Sherali-Adams gaps, flow-cover inequalities and generalized configurations for capacity-constrained Facility Location

Metric facility location is a well-studied problem for which linear programming methods have been used with great success in deriving approximation algorithms. The capacity-constrained generalizations, such as capacitated facility location (CFL) and lower-bounded facility location (LBFL), have proved notorious as far as LP-based approximation is concerned: while there are local-search-based constant-factor approximations, there is no known linear relaxation with constant integrality gap. According to Williamson and Shmoys devising a relaxation-based approximation for \cfl\ is among the top 10 open problems in approximation algorithms. This paper advances significantly the state-of-the-art on the effectiveness of linear programming for capacity-constrained facility location through a host of impossibility results for both CFL and LBFL. We show that the relaxations obtained from the natural LP at $\Omega(n)$ levels of the Sherali-Adams hierarchy have an unbounded gap, partially answering an open question of \cite{LiS13, AnBS13}. Here, $n$ denotes the number of facilities in the instance. Building on the ideas for this result, we prove that the standard CFL relaxation enriched with the generalized flow-cover valid inequalities \cite{AardalPW95} has also an unbounded gap. This disproves a long-standing conjecture of \cite{LeviSS12}. We finally introduce the family of proper relaxations which generalizes to its logical extreme the classic star relaxation and captures general configuration-style LPs. We characterize the behavior of proper relaxations for CFL and LBFL through a sharp threshold phenomenon.Comment: arXiv admin note: substantial text overlap with arXiv:1305.599

Extended Formulation Lower Bounds via Hypergraph Coloring?

Exploring the power of linear programming for combinatorial optimization problems has been recently receiving renewed attention after a series of breakthrough impossibility results. From an algorithmic perspective, the related questions concern whether there are compact formulations even for problems that are known to admit polynomial-time algorithms. We propose a framework for proving lower bounds on the size of extended formulations. We do so by introducing a specific type of extended relaxations that we call product relaxations and is motivated by the study of the Sherali-Adams (SA) hierarchy. Then we show that for every approximate relaxation of a polytope P, there is a product relaxation that has the same size and is at least as strong. We provide a methodology for proving lower bounds on the size of approximate product relaxations by lower bounding the chromatic number of an underlying hypergraph, whose vertices correspond to gap-inducing vectors. We extend the definition of product relaxations and our methodology to mixed integer sets. However in this case we are able to show that mixed product relaxations are at least as powerful as a special family of extended formulations. As an application of our method we show an exponential lower bound on the size of approximate mixed product formulations for the metric capacitated facility location problem, a problem which seems to be intractable for linear programming as far as constant-gap compact formulations are concerned

LP-Based Algorithms for Capacitated Facility Location

Linear programming has played a key role in the study of algorithms for combinatorial optimization problems. In the field of approximation algorithms, this is well illustrated by the uncapacitated facility location problem. A variety of algorithmic methodologies, such as LP-rounding and primal-dual method, have been applied to and evolved from algorithms for this problem. Unfortunately, this collection of powerful algorithmic techniques had not yet been applicable to the more general capacitated facility location problem. In fact, all of the known algorithms with good performance guarantees were based on a single technique, local search, and no linear programming relaxation was known to efficiently approximate the problem. In this paper, we present a linear programming relaxation with constant integrality gap for capacitated facility location. We demonstrate that the fundamental theories of multi-commodity flows and matchings provide key insights that lead to the strong relaxation. Our algorithmic proof of integrality gap is obtained by finally accessing the rich toolbox of LP-based methodologies: we present a constant factor approximation algorithm based on LP-rounding.Comment: 25 pages, 6 figures; minor revision

Lift & Project Systems Performing on the Partial-Vertex-Cover Polytope

We study integrality gap (IG) lower bounds on strong LP and SDP relaxations derived by the Sherali-Adams (SA), Lovasz-Schrijver-SDP (LS+), and Sherali-Adams-SDP (SA+) lift-and-project (L&P) systems for the t-Partial-Vertex-Cover (t-PVC) problem, a variation of the classic Vertex-Cover problem in which only t edges need to be covered. t-PVC admits a 2-approximation using various algorithmic techniques, all relying on a natural LP relaxation. Starting from this LP relaxation, our main results assert that for every epsilon > 0, level-Theta(n) LPs or SDPs derived by all known L&P systems that have been used for positive algorithmic results (but the Lasserre hierarchy) have IGs at least (1-epsilon)n/t, where n is the number of vertices of the input graph. Our lower bounds are nearly tight. Our results show that restricted yet powerful models of computation derived by many L&P systems fail to witness c-approximate solutions to t-PVC for any constant c, and for t = O(n). This is one of the very few known examples of an intractable combinatorial optimization problem for which LP-based algorithms induce a constant approximation ratio, still lift-and-project LP and SDP tightenings of the same LP have unbounded IGs. We also show that the SDP that has given the best algorithm known for t-PVC has integrality gap n/t on instances that can be solved by the level-1 LP relaxation derived by the LS system. This constitutes another rare phenomenon where (even in specific instances) a static LP outperforms an SDP that has been used for the best approximation guarantee for the problem at hand. Finally, one of our main contributions is that we make explicit of a new and simple methodology of constructing solutions to LP relaxations that almost trivially satisfy constraints derived by all SDP L&P systems known to be useful for algorithmic positive results (except the La system).Comment: 26 page

Exact Algorithms for Mixed-Integer Multilevel Programming Problems

We examine multistage optimization problems, in which one or more decision makers solve a sequence of interdependent optimization problems. In each stage the corresponding decision maker determines values for a set of variables, which in turn parameterizes the subsequent problem by modifying its constraints and objective function. The optimization literature has covered multistage optimization problems in the form of bilevel programs, interdiction problems, robust optimization, and two-stage stochastic programming. One of the main differences among these research areas lies in the relationship between the decision makers. We analyze the case in which the decision makers are self-interested agents seeking to optimize their own objective function (bilevel programming), the case in which the decision makers are opponents working against each other, playing a zero-sum game (interdiction), and the case in which the decision makers are cooperative agents working towards a common goal (two-stage stochastic programming). Traditional exact approaches for solving multistage optimization problems often rely on strong duality either for the purpose of achieving single-level reformulations of the original multistage problems, or for the development of cutting-plane approaches similar to Benders\u27 decomposition. As a result, existing solution approaches usually assume that the last-stage problems are linear or convex, and fail to solve problems for which the last-stage is nonconvex (e.g., because of the presence of discrete variables). We contribute exact finite algorithms for bilevel mixed-integer programs, three-stage defender-attacker-defender problems, and two-stage stochastic programs. Moreover, we do not assume linearity or convexity for the last-stage problem and allow the existence of discrete variables. We demonstrate how our proposed algorithms significantly outperform existing state-of-the-art algorithms. Additionally, we solve for the first time a class of interdiction and fortification problems in which the third-stage problem is NP-hard, opening a venue for new research and applications in the field of (network) interdiction

Discrete location models for content distribution

Cataloged from PDF version of article.The advances in information and computer technology has tremendously eased the way to reach electronic information. This, however, also brought forth many problems regarding the distribution of electronic content. This is especially true in the Internet, where there is a phenomenal growth of demand for any kind of electronic information, placing a high burden on the underlying infrastructure. In this dissertation, we study problems arising in distribution of electronic content. The first problem studied here is related to Content Distribution Networks (CDNs), which have emerged as a new technology to overcome the problems arising on the Internet due to the fast growth of the web-related traffic, such as slow response times and heavy server loads. They aim at increasing the effectiveness of the network by locating identical or partial copies of the origin server(s) throughout the network, which are referred to as proxy servers. In order for such structures to run efficiently, the CDN must be designed such that system resource are properly managed. To this purpose, we develop integer programming models for the problem of designing CDNs and investigate exact and heuristic algorithms for their solution. The second problem considered in this dissertation is Video Placement and Routing, which is related to the so-called Video-on-Demand (VoD) services. Such services are used to deliver programs to the users on request and find many applications in education, entertainment and business. Although bearing similarities with the CDN phenomena, VoD services have special characteristics with respect to the structure of the network and the type of content distributed. We study the problem of Video Placement and Routing for such networks and offer an optimization based solution algorithm for the associated integer programming model. The third problem studied here is the problem of allocating databases in distributed computing systems. In this context, we specifically focus on the well-known multidimensional Knapsack Problem (mKP). The mKP arises as a subproblem in solving the database location problem. We concentrate on the well known cover inequalities that are known to be important for the solution of the mKP. We then propose a novel separation procedure to identify violated cover inequalities and utilize this procedure in a branch-and-cut framework devised for the solution of the mKP.Bektaş, TolgaPh.D

An Optimisation-based Framework for Complex Business Process: Healthcare Application

The Irish healthcare system is currently facing major pressures due to rising demand, caused by population growth, ageing and high expectations of service quality. This pressure on the Irish healthcare system creates a need for support from research institutions in dealing with decision areas such as resource allocation and performance measurement. While approaches such as modelling, simulation, multi-criteria decision analysis, performance management, and optimisation can – when applied skilfully – improve healthcare performance, they represent just one part of the solution. Accordingly, to achieve significant and sustainable performance, this research aims to develop a practical, yet effective, optimisation-based framework for managing complex processes in the healthcare domain. Through an extensive review of the literature on the aforementioned solution techniques, limitations of using each technique on its own are identified in order to define a practical integrated approach toward developing the proposed framework. During the framework validation phase, real-time strategies have to be optimised to solve Emergency Department performance issues in a major hospital. Results show a potential of significant reduction in patients average length of stay (i.e. 48% of average patient throughput time) whilst reducing the over-reliance on overstretched nursing resources, that resulted in an increase of staff utilisation between 7% and 10%. Given the high uncertainty in healthcare service demand, using the integrated framework allows decision makers to find optimal staff schedules that improve emergency department performance. The proposed optimum staff schedule reduces the average waiting time of patients by 57% and also contributes to reduce number of patients left without treatment to 8% instead of 17%. The developed framework has been implemented by the hospital partner with a high level of success

Operational Research: Methods and Applications

Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order. The authors dedicate this paper to the 2023 Turkey/Syria earthquake victims. We sincerely hope that advances in OR will play a role towards minimising the pain and suffering caused by this and future catastrophes