Layered Graph Approaches to the Hop Constrained Connected Facility Location Problem

Abstract

G iven 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-based layered graph approach for the Steiner tree problem with revenues, budget and hop-constraints

“…The power of layered graphs has been recently demonstrated for many problems, including hop-and diameter-constrained spanning trees [19], hop-constrained connected facility location [23], or for problems that involve more general hop-or diameter-constraints (see, e.g., [16,17]). In this paper, we proposed a new extended formulation based on a layered graph for hop-constrained spanning/Steiner tree problems.…”

“…To this end, hop-constrained problems are formulated on G L by associating variables to the arcs A L of the layered graph, e.g., x h i j is one, if arc {i, j} is used on layer h (see, e.g., [19,23]). While this usually gives models with strong LP-bounds, the size of the resulting MIP formulations soon becomes prohibitive.…”

“…In state-of-the-art approaches, connectivity constraints are therefore modeled using cutset inequalities on layered graphs (see, e.g. [19,23]). …”

“…For a similar approach, see also [23]. By definition of dist(v, P), if v ∈ S, we must cross at least dist(v, P) − 1 layers in order to reach a node in P from v. It follows that all variables y h…”

An (MI)LP-Based Primal Heuristic for 3-Architecture Connected Facility Location in Urban Access Network Design

“…The ConFL has been introduced and proven to be NP-Hard in [11]. A hop-constrained version of ConFL that is related to the design of single-architecture access network has been studied in [12]. The canonical ConFL considers a single architecture and can be associated to the design of an urban access network using a single technology (i.e., either optical fiber or copper connections).…”