Mind the Gap: Edge Facility Location Problems in Theory and Practice

Abstract

Motivated by applications in urban planning, network analysis, and data visualization, we introduce center selection problems in graphs where the centers are represented by edges. This is in contrast to classic center selection problems where centers are usually placed at the nodes of a graph. Given a weighted graph G(V, E) and a budget $k \in \mathbb {N}$, the goal is to select k edges from E such that the maximum distance from any point of interest in the graph to its nearest center is minimized. We consider three different problem variants, based on defining the points of interest either as the edges of G, or the nodes, or all points on the edges. We provide a variety of hardness results and approximation algorithms. A key difficulty of edge center selection is that the underlying distance function may not satisfy the triangle inequality, which is crucially used in approximation algorithms for node center selection. In addition, we introduce efficient heuristics that produce solutions of good quality even in large graphs, as demonstrated in our experimental evaluation.

Appendix

1.1 Appendix A: Omitted Proof

Theorem 4. There is a 4-approximation algorithm for PECS.

Proof

We use an adjusted version of the parametric pruning algorithm described in Subsect. 4.1. We first need to argue that the set R of possible solution distance values is still polynomially bounded despite for PECS all points on edges are considered as locations of interest. To accomplish that we use the following observation: For any edge $e=\{v,w\} \in E$ the maximum distance of a point p on e towards the nearest center is determined either by c(p, v) or c(p, w) plus the cost of the respective end point towards its nearest center. Hence for any pair of edges $e_v ,e_w \in E$ (including $e_v=e_w$) where $e_v$ is assumed to be the nearest center for v and $e_w$ the nearest center for w, the point on e which has maximum distance to $e_v$ and $e_w$ can be easily computed and the respective distance is added to R. Accordingly, we get $|R| \in \mathcal {O}(n^3)$.

Next, we make the crucial observation that if $\textrm{OPT}=r$, then all edges $e \in E$ with $c(e) >2r$ have to be included in the PECS solution, as otherwise the midpoint of the edge would already have a too large distance to the end points. Hence only if the number of these heavy edges does not already exceed k, a PECS solution with the requested size can exist. Based on this observation, we propose the following modification: We identify the set of edges $H := \{e \in E \mid c(e) > 2r\}$. After computing F we find the smallest edge cover that contains all edges $H' := \{e' \mid e \in H\}$ by first removing all nodes incident to $H'$ from F, computing a smallest edge cover on the resulting graph, adding $H'$ and adding any incident edge for uncovered nodes (nodes are still not covered iff they became isolated by removing $H'$ from F). As before, any node $v\in V$ has distance at most 3r from the set S containing the corresponding edges, and any point on an edge has distance at most r to its nearest node. Hence, this is a 4-approximation algorithm.    $\square$

1.2 Appendix B: Omitted Experimental Results

First, a remark about the benchmark instances: The $\psi$-values for the shown road networks are huge. It is important to note, though, that a small percentage of the edges is very short (length of 1) or very long ($L=\psi$) in these graphs, see Fig. 3 for an illustration of the edge weight distribution in NY. Considering the applications mentioned in the introduction, as placing parks in an urban area, such overly short or long edges would be non-sensical facilities. Hence when modelling suitable instances for such applications, merging short edges on degree-2 chains or subdividing long edges would be meaningful, and accordingly the $\psi$ value would decrease. The grid instances hence better reflect the kind of input we would expect for facility placement in a city road network.

In the main text we observe that GREEDY works quite fast even on large graphs given that the edge budget is sufficiently small. And for graphs with small $\psi$ value (e.g. grid instances), it also comes with a proper quality guarantee. For example, for the 200$\,\times \,$150 grid instance, the approximation factor is at most 3.25.

But we also observe a disadvantage of using GREEDY for road networks. As the algorithm always selects the edge with the furthest distance to the previously chosen ones, it often chooses centers in dead-ends, as illustrated in Fig. 4. (The same would happen with node centers in the classical setting.) Obviously, if the path to the dead-end is not too long, then it would make more sense to select the center at the beginning of that path instead of the end, as this would decrease the driving distance from all non-path edges towards the center. One could hence improve the solution with a postprocessing step or integrate the observation directly into the algorithm.

Fig. 4.

Cutout of a road network with edge centers (blue) chosen by the greedy approximation algorithm. Note that many chosen edges in this cutout are at the end of dead-end streets. (Color figure online)