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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The traveling salesman problem (TSP) is among the most famous NP-hard optimization problems. We design for this problem an algorithm that for any fixed @math computes in randomized polynomial time a @math -approximation to the optimal tour in TSP instances that form an arbitrary metric space with bounded intrinsic dimension. The celebrated results of Arora [J. ACM, 45 (1998), pp. 753--782] and Mitchell [SIAM J. Comput., 28 (1999), pp. 1298--1309] prove that the above result holds in the special case of TSP in a fixed-dimensional Euclidean space. Thus, our algorithm demonstrates that the algorithmic tractability of metric TSP depends on the dimensionality of the space and not on its specific geometry. This result resolves a problem that has been open since the quasi-polynomial time algorithm of Talwar [Proceedings of the 36th Annual ACM Symposium on Theory of Computing, 2004, pp. 281--290]. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c > 1 and given any n nodes in R 2 , a randomized version of the scheme finds a (1 + 1 c )-approximation to the optimum traveling salesman tour in O(n (log n ) O(c) ) time. When the nodes are in R d , the running time increases to O(n (log n ) (O( d c)) d-1 ). For every fixed c, d the running time is n • poly(log n ), that is nearly linear in n . The algorithmm can be derandomized, but this increases the running time by a factor O(n d ). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3 2-aproximation in polynomial time. We also give similar approximation schemes for some other NP-hard Euclidean problems: Minimum Steiner Tree, k -TSP, and k -MST. (The running times of the algorithm for k -TSP and k -MST involve an additional multiplicative factor k .) The previous best approximation algorithms for all these problems achieved a constant-factor approximation. We also give efficient approximation schemes for Euclidean Min-Cost Matching, a problem that can be solved exactly in polynomial time. All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as e p for p ≥ 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.