Some 0/1 polytopes need exponential size extended formulations

We prove that there are 0/1 polytopes P⊆R[superscript n] that do not admit a compact LP formulation. More precisely we show that for every n there is a set X⊆{0,1}[superscript n] such that conv(X) must have extension complexity at least 2[superscript n/2⋅(1−o(1)] . In other words, every polyhedron Q that can be linearly projected on conv(X) must have exponentially many facets. In fact, the same result also applies if conv(X) is restricted to be a matroid polytope. Conditioning on NP⊈P[subscript /poly], our result rules out the existence of a compact formulation for any NP -hard optimization problem even if the formulation may contain arbitrary real numbers

Time and multiple objectives in scheduling and routing problems

Many optimization problems encountered in practice are multi-objective by nature, i.e., different objectives are conflicting and equally important. Many times, it is not desirable to drop some of them or to optimize them in a composite single objective or hierarchical manner. Furthermore, cost parameters change over time which makes optimization problems harder. For instance, in the transport sector, travel costs are a function of travel time which changes depending on the time of the day a vehicle is travelling (e.g., due to road congestion). Road congestion results in tremendous delays which lead to a decrease in the service quality and the responsiveness of logistic service providers. In Chapter 2, we develop a generic approach to deal with Multi-Objective Scheduling Problems (MOSPs) with State-Dependent Cost Parameters. The aim is to determine the set of Pareto solutions that capture the trade offs between the different conflicting objectives. Due to the complexity of MOSPs, an efficient approximation based on dynamic programming is developed. The approximation has a provable worse case performance guarantee. Even though the generated approximate Pareto front consist of fewer solutions, it still represents a good coverage of the true Pareto front. Furthermore, considerable gains in computation times are achieved. In Chapter 3, the developed methodology is validated on the multi-objective timedependent knapsack problem. In the classical knapsack problem, the input consists of a knapsack with a finite capacity and a set of items, each with a certain weight and a cost. A feasible solution to the knapsack problem is a selection of items such that their total weight does not exceed the knapsack capacity. The goal is to maximize the single objective function consisting of the total pro t of the selected items. We extend the classical knapsack problem in two ways. First, we consider time-dependent profits (e.g., in a retail environment profit depends on whether it is Christmas or not)

Proceedings of the XIII Global Optimization Workshop: GOW'16

[Excerpt] Preface: Past Global Optimization Workshop shave been held in Sopron (1985 and 1990), Szeged (WGO, 1995), Florence (GO’99, 1999), Hanmer Springs (Let’s GO, 2001), Santorini (Frontiers in GO, 2003), San José (Go’05, 2005), Mykonos (AGO’07, 2007), Skukuza (SAGO’08, 2008), Toulouse (TOGO’10, 2010), Natal (NAGO’12, 2012) and Málaga (MAGO’14, 2014) with the aim of stimulating discussion between senior and junior researchers on the topic of Global Optimization. In 2016, the XIII Global Optimization Workshop (GOW’16) takes place in Braga and is organized by three researchers from the University of Minho. Two of them belong to the Systems Engineering and Operational Research Group from the Algoritmi Research Centre and the other to the Statistics, Applied Probability and Operational Research Group from the Centre of Mathematics. The event received more than 50 submissions from 15 countries from Europe, South America and North America. We want to express our gratitude to the invited speaker Panos Pardalos for accepting the invitation and sharing his expertise, helping us to meet the workshop objectives. GOW’16 would not have been possible without the valuable contribution from the authors and the International Scientific Committee members. We thank you all. This proceedings book intends to present an overview of the topics that will be addressed in the workshop with the goal of contributing to interesting and fruitful discussions between the authors and participants. After the event, high quality papers can be submitted to a special issue of the Journal of Global Optimization dedicated to the workshop. [...

Approximation Algorithms for NP-Hard Problems

The workshop was concerned with the most important recent developments in the area of efficient approximation algorithms for NP-hard optimization problems as well as with new techniques for proving intrinsic lower bounds for efficient approximation

Algorithmes d'approximation pour des programmes linéaires et les problèmes de Packing avec des contraintes géometriques

In this thesis we approach several problems with approximation algorithms; these are feasibility problems as well as optimization problems. In Chapter 1 we give a brief introduction into the general paradigm of approximation algorithms, motivate the problems, and give an outline of the thesis. In Chapter 2, we discuss two algorithms to approximately generate a feasible solution of the mixed packing and covering problem which is a model from convex optimization. This problem includes a large class of linear programs. The algorithms generate approximately feasible solutions within O(M(ln M+epsilon^{-2} ln epsilon^{-1})) and O(M epsilon{-2} ln (M epsilon^{-1}))$iterations, respectively, where in each iteration a block problem which depends on the specific application has to be solved. Both algorithms, applied to linear programs, can result in column generation algorithms. In Chapter 3, we implement an algorithm for the so-called max-min-resource sharing problem. This is a certain convex optimization problem which, similar to the problem in Chapter 1, includes a large class of linear programs. The implementation, which is included in the appendix, is done in C++. We use the implementation in the context of an AFPTAS for Strip Packing in order to evaluate dynamic optimization of a parameter in the algorithm, namely the step length used for interpolation. We compare our choice to the static step length proposed in the analysis of the algorithm and conclude that dynamic optimization of the step length significantly reduces the number of iterations. In Chapter 4, we study two closely related scheduling problems, namely non-preemptive scheduling with fixed jobs and scheduling with non-availability for sequential jobs on m identical machines under the makespan objective, where m is constant. For the first problem, which does not admit an FPTAS unless P=NP, we obtain a new PTAS. For the second problem, we show that a suitable restriction (namely the permanent availability of one machine) is necessary to obtain a bounded approximation ratio. For this restriction, which does not admit an FPTAS unless P=NP, we present a PTAS; we also discuss the complexity of various special cases. In total, the results are basically best possible. In Chapter 5, we continue the studies from Chapter 4 where now the number m of machines is part of the input, which makes the problem algorithmically harder. Scheduling with fixed jobs does not admit an approximation ratio better than 3/2, unless P=NP; here we obtain an approximation ratio of 3/2+epsilon for any epsilon>0. For scheduling with non-availability, we require a constant percentage of the machines to be permanently available. This restriction also does not admit an approximation ratio better than 3/2 unless P=NP; we also obtain an approximation ratio of$3/2+\epsilon$ for any epsilon>0. With an interesting argument, the approximation ratio for both problems is refined to exactly 3/2. We also point out an interesting relation of scheduling with fixed jobs to Bin Packing. As in Chapter 4, the results are in a certain sense best possible. Finally, in Chapter 6, we conclude with some remarks and open research problems

Dagstuhl Reports : Volume 1, Issue 2, February 2011

Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn