The Metric Traveling Salesman Problem (TSP) is a classical NP-hard optimization problem. The double-tree shortcutting method for Metric TSP yields an exponentially-sized space of TSP tours, each of which approximates the optimal solution within at most a factor of 2. We consider the problem of finding among these tours the one that gives the closest approximation, i.e.\ the \emph{minimum-weight double-tree shortcutting}. Previously, we gave an efficient algorithm for this problem, and carried out its experimental analysis. In this paper, we address the related question of the worst-case approximation ratio for the minimum-weight double-tree shortcutting method. In particular, we give lower bounds on the approximation ratio in some specific metric spaces: the ratio of 2 in the discrete shortest path metric, 1.622 in the planar Euclidean metric, and 1.666 in the planar Minkowski metric. The first of these lower bounds is tight; we conjecture that the other two bounds are also tight, and in particular that the minimum-weight double-tree method provides a 1.622-approximation for planar Euclidean TSP.
Deep Dive into Minimum-weight double-tree shortcutting for Metric TSP: Bounding the approximation ratio.
The Metric Traveling Salesman Problem (TSP) is a classical NP-hard optimization problem. The double-tree shortcutting method for Metric TSP yields an exponentially-sized space of TSP tours, each of which approximates the optimal solution within at most a factor of 2. We consider the problem of finding among these tours the one that gives the closest approximation, i.e.\ the \emph{minimum-weight double-tree shortcutting}. Previously, we gave an efficient algorithm for this problem, and carried out its experimental analysis. In this paper, we address the related question of the worst-case approximation ratio for the minimum-weight double-tree shortcutting method. In particular, we give lower bounds on the approximation ratio in some specific metric spaces: the ratio of 2 in the discrete shortest path metric, 1.622 in the planar Euclidean metric, and 1.666 in the planar Minkowski metric. The first of these lower bounds is tight; we conjecture that the other two bounds are also tight, and
The Metric Travelling Salesman Problem (TSP) is a classical combinatorial optimization problem. We represent a set of n points in a metric space by a complete weighted graph on n nodes, where the weight of an edge is defined by the distance between the corresponding points. The objective of Metric TSP is to find in this graph a minimum-weight Hamiltonian cycle (equivalently, a minimum-weight tour visiting every node at least once). The most common example of Metric TSP is the planar Euclidean TSP, where the points lie in the two-dimensional Euclidean plane, and the distances are measured according to the Euclidean metric.
Metric TSP, even restricted to planar Euclidean TSP, is well-known to be NP-hard [10]. Metric TSP is also known to be NP-hard to approximate to within a ratio 1.00456, but polynomial-time approximable to within a ratio 1.5. Fixed-dimension Euclidean TSP is known to have a PTAS (i.e. a family of algorithms with approximation ratio arbitrarily close to 1) [1]; this generalises to any metric defined by a fixed-dimension Minkowski vector norm.
Two simple methods, double-tree shortcutting [12] and Christofides’ [4,13], allow one to approximate the solution of Metric TSP within a factor of 2 and 1.5, respectively. Both these methods belong to the class of tourconstructing heuristics, i.e. “heuristics that incrementally construct a tour and stop as soon as a valid tour is created” [7]. In both methods, we build an Eulerian graph on the given point set, select an Euler tour of the graph, and then perform shortcutting on this tour by removing repeated nodes, until all node repetitions are removed. In general, it is not prescribed which one of several occurrences of a particular node to remove. Therefore, the methods yield an exponentially-sized space of TSP tours (shortcuttings of a specific Euler tour in a specific Eulerian graph), each of which approximates the optimal solution within at most a factor of 2 (respectively, 1.5).
The two methods differ in the way the initial weighted Eulerian graph is constructed. Both start by finding the graph’s minimum-weight spanning tree (MST). The double-tree method then doubles every edge in the MST, while the Christofides method adds to the MST a minimum-weight matching built on the set of odd-degree nodes. The weight of the resulting Euler tour is higher than the optimal TSP tour at most by a factor of 2 (respectively, 1.5), and the subsequent shortcutting can only decrease the tour weight.
While any tour obtained by shortcutting of the original Euler tour approximates the optimal solution within at most a factor of 2 (respectively, 1.5), clearly, it is still desirable to find the shortcutting that gives the closest approximation. Given an Eulerian graph on a set of points, we will consider its minimum-weight shortcutting across all shortcuttings of all possible Euler tours of the graph. We shall correspondingly speak about the minimum-weight double-tree and the minimum-weight Christofides methods.
Unfortunately, for the general Metric TSP (i.e. an arbitrary complete weighted graph with the triangle inequality), the corresponding double-tree and Christofides minimum-weight shortcutting problems are both NP-hard. The minimum-weight double-tree shortcutting problem was also believed for a long time to be NP-hard for planar Euclidean TSP, until a polynomialtime algorithm was given by Burkard et al. [3]. In [6], we gave an improved algorithm running in time O(4 d n 2 ), where d is the maximum node degree in the rooted minimum spanning tree (e.g. in the non-degenerate planar Euclidean case, d ≤ 4). In contrast, the Christofides version of the problem remains NP-hard even for planar Euclidean TSP [11].
A natural question about the properties of the two approximation methods and their variants is whether the approximation ratios 2 and 1.5 are tight, i.e. whether there is a problem instance where the approximate solution has approximation ratio 2 (respectively, 1.5), or a family of problem instances where the approximate solutions approach these ratios arbitrarily closely.
For the minimum-weight double-tree method, the answer to this question is unknown, as observed e.g. in [9]. The only existing lower bounds for the double-tree method apply to a shortcutting that is performed in some suboptimal, easily computable order. An example of such an order is depthfirst tree traversal; we shall call the resulting method depth-first double-tree shortcutting. A tight lower bound for this method is given by the standard Euclidean lower-bound construction shown in Figure 1, whichadapted from [8]. Figure 1a shows an instance point set and the (unique) minimum spanning tree. We assume that ǫ = o(1); for example, we can take ǫ = 1/n. The vertical size of the instance set is 1, and the horizontal size is 1 + o(1) n. The weight of the unique MST is 2 + o(1) n; the double-tree weight is 4 + o(1) n. The double tree undergoes no significant shortcutting, and the resulting tour
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