Note on combinatorial optimization with max-linear objective functions

AbstractWe consider combinatorial optimization problems with a feasible solution set S⊆{0,1}n specified by a system of linear constraints in 0–1 variables. Additionally, several cost functions c1,…,cp are given. The max-linear objective function is defined by f(x):=max{c1x,…,cpx: x∈S}; where cq:=(cq1,…,cqn) is for q=1,…,p an integer row vector in Rn.The problem of minimizing f(x) over S is called the max-linear combinatorial optimization (MLCO) problem.We will show that MLCO is NP-hard even for the simplest case of S⊆{0,1}n and p=2, and strongly NP-hard for general p. We discuss the relation to multi-criteria optimization and develop some bounds for MLCO

Online Contention Resolution Schemes

We introduce a new rounding technique designed for online optimization problems, which is related to contention resolution schemes, a technique initially introduced in the context of submodular function maximization. Our rounding technique, which we call online contention resolution schemes (OCRSs), is applicable to many online selection problems, including Bayesian online selection, oblivious posted pricing mechanisms, and stochastic probing models. It allows for handling a wide set of constraints, and shares many strong properties of offline contention resolution schemes. In particular, OCRSs for different constraint families can be combined to obtain an OCRS for their intersection. Moreover, we can approximately maximize submodular functions in the online settings we consider. We, thus, get a broadly applicable framework for several online selection problems, which improves on previous approaches in terms of the types of constraints that can be handled, the objective functions that can be dealt with, and the assumptions on the strength of the adversary. Furthermore, we resolve two open problems from the literature; namely, we present the first constant-factor constrained oblivious posted price mechanism for matroid constraints, and the first constant-factor algorithm for weighted stochastic probing with deadlines.Comment: 33 pages. To appear in SODA 201

Algorithms for weighted multidimensional search and perfect phylogeny

This dissertation is a collection of papers from two independent areas: convex optimization problems in R[superscript]d and the construction of evolutionary trees;The paper on convex optimization problems in R[superscript]d gives improved algorithms for solving the Lagrangian duals of problems that have both of the following properties. First, in absence of the bad constraints, the problems can be solved in strongly polynomial time by combinatorial algorithms. Second, the number of bad constraints is fixed. As part of our solution to these problems, we extend Cole\u27s circuit simulation approach and develop a weighted version of Megiddo\u27s multidimensional search technique;The papers on evolutionary tree construction deal with the perfect phylogeny problem, where species are specified by a set of characters and each character can occur in a species in one of a fixed number of states. This problem is known to be NP-complete. The dissertation contains the following results on the perfect phylogeny problem: (1) A linear time algorithm when all the characters have two states. (2) A polynomial time algorithm when the number of character states is fixed. (3) A polynomial time algorithm when the number of characters is fixed

A cooperative local search-based algorithm for the Multiple-Scenario Max-Min Knapsack Problem

8International audienceThe purpose of this article is to present a novel method to approximately solve the Multiple-Scenario Max-Min Knapsack Problem (MSM2KP). This problem models many real world situations, e.g. when for many scenarios noted $\pi \in \mathcal P=\{1,\ldots,P\}$, the aim is to identify the one offering a better alternative in term of maximizing the worst possible outcome. Herein is presented a cooperative approach based on two local search algorithms: (i) a limited-area local search applied in the elite neighborhood and which accepts the first solution with some deterioration threshold of the current solution, (ii) a wide range local search is applied to perform a sequence of paths exchange to improve the current solution. Results have been analyzed by means state-of-the art methods and via problem instances obtained by a generator code taken from the literature. The tests were executed in compeltely comparable scenarios to those of the literature. The results are promising and the efficiency of the proposed approach is also shown

Greedy and dynamic programming by calculation

Dissertação mestrado integrado em Informatics EngineeringThe mathematical study of the greedy algorithm provides a blueprint for the study of Dynamic Programming (DP), whose body of knowledge is largely unorganized, remaining obscure to a large part of the software engineering community. This study aims to structure this body of knowledge, narrowing the gap between a purely examplebased approach to DP and its scientific foundations. To that effect, matroid theory is leveraged through a pointfree relation algebra, which is applied to greedy and DP problems. A catalogue of such problems is compiled, and a broad characterization of DP algorithms is given. Alongside, the theory underlying the thinning relational operator is explored.O estudo matemático do algoritmo ganancioso («greedy») serve como guia para o estudo da programação dinâmica, cujo corpo de conhecimento permanece desorganizado e obscuro a uma grande parte da comunidade de engenharia de software. Este estudo visa estruturar esse corpo de conhecimento, fazendo a ponte entre a abordagem popular baseada em exemplos e os métodos mais teóricos da literatura científica. Para esse efeito, a teoria dos matroides é explorada pelo uso de uma álgebra de relações pointfree, e aplicada a problemas «greedy» e de programação dinâmica. Um catálogo de tais problemas é compilado, e é feita uma caraterização geral de algoritmos de programação dinâmica. Em paralelo, é explorada a teoria do combinador relacional de «thinning».This work is financed by National Funds through the Portuguese funding agency, FCT - Fundação para a Ciência e a Tecnologia, within project UIDB/50014/202

New perspectives and applications for greedy algorithms in machine learning

Approximating probability densities is a core problem in Bayesian statistics, where the inference involves the computation of a posterior distribution. Variational Inference (VI) is a technique to approximate posterior distributions through optimization. It involves specifying a set of tractable densities, out of which the final approximation is to be chosen. While VI is traditionally motivated with the goal of tractability, the focus in this dissertation is to use Bayesian approximation to obtain parsimonious distributions. With this goal in mind, we develop greedy algorithm variants and study their theoretical properties by establishing novel connections of the resulting optimization problems in parsimonious VI with traditional studies in the discrete optimization literature. Specific realizations lead to efficient solutions for many sparse probabilistic models like Sparse regression, Sparse PCA, Sparse Collective Matrix Factorization (CMF) etc. For cases where existing results are insufficient to provide acceptable approximation guarantees, we extend the optimization results for some large scale algorithms to a much larger class of functions.The developed methods are applied to both simulated and real world datasets, including high dimensional functional Magnetic Resonance Imaging (fMRI) datasets, and to the real world tasks of interpreting data exploration and model predictions.Electrical and Computer Engineerin

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)