Approximation Schemes for Node-Weighted Geometric Steiner Tree Problems

In this paper we introduce a new technique for approximation schemes for geometrical optimization problems. As an example problem, we consider the following variant of the geometric Steiner tree problem. Every point u which is not included in the tree costs a penalty of π(u) units. Furthermore, every Steiner point that we use costs c S units. The goal is to minimize the total length of the tree plus the penalties. Our technique yields a polynomial time approximation scheme for the problem, if the points lie in the plan

Travelling on Graphs with Small Highway Dimension

We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter was introduced by Abraham et al. [SODA 2010] as a model for transportation networks, on which TSP and STP naturally occur for various applications in logistics. It was previously shown [Feldmann et al. ICALP 2015] that these problems admit a quasi-polynomial time approximation scheme (QPTAS) on graphs of constant highway dimension. We demonstrate that a significant improvement is possible in the special case when the highway dimension is 1, for which we present a fully-polynomial time approximation scheme (FPTAS). We also prove that STP is weakly NP-hard for these restricted graphs. For TSP we show NP-hardness for graphs of highway dimension 6, which answers an open problem posed in [Feldmann et al. ICALP 2015]

Secluded Connectivity Problems

Consider a setting where possibly sensitive information sent over a path in a network is visible to every {neighbor} of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a path $P$ can be measured as the number of nodes adjacent to it, denoted by $N[P]$. A path is said to be secluded if its exposure is small. A similar measure can be applied to other connected subgraphs, such as Steiner trees connecting a given set of terminals. Such subgraphs may be relevant due to considerations of privacy, security or revenue maximization. This paper considers problems related to minimum exposure connectivity structures such as paths and Steiner trees. It is shown that on unweighted undirected $n$-node graphs, the problem of finding the minimum exposure path connecting a given pair of vertices is strongly inapproximable, i.e., hard to approximate within a factor of $O(2^{\log^{1-\epsilon}n})$ for any $\epsilon>0$ (under an appropriate complexity assumption), but is approximable with ratio $\sqrt{\Delta}+3$, where $\Delta$ is the maximum degree in the graph. One of our main results concerns the class of bounded-degree graphs, which is shown to exhibit the following interesting dichotomy. On the one hand, the minimum exposure path problem is NP-hard on node-weighted or directed bounded-degree graphs (even when the maximum degree is 4). On the other hand, we present a polynomial algorithm (based on a nontrivial dynamic program) for the problem on unweighted undirected bounded-degree graphs. Likewise, the problem is shown to be polynomial also for the class of (weighted or unweighted) bounded-treewidth graphs