10 research outputs found
Complexity of the robust weighted independent set problems on interval graphs
This paper deals with the max-min and min-max regret versions of the maximum
weighted independent set problem on interval graphswith uncertain vertex
weights. Both problems have been recently investigated by Nobibon and Leus
(2014), who showed that they are NP-hard for two scenarios and strongly NP-hard
if the number of scenarios is a part of the input. In this paper, new
complexity and approximation results on the problems under consideration are
provided, which extend the ones previously obtained. Namely, for the discrete
scenario uncertainty representation it is proven that if the number of
scenarios is a part of the input, then the max-min version of the problem
is not at all approximable. On the other hand, its min-max regret version is
approximable within and not approximable within for
any unless the problems in NP have quasi polynomial algorithms.
Furthermore, for the interval uncertainty representation it is shown that the
min-max regret version is NP-hard and approximable within 2
On the approximability of robust spanning tree problems
In this paper the minimum spanning tree problem with uncertain edge costs is
discussed. In order to model the uncertainty a discrete scenario set is
specified and a robust framework is adopted to choose a solution. The min-max,
min-max regret and 2-stage min-max versions of the problem are discussed. The
complexity and approximability of all these problems are explored. It is proved
that the min-max and min-max regret versions with nonnegative edge costs are
hard to approximate within for any unless
the problems in NP have quasi-polynomial time algorithms. Similarly, the
2-stage min-max problem cannot be approximated within unless the
problems in NP have quasi-polynomial time algorithms. In this paper randomized
LP-based approximation algorithms with performance ratio of for
min-max and 2-stage min-max problems are also proposed
Single machine scheduling problems with uncertain parameters and the OWA criterion
In this paper a class of single machine scheduling problems is discussed. It
is assumed that job parameters, such as processing times, due dates, or weights
are uncertain and their values are specified in the form of a discrete scenario
set. The Ordered Weighted Averaging (OWA) aggregation operator is used to
choose an optimal schedule. The OWA operator generalizes traditional criteria
in decision making under uncertainty, such as the maximum, average, median or
Hurwicz criterion. It also allows us to extend the robust approach to
scheduling by taking into account various attitudes of decision makers towards
the risk. In this paper a general framework for solving single machine
scheduling problems with the OWA criterion is proposed and some positive and
negative computational results for two basic single machine scheduling problems
are provided
Universal Sequencing on a Single Machine
We consider scheduling on an unreliable machine that may experience unexpected changes in processing speed or even full breakdowns. We aim for a universal solution that performs well without adaptation for any possible machine behavior. For the objective of minimizing the total weighted completion time, we design a polynomial time deterministic algorithm that finds a universal scheduling sequence with a solution value within 4 times the value of an optimal clairvoyant algorithm that knows the disruptions in advance. A randomized version of this algorithm attains in expectation a ratio of e. We also show that both results are best possible among all universal solutions. As a direct consequence of our results, we answer affirmatively the question of whether a constant approximation algorithm exists for the offline version of the problem when machine unavailability periods are known in advance. When jobs have individual release dates, the situation changes drastically. Even if all weights are equal, there are instances for which any universal solution is a factor of Ω(log n/ log log n) worse than an optimal sequence. Motivated by this hardness, we study the special case when the processing time of each job is proportional to its weight. We present a non-trivial algorithm with a small constant performance guarantee. © 2010 Springer-Verlag
Single machine scheduling with scenarios
In the field of robust optimization, the goal is to provide solutions to combinatorial problems that hedge against variations of the numerical parameters. This constitutes an effort to design algorithms that are applicable in the presence of uncertainty in the definition of the instance. We study the single machine scheduling problem with the objective of minimizing the weighted sum of completion times. We model uncertainty by replacing the vector of numerical values in the description of the instance by a set of possible vectors, called scenarios. The goal is to find a schedule of minimum value in the worst-case scenario. We first show that the general problem cannot be approximated within O(log1-εn) for any ε>0, unless NP has quasi-polynomial algorithms. We then study more tractable special cases and obtain a linear program (LP)-based 2-approximation algorithm for the unweighted case. We show that our analysis is tight by providing a matching lower bound on the integrality gap of the LP. Moreover, we prove that the unweighted version is NP-hard to approximate within a factor less than 6/5. We conclude by presenting a polynomial-time algorithm based on dynamic programming for the case when the number of scenarios and the values of the instance are bounded by some constant.©2012 Elsevier B.V. All rights reserved
Approximating Single Machine Scheduling with Scenarios
Abstract. In the field of robust optimization, the goal is to provide solutions to combinatorial problems that hedge against variations of the numerical parameters. This constitutes an effort to design algorithms that are applicable in the presence of uncertainty in the definition of the instance. We study the single machine scheduling problem with the objective to minimize the weighted sum of completion times. We model uncertainty by replacing the vector of numerical values in the description of the instance by a set of possible vectors, called scenarios. The goal is to find the schedule with minimum value in the worst-case scenario. We first show that the general problem is intractable by proving that it cannot be approximated within O(log 1−ε n) for any ε>0, unless NP has quasi-polynomial algorithms. We then study more tractable special cases and obtain an LP based 2-approximation algorithm for the unweighted case. We show that our analysis is tight by providing a matching lower bound on the integrality gap of the LP. Moreover, we prove that the unweighted version is NP-hard to approximate within a factor less than 6/5. We conclude by presenting a polynomial time algorithm based on dynamic programming for the case when the number of scenarios and the values of the instance are bounded by some constant.