The parameterized hardness of the k-center problem in transportation networks

In this paper we study the hardness of the k-Center problem on inputs that model transportation networks. For the problem, an edge-weighted graph G=(V,E) and an integer k are given and a center set C subseteq V needs to be chosen such that |C|<= k. The aim is to minimize the maximum distance of any vertex in the graph to the closest center. This problem arises in many applications of logistics, and thus it is natural to consider inputs that model transportation networks. Such inputs are often assumed to be planar graphs, low doubling metrics, or bounded highway dimension graphs. For each of these models, parameterized approximation algorithms have been shown to exist. We complement these results by proving that the k-Center problem is W[1]-hard on planar graphs of constant doubling dimension, where the parameter is the combination of the number of centers k, the highway dimension h, and even the treewidth t. Moreover, under the Exponential Time Hypothesis there is no f(k,t,h)* n^{o(t+sqrt{k+h})} time algorithm for any computable function f. Thus it is unlikely that the optimum solution to k-Center can be found efficiently, even when assuming that the input graph abides to all of the above models for transportation networks at once! Additionally we give a simple parameterized (1+{epsilon})-approximation algorithm for inputs of doubling dimension d with runtime (k^k/{epsilon}^{O(kd)})* n^{O(1)}. This generalizes a previous result, which considered inputs in D-dimensional L_q metrics

Parameterized complexity of the MINCCA problem on graphs of bounded decomposability

In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The "Minimum Changeover Cost Arborescence" (MINCCA) problem consists in finding an arborescence with a given root vertex such that the total changeover cost of the internal vertices is minimized. It has been recently proved by G\"oz\"upek et al. [TCS 2016] that the problem is FPT when parameterized by the treewidth and the maximum degree of the input graph. In this article we present the following results for the MINCCA problem: - the problem is W[1]-hard parameterized by the treedepth of the input graph, even on graphs of average degree at most 8. In particular, it is W[1]-hard parameterized by the treewidth of the input graph, which answers the main open problem of G\"oz\"upek et al. [TCS 2016]; - it is W[1]-hard on multigraphs parameterized by the tree-cutwidth of the input multigraph; - it is FPT parameterized by the star tree-cutwidth of the input graph, which is a slightly restricted version of tree-cutwidth. This result strictly generalizes the FPT result given in G\"oz\"upek et al. [TCS 2016]; - it remains NP-hard on planar graphs even when restricted to instances with at most 6 colors and 0/1 symmetric costs, or when restricted to instances with at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs.Comment: 25 pages, 11 figure

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]

On Sparse Hitting Sets: From Fair Vertex Cover to Highway Dimension

We consider the Sparse Hitting Set (Sparse-HS) problem, where we are given a set system (V,?,?) with two families ?,? of subsets of the universe V. The task is to find a hitting set for ? that minimizes the maximum number of elements in any of the sets of ?. This generalizes several problems that have been studied in the literature. Our focus is on determining the complexity of some of these special cases of Sparse-HS with respect to the sparseness k, which is the optimum number of hitting set elements in any set of ? (i.e., the value of the objective function). For the Sparse Vertex Cover (Sparse-VC) problem, the universe is given by the vertex set V of a graph, and ? is its edge set. We prove NP-hardness for sparseness k ? 2 and polynomial time solvability for k = 1. We also provide a polynomial-time 2-approximation algorithm for any k. A special case of Sparse-VC is Fair Vertex Cover (Fair-VC), where the family ? is given by vertex neighbourhoods. For this problem it was open whether it is FPT (or even XP) parameterized by the sparseness k. We answer this question in the negative, by proving NP-hardness for constant k. We also provide a polynomial-time (2-1/k)-approximation algorithm for Fair-VC, which is better than any approximation algorithm possible for Sparse-VC or the Vertex Cover problem (under the Unique Games Conjecture). We then switch to a different set of problems derived from Sparse-HS related to the highway dimension, which is a graph parameter modelling transportation networks. In recent years a growing literature has shown interesting algorithms for graphs of low highway dimension. To exploit the structure of such graphs, most of them compute solutions to the r-Shortest Path Cover (r-SPC) problem, where r > 0, ? contains all shortest paths of length between r and 2r, and ? contains all balls of radius 2r. It is known that there is an XP algorithm that computes solutions to r-SPC of sparseness at most h if the input graph has highway dimension h. However it was not known whether a corresponding FPT algorithm exists as well. We prove that r-SPC and also the related r-Highway Dimension (r-HD) problem, which can be used to formally define the highway dimension of a graph, are both W[1]-hard. Furthermore, by the result of Abraham et al. [ICALP 2011] there is a polynomial-time O(log k)-approximation algorithm for r-HD, but for r-SPC such an algorithm is not known. We prove that r-SPC admits a polynomial-time O(log n)-approximation algorithm

Hierarchy of Transportation Network Parameters and Hardness Results

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 k-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 Feldmann et al. [Andreas Emil Feldmann et al., 2015]. Finally we prove that on graphs G=(V,E) of skeleton dimension O(log^2 |V|) it is NP-hard to approximate the k-Center problem within a factor less than 2

Dagstuhl Reports : Volume 1, Issue 2, February 2011

Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

Station Location - Complexity and Approximation

We consider a geometric set covering problem. In its original form it consists of adding stations to an existing geometric transportation network so that each of a given set of settlements is not too far from a station. The problem is known to be NP-hard in general. However, special cases with certain properties have been shown to be efficiently solvable in theory and in practice, especially if the covering matrix has (almost) consecutive ones property. In this paper we are narrowing the gap between intractable and efficiently solvable cases of the problem. We also present an approximation algorithm for cases with almost consecutive ones property