1,966 research outputs found
Online Multistage Subset Maximization Problems
Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as subset maximization problems: One is given a ground set N={1,...,n}, a collection F subseteq 2^N of subsets thereof such that the empty set is in F, and an objective (profit) function p: F -> R_+. The task is to choose a set S in F that maximizes p(S). We consider the multistage version (Eisenstat et al., Gupta et al., both ICALP 2014) of such problems: The profit function p_t (and possibly the set of feasible solutions F_t) may change over time. Since in many applications changing the solution is costly, the task becomes to find a sequence of solutions that optimizes the trade-off between good per-time solutions and stable solutions taking into account an additional similarity bonus. As similarity measure for two consecutive solutions, we consider either the size of the intersection of the two solutions or the difference of n and the Hamming distance between the two characteristic vectors.
We study multistage subset maximization problems in the online setting, that is, p_t (along with possibly F_t) only arrive one by one and, upon such an arrival, the online algorithm has to output the corresponding solution without knowledge of the future.
We develop general techniques for online multistage subset maximization and thereby characterize those models (given by the type of data evolution and the type of similarity measure) that admit a constant-competitive online algorithm. When no constant competitive ratio is possible, we employ lookahead to circumvent this issue. When a constant competitive ratio is possible, we provide almost matching lower and upper bounds on the best achievable one
A Practical Guide to Robust Optimization
Robust optimization is a young and active research field that has been mainly
developed in the last 15 years. Robust optimization is very useful for
practice, since it is tailored to the information at hand, and it leads to
computationally tractable formulations. It is therefore remarkable that
real-life applications of robust optimization are still lagging behind; there
is much more potential for real-life applications than has been exploited
hitherto. The aim of this paper is to help practitioners to understand robust
optimization and to successfully apply it in practice. We provide a brief
introduction to robust optimization, and also describe important do's and
don'ts for using it in practice. We use many small examples to illustrate our
discussions
Changing Bases: Multistage Optimization for Matroids and Matchings
This paper is motivated by the fact that many systems need to be maintained
continually while the underlying costs change over time. The challenge is to
continually maintain near-optimal solutions to the underlying optimization
problems, without creating too much churn in the solution itself. We model this
as a multistage combinatorial optimization problem where the input is a
sequence of cost functions (one for each time step); while we can change the
solution from step to step, we incur an additional cost for every such change.
We study the multistage matroid maintenance problem, where we need to maintain
a base of a matroid in each time step under the changing cost functions and
acquisition costs for adding new elements. The online version of this problem
generalizes online paging. E.g., given a graph, we need to maintain a spanning
tree at each step: we pay for the cost of the tree at time
, and also for the number of edges changed at
this step. Our main result is an -approximation, where is
the number of elements/edges and is the rank of the matroid. We also give
an approximation for the offline version of the problem. These
bounds hold when the acquisition costs are non-uniform, in which caseboth these
results are the best possible unless P=NP.
We also study the perfect matching version of the problem, where we must
maintain a perfect matching at each step under changing cost functions and
costs for adding new elements. Surprisingly, the hardness drastically
increases: for any constant , there is no
-approximation to the multistage matching maintenance
problem, even in the offline case
K-Adaptability in Two-Stage Distributionally Robust Binary Programming
We propose to approximate two-stage distributionally robust programs with binary recourse decisions by their associated K-adaptability problems, which pre-select K candidate secondstage policies here-and-now and implement the best of these policies once the uncertain parameters have been observed. We analyze the approximation quality and the computational complexity of the K-adaptability problem, and we derive explicit mixed-integer linear programming reformulations. We also provide efficient procedures for bounding the probabilities with which each of the K second-stage policies is selected
- …