3,045 research outputs found
Complexity results and exact algorithms for robust knapsack problems.
This paper studies the robust knapsack problem, for which solutions are, up to a certain point, immune to data uncertainty. We complement the works found in the literature where uncertainty affects only the profits or only the weights of the items by studying the complexity and approximation of the general setting with uncertainty regarding both the profits and the weights, for three different objective functions. Furthermore, we develop a scenario-relaxation algorithm for solving the general problem and present computational results.Knapsack problem; Robustness; Scenario-relaxation algorithm; NP-hard; Approximation;
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
Counting approximately-shortest paths in directed acyclic graphs
Given a directed acyclic graph with positive edge-weights, two vertices s and
t, and a threshold-weight L, we present a fully-polynomial time
approximation-scheme for the problem of counting the s-t paths of length at
most L. We extend the algorithm for the case of two (or more) instances of the
same problem. That is, given two graphs that have the same vertices and edges
and differ only in edge-weights, and given two threshold-weights L_1 and L_2,
we show how to approximately count the s-t paths that have length at most L_1
in the first graph and length at most L_2 in the second graph. We believe that
our algorithms should find application in counting approximate solutions of
related optimization problems, where finding an (optimum) solution can be
reduced to the computation of a shortest path in a purpose-built auxiliary
graph
Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints
We investigate two new optimization problems -- minimizing a submodular
function subject to a submodular lower bound constraint (submodular cover) and
maximizing a submodular function subject to a submodular upper bound constraint
(submodular knapsack). We are motivated by a number of real-world applications
in machine learning including sensor placement and data subset selection, which
require maximizing a certain submodular function (like coverage or diversity)
while simultaneously minimizing another (like cooperative cost). These problems
are often posed as minimizing the difference between submodular functions [14,
35] which is in the worst case inapproximable. We show, however, that by
phrasing these problems as constrained optimization, which is more natural for
many applications, we achieve a number of bounded approximation guarantees. We
also show that both these problems are closely related and an approximation
algorithm solving one can be used to obtain an approximation guarantee for the
other. We provide hardness results for both problems thus showing that our
approximation factors are tight up to log-factors. Finally, we empirically
demonstrate the performance and good scalability properties of our algorithms.Comment: 23 pages. A short version of this appeared in Advances of NIPS-201
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