arXiv subjects

Adam Kasperski

Publications and source records attributed to Adam Kasperski.

33 records · Page 2Linked to original sources

arXiv2018

Robust recoverable 0-1 optimization problems under polyhedral uncertainty

This paper deals with a robust recoverable approach to 0-1 programming problems. It is assumed that a solution constructed in the first stage can be modified to some extent in the second stage. This modification consists in choosing a solution in some prescribed neighborhood of the current solution. The second stage solution cost can be uncertain and a polyhedral structure of uncertainty is used. The resulting robust recoverable problem is a min-max-min problem, which can be hard to solve when the number of variables is large. In this paper we provide a framework for solving robust recoverable 0-1 programming problems with a specified polyhedral uncertainty and propose several lower bounds and approximate solutions, which can be used for a wide class of 0-1 optimization problems. The results of computational tests for two problems, namely the assignment and the knapsack ones, are also presented.

cs.DS↗

arXiv2018

Approximating multiobjective combinatorial optimization problems with the OWA criterion

The paper deals with a multiobjective combinatorial optimization problem with $K$ linear cost functions. The popular Ordered Weighted Averaging (OWA) criterion is used to aggregate the cost functions and compute a solution. It is well known that minimizing OWA for most basic combinatorial problems is weakly NP-hard even if the number of objectives $K$ equals two, and strongly NP-hard when $K$ is a part of the input. In this paper, the problem with nonincreasing weights in the OWA criterion and a large $K$ is considered. A method of reducing the number of objectives by appropriately aggregating the objective costs before solving the problem is proposed. It is shown that an optimal solution to the reduced problem has a guaranteed worst-case approximation ratio. Some new approximation results for the Hurwicz criterion, which is a special case of OWA, are also presented.

cs.DS↗

arXiv2017

Risk averse single machine scheduling - complexity and approximation

In this paper a class of single machine scheduling problems is considered. It is assumed that job processing times and due dates can be uncertain and they are specified in the form of discrete scenario set. A probability distribution in the scenario set is known. In order to choose a schedule some risk criteria such as the value at risk (VaR) an conditional value at risk (CVaR) are used. Various positive and negative complexity results are provided for basic single machine scheduling problems. In this paper new complexity results are shown and some known complexity results are strengthen.

cs.DS↗

arXiv2017

Robust recoverable and two-stage selection problems

In this paper the following selection problem is discussed. A set of $n$ items is given and we wish to choose a subset of exactly $p$ items of the minimum total cost. This problem is a special case of 0-1 knapsack in which all the item weights are equal to~1. Its deterministic version has a trivial $O(n)$-time algorithm, which consists in choosing $p$ items of the smallest costs. In this paper it is assumed that the item costs are uncertain. Two robust models, namely two-stage and recoverable ones, under discrete and interval uncertainty representations, are discussed. Several positive and negative complexity results for both of them are provided.

cs.DS↗

arXiv2017

On Recoverable and Two-Stage Robust Selection Problems with Budgeted Uncertainty

In this paper the problem of selecting $p$ out of $n$ available items is discussed, such that their total cost is minimized. We assume that costs are not known exactly, but stem from a set of possible outcomes. Robust recoverable and two-stage models of this selection problem are analyzed. In the two-stage problem, up to $p$ items is chosen in the first stage, and the solution is completed once the scenario becomes revealed in the second stage. In the recoverable problem, a set of $p$ items is selected in the first stage, and can be modified by exchanging up to $k$ items in the second stage, after a scenario reveals. We assume that uncertain costs are modeled through bounded uncertainty sets, i.e., the interval uncertainty sets with an additional linear (budget) constraint, in their discrete and continuous variants. Polynomial algorithms for recoverable and two-stage selection problems with continuous bounded uncertainty, and compact mixed integer formulations in the case of discrete bounded uncertainty are constructed.

math.OC↗

arXiv2016

Recoverable robust spanning tree problem under interval uncertainty representations

This paper deals with the recoverable robust spanning tree problem under interval uncertainty representations. A polynomial time, combinatorial algorithm for the recoverable spanning tree problem is first constructed. This problem generalizes the incremental spanning tree problem, previously discussed in literature. The algorithm built is then applied to solve the recoverable robust spanning tree problem, under the traditional interval uncertainty representation, in polynomial time. Moreover, the algorithm allows to obtain, under some mild assumptions about the uncertainty intervals,several approximation results for the recoverable robust spanning tree problem under the Bertsimas and Sim interval uncertainty representation and the interval uncertainty representation with a budget constraint.

cs.DS↗

arXiv2016

The robust recoverable spanning tree problem with interval costs is polynomially solvable

In this paper the robust recoverable spanning tree problem with interval edge costs is considered. The complexity of this problem has remained open to date. It is shown that the problem is polynomially solvable, by using an iterative relaxation method. A generalization of this idea to the robust recoverable matroid basis problem is also presented. Polynomial algorithms for both robust recoverable problems are proposed.

cs.DS↗

arXiv2015

Using the WOWA operator in robust discrete optimization problems

In this paper a class of discrete optimization problems with uncertain costs is discussed. The uncertainty is modeled by introducing a scenario set containing a finite number of cost scenarios. A probability distribution in the scenario set is available. In order to choose a solution the weighted OWA criterion (WOWA) is applied. This criterion allows decision makers to take into account both probabilities for scenarios and the degree of pessimism/ optimism. In this paper the complexity of the considered class of discrete optimization problems is described and some exact and approximation algorithms for solving it are proposed. An application to a selection problem, together with results of computational tests are shown.

cs.DS↗

arXiv2014

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.

cs.DS↗

arXiv2014

Combinatorial optimization problems with uncertain costs and the OWA criterion

In this paper a class of combinatorial optimization problems with uncertain costs is discussed. The uncertainty is modeled by specifying a discrete scenario set containing $K$ distinct cost scenarios. The Ordered Weighted Averaging (OWA for short) aggregation operator is applied to choose a solution. The well-known criteria such as: the maximum, minimum, average, Hurwicz and median are special cases of OWA. By using OWA, the traditional min-max approach to combinatorial optimization problems with uncertain costs, often regarded as too conservative, can be generalized. The computational complexity and approximability of the problem of minimizing OWA for the considered class of problems are investigated and some new positive and negative results in this area are provided. These results remain valid for many important problems, such as network or resource allocation problems.

cs.DS↗

arXiv2014

Approximability of the robust representatives selection problem

In this paper new complexity and approximation results on the robust versions of the representatives selection problem, under the scenario uncertainty representation, are provided, which extend the results obtained in the recent papers by Dolgui and Kovalev (2012), and Deineko and Woeginger (2013). Namely, it is shown that if the number of scenarios is a part of input, then the min-max (regret) representatives selection problem is not approximable within a ratio of $O(\log^{1-ε}K)$ for any $ε>0$, where $K$ is the number of scenarios, unless the problems in NP have quasi-polynomial time algorithms. An approximation algorithm with an approximation ratio of $O(\log K/ \log \log K)$ for the min-max version of the problem is also provided.

cs.DS↗

arXiv2014

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 $K$ 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 $K$ and not approximable within $O(\log^{1-ε}K)$ for any $ε>0$ 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.

cs.DS↗

arXiv2013

Bottleneck combinatorial optimization problems with uncertain costs and the OWA criterion

In this paper a class of bottleneck combinatorial optimization problems with uncertain costs is discussed. The uncertainty is modeled by specifying a discrete scenario set containing a finite number of cost vectors, called scenarios. In order to choose a solution the Ordered Weighted Averaging aggregation operator (shortly OWA) is applied. The OWA operator generalizes traditional criteria in decision making under uncertainty such as the maximum, minimum, average, median, or Hurwicz criterion. New complexity and approximation results in this area are provided. These results are general and remain valid for many problems, in particular for a wide class of network problems.

cs.DS↗

arXiv2010

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 $O(\log^{1-ε} n)$ for any $ε>0$ unless the problems in NP have quasi-polynomial time algorithms. Similarly, the 2-stage min-max problem cannot be approximated within $O(\log n)$ unless the problems in NP have quasi-polynomial time algorithms. In this paper randomized LP-based approximation algorithms with performance ratio of $O(\log^2 n)$ for min-max and 2-stage min-max problems are also proposed.

cs.CC↗

arXiv2008

On the approximability of minmax (regret) network optimization problems

In this paper the minmax (regret) versions of some basic polynomially solvable deterministic network problems are discussed. It is shown that if the number of scenarios is unbounded, then the problems under consideration are not approximable within $\log^{1-ε} K$ for any $ε>0$ unless NP $\subseteq$ DTIME$(n^{\mathrm{poly} \log n})$, where $K$ is the number of scenarios.

cs.CC↗