The graph parameters highway dimension and skeleton dimension were introduced to capture the properties of transportation networks. As many important optimization problems like Travelling Salesperson, Steiner Tree or @math -Center arise in such networks, it is worthwhile to study them on graphs of bounded highway or skeleton dimension. We investigate the relationships between mentioned parameters and how they are related to other important graph parameters that have been applied successfully to various optimization problems. We show that the skeleton dimension is incomparable to any of the parameters distance to linear forest, bandwidth, treewidth and highway dimension and hence, it is worthwhile to study mentioned problems also on graphs of bounded skeleton dimension. Moreover, we prove that the skeleton dimension is upper bounded by the max leaf number and that for any graph on at least three vertices there are edge weights such that both parameters are equal. Then we show that computing the highway dimension according to most recent definition is NP-hard, which answers an open question stated by Finally we prove that on graphs @math of skeleton dimension @math it is NP-hard to approximate the @math -Center problem within a factor less than @math .
We now briefly sum-up some algorithmic results in the context of optimization problems in transportation networks. Arora @cite_25 developed a general framework that enables PTASs for several geometric problems where the network is embedded in the Euclidean plane. Building upon the work of Arora, Talwar @cite_26 developed QPTASs for TSP , Steiner Tree , @math -Median and Facility Location on graphs of low (for a formal definition, see Definition ). This was improved by @cite_15 , who obtained a PTAS for TSP . As the skeleton dimension of a graph upper bounds its doubling dimension (cf. ) the aforementioned results immediately imply a PTAS for TSP and QPTASs for Steiner Tree , @math -Median and Facility Location .
@cite_15 @cite_26 @cite_25
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Given a directed graph @math and a list @math of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of @math that contains a directed @math path for every @math . The special case Directed Steiner Tree (when we ask for paths from a root @math to terminals @math ) is known to be fixed-parameter tractable parameterized by the number of terminals, while the special case Strongly Connected Steiner Subgraph (when we ask for a path from every @math to every other @math ) is known to be W[1]-hard. We systematically explore the complexity landscape of directed Steiner problems to fully understand which other special cases are FPT or W[1]-hard. Formally, if @math is a class of directed graphs, then we look at the special case of Directed Steiner Network where the list @math of requests form a directed graph that is a member of @math . Our main result is a complete characterization of the classes @math resulting in fixed-parameter tractable special cases: we show that if every pattern in @math has the combinatorial property of being "transitively equivalent to a bounded-length caterpillar with a bounded number of extra edges," then the problem is FPT, and it is W[1]-hard for every recursively enumerable @math not having this property. This complete dichotomy unifies and generalizes the known results showing that Directed Steiner Tree is FPT [Dreyfus and Wagner, Networks 1971], @math -Root Steiner Tree is FPT for constant @math [Such 'y, WG 2016], Strongly Connected Steiner Subgraph is W[1]-hard [, SIAM J. Discrete Math. 2011], and Directed Steiner Network is solvable in polynomial-time for constant number of terminals [Feldman and Ruhl, SIAM J. Comput. 2006], and moreover reveals a large continent of tractable cases that were not known before. We give the first polynomial-time approximation schemes (PTASs) for the following problems: (1) uniform facility location in edge-weighted planar graphs, (2) k-median and k-means in edge-weighted planar graphs, (3) k-means in Euclidean space of bounded dimension. Our first and second results extend to minor-closed families of graphs. All our results extend to cost functions that are the pth power of the shortest-path distance. The algorithm is local search where the local neighborhood of a solution S consists of all solutions obtained from S by removing and adding 1 eO(1) centers. We consider the Steiner tree problem in graphs under uncertainty, the so-called two-stage stochastic Steiner tree problem (SSTP). The problem consists of two stages: In the first stage, we do not know which nodes need to be connected. Instead, we know costs at which we may buy edges, and a set of possible scenarios one of which will arise in the second stage. Each scenario consists of its own terminal set, a probability, and second-stage edge costs. We want to find a selection of first-stage edges and second-stage edges for each scenario that minimizes the expected costs and satisfies all connectivity requirements. We show that SSTP is in the class of fixed-parameter tractable problems (FPT), parameterized by the number of terminals. Additionally, we transfer our results to the directed and the prize-collecting variant of SSTP. Given a graph @math and a subset @math of terminals, a of @math is a tree that spans @math . In the vertex-weighted Steiner tree (VST) problem, each vertex is assigned a non-negative weight, and the goal is to compute a minimum weight Steiner tree of @math . We study a natural generalization of the VST problem motivated by multi-level graph construction, the (V-GSST), which can be stated as follows: given a graph @math and terminals @math , where each terminal @math requires a facility of a minimum grade of service @math , compute a Steiner tree @math by installing facilities on a subset of vertices, such that any two vertices requiring a certain grade of service are connected by a path in @math with the minimum grade of service or better. Facilities of higher grade are more costly than facilities of lower grade. Multi-level variants such as this one can be useful in network design problems where vertices may require facilities of varying priority. While similar problems have been studied in the edge-weighted case, they have not been studied as well in the more general vertex-weighted case. We first describe a simple heuristic for the V-GSST problem whose approximation ratio depends on @math , the number of grades of service. We then generalize the greedy algorithm of [Klein & Ravi, 1995] to show that the V-GSST problem admits a @math -approximation, where @math is the set of terminals requiring some facility. This result is surprising, as it shows that the (seemingly harder) multi-grade problem can be approximated as well as the VST problem, and that the approximation ratio does not depend on the number of grades of service.