A structural approach to kernels for ILPs: Treewidth and Total Unimodularity

Kernelization is a theoretical formalization of efficient preprocessing for NP-hard problems. Empirically, preprocessing is highly successful in practice, for example in state-of-the-art ILP-solvers like CPLEX. Motivated by this, previous work studied the existence of kernelizations for ILP related problems, e.g., for testing feasibility of Ax <= b. In contrast to the observed success of CPLEX, however, the results were largely negative. Intuitively, practical instances have far more useful structure than the worst-case instances used to prove these lower bounds. In the present paper, we study the effect that subsystems with (Gaifman graph of) bounded treewidth or totally unimodularity have on the kernelizability of the ILP feasibility problem. We show that, on the positive side, if these subsystems have a small number of variables on which they interact with the remaining instance, then we can efficiently replace them by smaller subsystems of size polynomial in the domain without changing feasibility. Thus, if large parts of an instance consist of such subsystems, then this yields a substantial size reduction. We complement this by proving that relaxations to the considered structures, e.g., larger boundaries of the subsystems, allow worst-case lower bounds against kernelization. Thus, these relaxed structures can be used to build instance families that cannot be efficiently reduced, by any approach.Comment: Extended abstract in the Proceedings of the 23rd European Symposium on Algorithms (ESA 2015

Изучение структуры цинктитановых боросиликатных стекол по данным рассеяния рентгеновских лучей под малыми углами

В статті вивчено мікронеоднорідну будову цинктитанових боросилікатних стекол та процесів фазового розділу в них за даним розсіяння нейтронів під малими кутами. Зроблено висновок про характер розподілення часток, що виділяються за розмірами, які змінюються в дослідних стеклах в залежності від вмісту в них TiO₂ та ZnO. Встановлено вплив наявності мікронеоднорідностей після варки на характер їх фазовогорозподілення.In paper the micronon-uniform structure zinc-titanium borosilicate glass and processes of phase separation in them according to diffusing under vanishing angles of neutrons is investigated. It is drawn a In paper the micronon-uniform structure zinc-titanium borosilicate glass and processes of phase separation in them according to diffusing under vanishing angles of neutrons is investigated. It is drawn a leading-out on distribution of depositing corpuscles character on sizes which changes in studied glasses depending on the contents in them TiO₂ and ZnO. Effect of presence micro micronon-uniforms after melting on character of their phase separationis established

A computational analysis of lower bounds for big bucket production planning problems

In this paper, we analyze a variety of approaches to obtain lower bounds for multi-level production planning problems with big bucket capacities, i.e., problems in which multiple items compete for the same resources. We give an extensive survey of both known and new methods, and also establish relationships between some of these methods that, to our knowledge, have not been presented before. As will be highlighted, understanding the substructures of difficult problems provide crucial insights on why these problems are hard to solve, and this is addressed by a thorough analysis in the paper. We conclude with computational results on a variety of widely used test sets, and a discussion of future research

A decomposition algorithm for robust lot sizing problem with remanufacturing option

In this paper, we propose a decomposition procedure for constructing robust optimal production plans for reverse inventory systems. Our method is motivated by the need of overcoming the excessive computational time requirements, as well as the inaccuracies caused by imprecise representations of problem parameters. The method is based on a min-max formulation that avoids the excessive conservatism of the dualization technique employed by Wei et al. (2011). We perform a computational study using our decomposition framework on several classes of computer generated test instances and we report our experience. Bienstock and Özbay (2008) computed optimal base stock levels for the traditional lot sizing problem when the production cost is linear and we extend this work here by considering return inventories and setup costs for production. We use the approach of Bertsimas and Sim (2004) to model the uncertainties in the input

Algorithm Engineering in Robust Optimization

Robust optimization is a young and emerging field of research having received a considerable increase of interest over the last decade. In this paper, we argue that the the algorithm engineering methodology fits very well to the field of robust optimization and yields a rewarding new perspective on both the current state of research and open research directions. To this end we go through the algorithm engineering cycle of design and analysis of concepts, development and implementation of algorithms, and theoretical and experimental evaluation. We show that many ideas of algorithm engineering have already been applied in publications on robust optimization. Most work on robust optimization is devoted to analysis of the concepts and the development of algorithms, some papers deal with the evaluation of a particular concept in case studies, and work on comparison of concepts just starts. What is still a drawback in many papers on robustness is the missing link to include the results of the experiments again in the design

An FPTAS for optimizing a class of low-rank functions over a polytope

We present a fully polynomial time approximation scheme (FPTAS) for optimizing a very general class of non-linear functions of low rank over a polytope. Our approximation scheme relies on constructing an approximate Pareto-optimal front of the linear functions which constitute the given low-rank function. In contrast to existing results in the literature, our approximation scheme does not require the assumption of quasi-concavity on the objective function. For the special case of quasi-concave function minimization, we give an alternative FPTAS, which always returns a solution which is an extreme point of the polytope. Our technique can also be used to obtain an FPTAS for combinatorial optimization problems with non-linear objective functions, for example when the objective is a product of a fixed number of linear functions. We also show that it is not possible to approximate the minimum of a general concave function over the unit hypercube to within any factor, unless P = NP. We prove this by showing a similar hardness of approximation result for supermodular function minimization, a result that may be of independent interest

A branch-and-cut algorithm for a multi-item inventory distribution problem

This paper considers a multi-item inventory distribution problem motivated by a practical case occurring in the logistic operations of an hospital. There, a single warehouse supplies several nursing wards. The goal is to define a weekly distribution plan of medical products that minimizes the visits to wards, while respecting inventory capacities and safety stock levels. A mathematical formulation is introduced and several improvements such as tightening constraints, valid inequalities and an extended reformulation are discussed. In order to deal with real size instances, an hybrid heuristic based on mathematical models is introduced and the improvements are discussed. A branch-and-cut algorithm using all the discussed improvements is proposed. Finally, a computational experimentation is reported to show the relevance of the model improvements and the quality of the heuristic scheme