A characterization of irreducible infeasible subsystems in flow networks

Infeasible network flow problems with supplies and demands can be characterized via violated cut-inequalities of the classical Gale-Hoffman theorem. Written as a linear program, irreducible infeasible subsystems (IISs) provide a different means of infeasibility characterization. In this article, we answer a question left open in the literature by showing a one-to-one correspondence between IISs and Gale-Hoffman-inequalities in which one side of the cut has to be weakly connected. We also show that a single max-flow computation allows one to compute an IIS. Moreover, we prove that finding an IIS of minimal cardinality in this special case of flow networks is strongly NP-hard

Identifying infeasible subsets of linear inequalities that are irreducible with respect to a given subset of the inequalities

A classical problem in the study of an infeasible system of linear inequalities is to determine irreducible infeasible subsets of inequalities (IIS), i.e. infeasible subsets of inequalities whose proper subsets are feasible. In this article, we examine a particular situation where only a given subsystem is of interest for the analysis of infeasibility. For this, we define B-IISs as infeasible subsets of inequalities that are irreducible with respect to a given subsystem. It is a generalization of the definition of an IIS, since an IIS is irreducible with respect to the full system. We provide a practical characterization of infeasible subsets irreducible with respect to a subsystem, making the link with the dual polytope commonly used in the detection of IISs. We then turn to the study of the BIISs that can be obtained from the Phase I of the simplex algorithm. We answer an open question regarding the covering of the clusters of such B-IISs and deduce a practical algorithm to find these covering B-IISs. Our findings are numerically illustratedon the Netlib infeasible linear programs

Graphs for Pattern Recognition

This monograph deals with mathematical constructions that are foundational in such an important area of data mining as pattern recognition. By using combinatorial and graph theoretic techniques, a closer look is taken at infeasible systems of linear inequalities, whose generalized solutions act as building blocks of geometric decision rules for pattern recognition. Infeasible systems of linear inequalities prove to be a key object in pattern recognition problems described in geometric terms thanks to the committee method. Such infeasible systems of inequalities represent an important special subclass of infeasible systems of constraints with a monotonicity property – systems whose multi-indices of feasible subsystems form abstract simplicial complexes (independence systems), which are fundamental objects of combinatorial topology. The methods of data mining and machine learning discussed in this monograph form the foundation of technologies like big data and deep learning, which play a growing role in many areas of human-technology interaction and help to find solutions, better solutions and excellent solutions

On the computational complexity of Ham-Sandwich cuts, Helly sets, and related problems

We study several canonical decision problems arising from some well-known theorems from combinatorial geometry. Among others, we show that computing the minimum size of a Caratheodory set and a Helly set and certain decision versions of the hs cut problem are W[1]-hard (and NP-hard) if the dimension is part of the input. This is done by fpt-reductions (which are actually ptime-reductions) from the d-Sum problem. Our reductions also imply that the problems we consider cannot be solved in time n^{o(d)} (where n is the size of the input), unless the Exponential-Time Hypothesis (ETH) is false. The technique of embedding d-Sum into a geometric setting is conceptually much simpler than direct fpt-reductions from purely combinatorial W[1]-hard problems (like the clique problem) and has great potential to show (parameterized) hardness and (conditional) lower bounds for many other problems

A Randomized Algorithm for the MaxFS Problem

We consider the NP-hard combinatorial optimization problem of finding a feasible subsystem of maximum cardinality among a given set of linear inequalities. In some new and challenging applications in digital broadcasting and in the modelling of protein folding potentials one faces very large MaxFS instances with up to millions of inequalities in thousands of variables. We introduce and analyze a randomized algorithm that is surprisingly successful at solving MaxFS instances that arise in these contexts and exhibit some numerical results.\ud \ud Raphael Hauser was supported through grant NAL/00720/G from the Nuffield Foundation and through grant GR/M30975 from the Engineering and Physical Sciences Research Council of the U

New Solution Methods for Joint Chance-Constrained Stochastic Programs with Random Left-Hand Sides

We consider joint chance-constrained programs with random lefthand sides. The motivation of this project is that this class of problem has many important applications, but there are few existing solution methods. For the most part, we deal with the subclass of problems for which the underlying parameter distributions are discrete. This assumption allows the original problem to be formulated as a deterministic equivalent mixed-integer program. We rst approach the problem as a mixed-integer program and derive a class of optimality cuts based on irreducibly infeasible subsets of the constraints of the scenarios of the problem. The IIS cuts can be computed effciently by means of a linear program. We give a method for improving the upper bound of the problem when no IIS cut can be identifi ed. We also give an implementation of an algorithm incorporating these ideas and finish with some computational results. We present a tabu search metaheuristic for fi nding good feasible solutions to the mixed-integer formulation of the problem. Our heuristic works by de ning a sufficient set of scenarios with the characteristic that all other scenarios do not have to be considered when generating upper bounds. We then use tabu search on the one-opt neighborhood of the problem. We give computational results that show our metaheuristic outperforming the state-of-the-art industrial solvers. We then show how to reformulate the problem so that the chance-constraints are monotonic functions. We then derive a convergent global branch-and-bound algorithm using the principles of monotonic optimization. We give a finitely convergent modi cation of the algorithm. Finally, we give a discussion on why this algorithm is computationally ine ffective. The last section of this dissertation details an application of joint chance-constrained stochastic programs to a vaccination allocation problem. We show why it is necessary to formulate the problem with random parameters and also why chance-constraints are a good framework for de fining an optimal policy. We give an example of the problem formulated as a chance constraint and a short numerical example to illustrate the concepts