5,704 research outputs found
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]
A PTAS for Bounded-Capacity Vehicle Routing in Planar Graphs
The Capacitated Vehicle Routing problem is to find a minimum-cost set of
tours that collectively cover clients in a graph, such that each tour starts
and ends at a specified depot and is subject to a capacity bound on the number
of clients it can serve. In this paper, we present a polynomial-time
approximation scheme (PTAS) for instances in which the input graph is planar
and the capacity is bounded. Previously, only a quasipolynomial-time
approximation scheme was known for these instances. To obtain this result, we
show how to embed planar graphs into bounded-treewidth graphs while preserving,
in expectation, the client-to-client distances up to a small additive error
proportional to client distances to the depot
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
A survey of approximation algorithms for capacitated vehicle routing problems
Finding the shortest travelling tour of vehicles with capacity k from the
depot to the customers is called the Capacity vehicle routing problem (CVRP).
CVRP plays an essential position in logistics systems, and it is the most
intensively studied problem in combinatorial optimization. In complexity, CVRP
with k 3 is an NP-hard problem, and it is APX-hard as well. We already
knew that it could not be approximated in metric space. Moreover, it is the
first problem resisting Arora's famous approximation framework. So, whether
there is, a polynomial-time (1+)-approximation for the Euclidean CVRP
for any is still an open problem. This paper will summarize the
research progress from history to up-to-date developments. The survey will be
updated periodically.Comment: First submissio
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
Distances and shortest paths on graphs of bounded highway dimension: simple, fast, dynamic
Dijkstra's algorithm is the standard method for computing shortest paths on
arbitrary graphs. However, it is slow for large graphs, taking at least linear
time. It has been long known that for real world road networks, creating a
hierarchy of well-chosen shortcuts allows fast distance and path computation,
with exact distance queries seemingly being answered in logarithmic time.
However, these methods were but heuristics until the work of Abraham et
al.~[JACM 2016], where they defined a graph parameter called highway dimension
which is constant for real-world road networks, and showed that in graphs of
constant highway dimension, a shortcut hierarchy exists that guarantees
shortest distance computation takes time and
space, where is the ratio of the smallest to largest edge, and is the
number of vertices. The problem is that they were unable to efficiently compute
the hierarchy of shortcuts. Here we present a simple and efficient algorithm to
compute the needed hierarchy of shortcuts in time and space ,
as well as supporting updates in time
A (1+ε)-embedding of low highway dimension graphs into bounded treewidth graphs
Graphs with bounded highway dimension were introduced by Abraham et al. [Proceedings of SODA 2010, pp. 782–793] as a model of transportation networks. We show that any such graph can be embedded into a distribution over bounded treewidth graphs with arbitrarily small distortion. More concretely, given a weighted graph G = (V, E) of constant highway dimension, we show how to randomly compute a weighted graph H = (V, E) that distorts shortest path distances of G by at most a 1 + ε factor in expectation, and whose treewidth is polylogarithmic in the aspect ratio of G. Our probabilistic embedding implies quasi-polynomial time approximation schemes for a number of optimization problems that naturally arise in transportation networks, including Travelling Salesman, Steiner Tree, and Facility Location. To construct our embedding for low highway dimension graphs we extend Talwar’s [Proceedings of STOC 2004, pp. 281–290] embedding of low doubling dimension metrics into bounded treewidth graphs, which generalizes known results for Euclidean metrics. We add several nontrivial ingredients to Talwar’s techniques, and in particular thoroughly analyze the structure of low highway dimension graphs. Thus we demonstrate that the geometric toolkit used for Euclidean metrics extends beyond the class of low doubling metrics
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