Max TSP Research Articles

Efficient PTAS for the maximum traveling salesman problem in a metric space of fixed doubling dimension

The maximum traveling salesman problem (Max TSP) consists of finding a Hamiltonian cycle with the maximum total weight of the edges in a given complete weighted graph. This problem is APX-hard in the general metric case but admits polynomial-time approximation schemes in the geometric setting, when the edge weights are induced by a vector norm in fixed-dimensional real space. We propose the first approximation scheme for Max TSP in an arbitrary metric space of fixed doubling dimension. The proposed algorithm implements an efficient PTAS which, for any fixed \(\varepsilon \in (0,1)\), computes a \((1-\varepsilon )\)-approximate solution of the problem in cubic time. Additionally, we suggest a cubic-time algorithm which finds asymptotically optimal solutions of the metric Max TSP in fixed and sublogarithmic doubling dimensions.

An Analysis on the Modeling of Container Terminal Operations

Objectives: Container terminals are essential intermodal interfaces in the global transportation network. Efficient container handling at terminals is important in reducing transportation costs and keeping shipping schedules. The present analysis describes these problems within the scope of container terminal modeling. Methods/Statistical Analysis: Basic formulation of the problem is stated as two-machine flow shop problem. The well-known maximum travelling salesman problem (Max TSP) has been applied in this study. Max TSP can be solved as a TSP by replacing each edge cost by its additive inverse, since, there is a different value for unloading stack i while loading stack j and loading stack i while unloading stack j; this model corresponds to the Asymmetric Travelling Salesman Problem (ATSP). Findings: It is found that, there is an essential need for improvements and optimization of all aspects in the container transportation chains. For the real life systems of this type, this problem has been solved optimally. Significant possibilities for time savings have been arrived in this study. For real life case, where the reloading is performed on two barges placed side by side (8 stacks in the bay, 11 bays, 4 containers in the stack) time savings as a function of the terminal length are presented. The time saving with respect to number of rows inside the container yard is presented. Simulation models of container cranes demonstrate significant time savings, if double cycling is applied. It is showed that application of the double cycling can result in time savings of 12 to 27 % depending on the system parameters. This analysis is based on discrete event simulation and analytical optimization methods. Application/Improvement: Good planning of container terminal operations reduces waiting time for liner ships. Reducing the waiting time increases customer satisfaction and improves the terminal productivity which gives the container terminal an advantage over its competitors.

Single approximation for the biobjective Max TSP

We mainly study the Max TSP with two objective functions. We propose an algorithm which returns a single Hamiltonian cycle with performance guarantee on both objectives. The algorithm is analyzed in three cases. When both (respectively, at least one) objective function(s) fulfill(s) the triangle inequality, the approximation ratio is 512−ε≈0.41 (respectively, 38−ε). When the triangle inequality is not assumed on any objective function, the algorithm is 1+2214−ε≈0.27-approximate.

Asymptotically optimal algorithms for geometric Max TSP and Max m-PSP

We consider the maximum traveling salesman problem (Max TSP) and the maximum m-peripatetic salesman problem (Max m-PSP) on assuming that the vertices of a graph lie in some geometric space. For the both problems we obtain approximation algorithms that find asymptotically optimal solutions in the case of a normed space with a bounded dimension and in the case of a polyhedral space with a bounded number of facets, respectively.

An asymptotically exact algorithm for the maximum traveling salesman problem in a finite-dimensional normed space

We consider the geometric maximum traveling salesman problem on assuming that the vertices of a graph lie in an arbitrary finite-dimensional normed space. For this problem we obtain an approximate algorithm with the relative error tending to zero as the number of vertices grows. The algorithm generalizes Serdyukov’s algorithm for the Euclidean Max TSP.

Deterministic [formula omitted]-approximation for the metric maximum TSP

We present the first 7 / 8 -approximation algorithm for the maximum Traveling Salesman Problem (MAX-TSP) with triangle inequality. Our algorithm is deterministic. This improves over both the randomized algorithm of Hassin and Rubinstein [R. Hassin, S. Rubinstein, A 7/8-approximation algorithm for metric Max TSP, Inf. Process. Lett. 81 (5) (2002) 247–251] with an expected approximation ratio of 7 / 8 − O ( n − 1 / 2 ) and the deterministic ( 7 / 8 − O ( n − 1 / 3 ) ) -approximation algorithm of Chen and Nagoya [Z.-Z. Chen, T. Nagoya, Improved approximation algorithms for metric max TSP, in: Proc. ESA’05, 2005, pp. 179–190]. In the new algorithm, we extend the approach of processing local configurations using the so-called loose-ends, which we introduced in [Ł. Kowalik, M. Mucha, 35/44-approximation for asymmetric maximum TSP with triangle inequality, in: Proc. 10th Workshop on Algorithms and Data Structures, WADS’07, 2007, pp. 590–601].

An Improved Randomized Approximation Algorithm for Max TSP

We present an O(n3)-time randomized approximation algorithm for the maximum traveling salesman problem whose expected approximation ratio is asymptotically \(\frac{251}{331}\), where n is the number of vertices in the input (undirected) graph. This improves the previous best.

Improved deterministic approximation algorithms for Max TSP

We present an O ( n 3 ) -time approximation algorithm for the maximum traveling salesman problem whose approximation ratio is asymptotically 61 81 , where n is the number of vertices in the input complete edge-weighted (undirected) graph. We also present an O ( n 3 ) -time approximation algorithm for the metric case of the problem whose approximation ratio is asymptotically 17 20 . Both algorithms improve on the previous bests.

A [formula omitted]-approximation algorithm for metric Max TSP

We present a randomized approximation algorithm for the metric version of undirected Max TSP. Its expected performance guarantee approaches 8 as n→∞, where n is the number of vertices in the graph.

Differential approximation results for the traveling salesman and related problems

This paper deals with the problem of constructing a Hamiltonian cycle of optimal weight, called TSP. We show that TSP is 2/3-differential approximable and cannot be differential approximable greater than 649/650. Next, we demonstrate that, when dealing with edge-costs 1 and 2, the same algorithm idea improves this ratio to 3/4 and we obtain a differential non-approximation threshold equal to 741/742. Remark that the 3/4-differential approximation result has been recently proved by a way more specific to the 1-, 2-case and with another algorithm in the recent conference, Symposium on Fundamentals of Computation Theory, 2001. Based upon these results, we establish new bounds for standard ratio: 5/6 for Max TSP[a,2a] and 7/8 for Max TSP[1,2] . We also derive some approximation results on partition graph problems by paths.

Better approximations for max TSP

We combine two known polynomial time approximation algorithms for the maximum traveling salesman problem to obtain a randomized algorithm which outputs a solution with expected value of at least r times the optimal one for any given r<25/33 .