex sets, least squares solutions can be completely recovered
from the limiting cycles given by compositions of two projection mappings. This
provides a partial answer to a question posed by C. Byrne. A novelty of our
approach is to use the notion of resolvent average and proximal average
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