8 research outputs found
Making Robust Decisions in Discrete Optimization Problems as a Game against Nature
In this paper a discrete optimization problem under uncertainty is discussed. Solving such a problem can be seen as a game against nature. In order to choose a solution, the minmax and minmax regret criteria can be applied. In this paper an extension of the known minmax (regret) approach is proposed. It is shown how different types of uncertainty can be simultaneously taken into account. Some exact and approximation algorithms for choosing a best solution are constructed.Discrete optimization, minmax, minmax regret, game against nature
Robust Inference of Trees
This paper is concerned with the reliable inference of optimal
tree-approximations to the dependency structure of an unknown distribution
generating data. The traditional approach to the problem measures the
dependency strength between random variables by the index called mutual
information. In this paper reliability is achieved by Walley's imprecise
Dirichlet model, which generalizes Bayesian learning with Dirichlet priors.
Adopting the imprecise Dirichlet model results in posterior interval
expectation for mutual information, and in a set of plausible trees consistent
with the data. Reliable inference about the actual tree is achieved by focusing
on the substructure common to all the plausible trees. We develop an exact
algorithm that infers the substructure in time O(m^4), m being the number of
random variables. The new algorithm is applied to a set of data sampled from a
known distribution. The method is shown to reliably infer edges of the actual
tree even when the data are very scarce, unlike the traditional approach.
Finally, we provide lower and upper credibility limits for mutual information
under the imprecise Dirichlet model. These enable the previous developments to
be extended to a full inferential method for trees.Comment: 26 pages, 7 figure
Computing Minimum Spanning Trees with Uncertainty
We consider the minimum spanning tree problem in a setting where information
about the edge weights of the given graph is uncertain. Initially, for each
edge of the graph only a set , called an uncertainty area, that
contains the actual edge weight is known. The algorithm can `update'
to obtain the edge weight . The task is to output the edge set of
a minimum spanning tree after a minimum number of updates. An algorithm is
-update competitive if it makes at most times as many updates as the
optimum. We present a 2-update competitive algorithm if all areas are
open or trivial, which is the best possible among deterministic algorithms. The
condition on the areas is to exclude degenerate inputs for which no
constant update competitive algorithm can exist. Next, we consider a setting
where the vertices of the graph correspond to points in Euclidean space and the
weight of an edge is equal to the distance of its endpoints. The location of
each point is initially given as an uncertainty area, and an update reveals the
exact location of the point. We give a general relation between the edge
uncertainty and the vertex uncertainty versions of a problem and use it to
derive a 4-update competitive algorithm for the minimum spanning tree problem
in the vertex uncertainty model. Again, we show that this is best possible
among deterministic algorithms
The update complexity of selection and related problems
We present a framework for computing with input data specified by intervals, representing uncertainty in the values of the input parameters. To compute a solution, the algorithm can query the input parameters that yield more refined estimates in form of sub-intervals and the objective is to minimize the number of queries. The previous approaches address the scenario where every query returns an exact value. Our framework is more general as it can deal with a wider variety of inputs and query responses and we establish interesting relationships between them that have not been investigated previously. Although some of the approaches of the previous restricted models can be adapted to the more general model, we
require more sophisticated techniques for the analysis and we also obtain improved algorithms for the previous model. We address selection problems in the generalized model and show that there exist 2-update competitive algorithms that do not depend on the lengths or distribution of the sub-intervals and hold against the worst case adversary. We also obtain similar bounds on the competitive ratio for the MST problem
in graphs
Minmax regret combinatorial optimization problems: an Algorithmic Perspective
Candia-Vejar, A (reprint author), Univ Talca, Modeling & Ind Management Dept, Curico, Chile.Uncertainty in optimization is not a new ingredient. Diverse models considering uncertainty have been developed over the last 40 years. In our paper we essentially discuss a particular uncertainty model associated with combinatorial optimization problems, developed in the 90's and broadly studied in the past years. This approach named minmax regret (in particular our emphasis is on the robust deviation criteria) is different from the classical approach for handling uncertainty, stochastic approach, where uncertainty is modeled by assumed probability distributions over the space of all possible scenarios and the objective is to find a solution with good probabilistic performance. In the minmax regret (MMR) approach, the set of all possible scenarios is described deterministically, and the search is for a solution that performs reasonably well for all scenarios, i.e., that has the best worst-case performance. In this paper we discuss the computational complexity of some classic combinatorial optimization problems using MMR. approach, analyze the design of several algorithms for these problems, suggest the study of some specific research problems in this attractive area, and also discuss some applications using this model
On the Complexity of the Robust Spanning Tree Problem With Interval Data
This paper studies the complexity of the robust spanning tree problem with interval data (RSTID). It settles the conjecture of Kouvelis and Yu [9] and shows that the problem remains NP-complete even when the underlying graph is complete or when the cost intervals are all [0; 1]. These results prove that the diculty of RSTID stems from two distinct aspects: the topology of the graph and the numerical properties of the cost intervals. As a consequence, they suggest new directions for improving and evaluating existing search algorithms [2,15,20] for this problem, since they have so far focused only on one of these aspects