Stefan Gollowitzer

MIP models for connected facility location

This article comprises the first theoretical and computational study on mixed integer programming (MIP) models for the connected facility location problem (ConFL). ConFL combines facility location and Steiner trees: given a set of customers, a set of potential facility locations and some inter-connection nodes, ConFL searches for the minimum-cost way of assigning each customer to exactly one open facility, and connecting the open facilities via a Steiner tree. The costs needed for building the Steiner tree, facility opening costs and the assignment costs need to be minimized. We model ConFL using seven compact and three mixed integer programming formulations of exponential size. We also show how to transform ConFL into the Steiner arborescence problem. A full hierarchy between the models is provided. For two exponential size models we develop a branch-and-cut algorithm. An extensive computational study is based on two benchmark sets of randomly generated instances with up to 1300 nodes and 115,000 edges. We empirically compare the presented models with respect to the quality of obtained bounds and the corresponding running time. We report optimal values for all but 16 instances for which the obtained gaps are below 0.6%.

Layered Graph Approaches to the Hop Constrained Connected Facility Location Problem

Given a set of customers, a set of potential facility locations, and some interconnection nodes, the goal of the connected facility location problem ConFL is to find the minimum-cost way of assigning each customer to exactly one open facility and connecting the open facilities via a Steiner tree. The sum of costs needed for building the Steiner tree, facility opening costs, and the assignment costs needs to be minimized. If the number of edges between a prespecified node the so-called root and each open facility is limited, we speak of the hop constrained facility location problem HC ConFL. This problem is of importance in the design of data-management and telecommunication networks. In this article we provide the first theoretical and computational study for this new problem that has not been studied in the literature so far. We propose two disaggregation techniques that enable the modeling of HC ConFL: i as a directed asymmetric ConFL on layered graphs, or ii as the Steiner arborescence problem SA on layered graphs. This allows for usage of best-known mixed integer programming models for ConFL or SA to solve the corresponding hop constrained problem to optimality. In our polyhedral study, we compare the obtained models with respect to the quality of their linear programming lower bounds. These models are finally computationally compared in an extensive computational study on a set of publicly available benchmark instances. Optimal values are reported for instances with up to 1,300 nodes and 115,000 edges.

A node splitting technique for two level network design problems with transition nodes

The Two Level Network Design (TLND) problem arises when local broadband access networks are planned in areas, where no existing infrastructure can be used, i.e., in the so-called greenfield deployments. Mixed strategies of Fiber-To-The-Home and Fiber-To-The-Curb, i.e., some customers are served by copper cables, some by fiber optic lines, can be modeled by an extension of the TLND. We are given two types of customers (primary and secondary), an additional set of Steiner nodes and fixed costs for installing either a primary or a secondary technology on each edge. The TLND problem seeks a minimum cost connected subgraph obeying a tree-tree topology, i.e., the primary nodes are connected by a rooted primary tree; the secondary nodes can be connected using both primary and secondary technology. In this paper we study an important extension of TLND in which additional transition costs need to be paid for intermediate facilities placed at the transition nodes, i.e., nodes where the change of technology takes place. The introduction of transition node costs leads to a problem with a rich structure permitting us to put in evidence reformulation techniques such as modeling in higher dimensional graphs (which in this case are based on a node splitting technique). We first provide a compact way of modeling intermediate facilities. We then present several generalizations of the facility-based inequalities involving an exponential number of constraints. Finally we show how to model the problem in an extended graph based on node splitting. Our main result states that the connectivity constraints on the splitted graph, projected back into the space of the variables of the original model, provide a new family of inequalities that implies, and even strictly dominates, all previously described cuts. We also provide a polynomial time separation algorithm for the more general cuts by calculating maximum flows on the splitted graph. We compare the proposed models both theoretically and computationally.

Rounding on the standard simplex: regular grids for global optimization

Given a point on the standard simplex, we calculate a proximal point on the regular grid which is closest with respect to any norm in a large class, including all

A comparison of several models for the hamiltonian p-median problem

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New models for and numerical tests of the Hamiltonian p-median problem

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Enhanced formulations and branch-and-cut for the two level network design problem with transition facilities

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