Abstract. We consider labeled **Traveling** **Salesman** Problems, defined upon a complete graph of n vertices with colored edges. The objective is to find a tour of maximum (or minimum) number of colors. We derive results regarding hardness of approximation, and analyze approxima- tion algorithms for both versions of the problem. For the maximization version we give a 1

Dierential approximation results for **traveling** **salesman** problem
Abstra
t
We prove that both minimum and maximum **traveling** **salesman** problems
an be approx- imately solved, in polynomial time within approximation ratio bounded above by 1/2. We next prove that, when dealing with edge-distan
es 1 and 2, both versions are approximable within 3/4. Based upon this result, we then improve the standard approximation ratio known for maximum **traveling** **salesman** with distan
es 1 and 2 from 5/7 to 7/8. Finally , we prove that, for any > 0, it is NP-hard to approximate both problems within better than 3475=3476 + .

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Dierential approximation results for **traveling** **salesman** problem
Abstra
t
We prove that both minimum and maximum **traveling** **salesman** problems
an be approx- imately solved, in polynomial time within approximation ratio bounded above by 1/2. We next prove that, when dealing with edge-distan
es 1 and 2, both versions are approximable within 3/4. Based upon this result, we then improve the standard approximation ratio known for maximum **traveling** **salesman** with distan
es 1 and 2 from 5/7 to 7/8. Finally , we prove that, for any > 0, it is NP-hard to approximate both problems within better than 3475=3476 + .

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Abstract
We prove that both minimum and maximum **traveling** **salesman** problems on complete graphs with edge-distances 1 and 2 (denoted by min TSP12 and max TSP12, respectively) are approximable within 3/4. Based upon this result, we improve the standard approximation ratio known for maximum **traveling** **salesman** with distances 1 and 2 from 3/4 to 7/8. Finally, we prove that, for any ² > 0, it is NP-hard to approximate both problems better than within 741/742 + ². The same results hold when dealing with a generalization of min and max TSP12, where instead of 1 and 2, edges are valued by a and b.

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Aubière Cedex, France alain.quilliot@isima.fr
ABSTRACT
Delivery of goods into urban areas constitutes an important issue for logistics service providers. One recent evolution in urban logistics involves the usage of drones in the delivery process. Delivery by drones offers new possibilities, but also induces new challenging routing problems. In this paper, we address a heuristic solution of the so-called Parallel Drone Scheduling **Traveling** **Salesman** Problem, recently introduced by Murray and Chu [19]. In this problem, deliveries are split between a vehicle and one or several drones. The vehicle performs a classical delivery tour from the depot, while the drones are constrained to perform back and forth trips. The objective is to minimize the completion time. We propose to solve the problem with an original iterative two-step heuristic, composed of: a coding step that transforms a solution into a customer sequence, and a decoding step that decomposes the customer sequence into a tour for the vehicle and series of trips for the drone(s). Decoding is expressed as a bicriteria shortest path problem and is carried out by dynamic programming. Experiments conducted on benchmark instances from the literature confirm the efficiency of the approach and give some insights on this drone delivery system.

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Keywords: Nested Monte-Carlo, Nested Rollout Policy Adaptation, Trav- eling **Salesman** Problem with Time Windows.
1 Introduction
In this paper we are interested in the minimization of the travel cost of the **Traveling** **Salesman** Problem with Time Windows. Recently, the use of a Nested Monte-Carlo algorithm (combined with expert knowledge and an evolutionary algorithm) gave good results on a set of state of the art problems [13]. However, as it has been pointed out by the authors, the effectiveness of the Nested Monte-Carlo algorithm decreases as the number of cities increases. When the number of cities is too large (greater than 30 for this set of problems), the algorithm is not able to find the state of the art solutions.

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Abstract
Dealing with multi-objective combinatorial optimization, this article proposes a new multi-objective set-based meta- heuristic named Perturbed Decomposition Algorithm (PDA). Combining ideas from decomposition methods, local search and data perturbation, PDA provides a 2-phase modular framework for finding an approximation of the Pareto front. The first phase decomposes the search into a number of linearly aggregated problems of the original multi- objective problem. The second phase conducts an iterative process: aggregated problems are first perturbed then selected and optimized by an efficient single-objective local search solver. Resulting solutions will serve as a starting point of a multi-objective local search procedure, called Pareto Local Search. After presenting a literature review of meta-heuristics on the multi-objective symmetric **Traveling** **Salesman** Problem (TSP), we conduct experiments on sev- eral instances of the bi-objective and tri-objective TSP. The experiments show that our proposed algorithm outperforms the best current methods on this problem.

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Mots-clé : voyageur du commerce, optimisation combinatoire, heuristique.
A new representation of **traveling** **salesman** tours and its use in heuristics
Abstract
A new approach is presented to the **traveling** **salesman** problem relying on a novel greedy representation of the solution space and leading to a different definition of neighborhood structures required in many local and random search approaches. Accordingly, a paralleliz- able search strategy is proposed based upon local search with random restarts that exploits the characteristics of the representation. Preliminary experimental results on several sets of test problems, among which very well-known benchmarks, show that the representation developed, matched with the search strategy proposed, attains high quality near-optimal solutions in moderate execution times.

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On the Shortest Spanning Subtree of a Graph and the **Traveling** **Salesman** Problem.
Received by the editors April 11, 1955.
[48] Several years ago a typewritten translation (of obscure origin ) of [1] raised some interest. This paper is devoted to the following theorem : If a (nite) connected graph has a positive real number attached to each edge (the length of the edge), and if these lengths are all distinct, then among the spanning 1 trees (German : Gerüst) of the graph there is only one, the

1 Introduction
The **traveling** **salesman** problem (TSP) is one of the most interesting and par- adigmatic optimization problems. In both minimization and maximization ver- sions, TSP has been widely studied and a large bibliography is available (see, for example, the books [8, 12, 13]). As it is well known, both versions of TSP are NP-hard but although in the case of MaxTSP the problem is approximable within constant ratio for all kinds of graphs [4, 10], in the case of MinTSP ap- proximation algorithms are known only for the metric case [5], i.e., when the graph distances satisfy the triangle inequality.

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1 Introduction
The **traveling** **salesman** problem (TSP) is one of the most interesting and paradig- matic optimization problems. In both minimization and maximization versions, TSP has been widely studied and a large bibliography is available (see, for exam- ple, the books [7, 11, 12]). As it is well known, both versions of TSP are NP-hard but although in the case of MaxTSP the problem is approximable within con- stant ratio for all kinds of graphs [4, 9], in the case of MinTSP approximation algorithms are known only for the metric case [5], i.e., when the graph distances satisfy the triangle inequality.

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16. R. M. Karp, Reductibility among combinatorial problems, R.E Miller and J.W. Thatche, Complexity of Computer Computations, 85 − 103, Plenum Press, NY, 1972.
17. S. R. Kosaraju, J. K. Park, and C. Stein, Long tours and short superstrings, Proc. F.O.C.S.’94 (1994), 166–177. 18. J. Monnot, S. Toulouse, and V. Th. Paschos, Differential approximation results for the **traveling** **salesman** problem

In this paper we wish to address the on-line version of the QTSP problem, named OL-QTSP. On line versions of other routing problems such as the travel- ing **salesman** problem [3], the **traveling** repairman problem [13, 16, 18], variants of the dial-a-ride problem [2, 13], have been studied in the literature in the re- cent years. The most updated results regarding these problems can be found in [17]. In the on-line version of QTSP we imagine that requests are given over time in a metric space and a server (the **traveling** **salesman**) has to decide which requests to serve and in what order to serve them, without yet knowing the whole sequence of requests, with the aim of fulfilling the quota by **traveling** the minimum possible amount of time. As it is common in the evaluation of on-line algorithms [14], the performance of the algorithm (cost of serving the requests needed to fulfill the quota), is matched against the performance of an optimum off-line server, that is a **traveling** **salesman** that knows all the requests ahead of time and decides which requests to serve and in what order, in order to fulfill the assigned quota. Clearly the off-line server cannot serve a request before its release time. The ratio between the former and the latter is called competitive ratio of the on-line algorithm (see [10]).

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possible, so as to diminish incompatibilities among different technologies. For the maximization case, consider the situation of designing a metropolitan peripheral ring road, where every color represents a different suburban area that a certain link would traverse. In order to maximize the number of suburban areas that such a peripheral ring covers, we seek a tour of a maximum number of colors. To the best of our knowledge, the only result known for labeled **traveling** **salesman** problems prior to ours is NP-hardness, shown by Broersma, Li and Woeginger in [2] for both

In this paper, we present a constructive method for optimizing exactly the traveling salesman problem (TSP) by solving a sequence of shortest route problems.. The t[r]

In this paper we wish to address the on-line version of the QTSP problem, named OL-QTSP. On line versions of other routing problems such as the travel- ing **salesman** problem [3], the **traveling** repairman problem [13, 16, 18], variants of the dial-a-ride problem [2, 13], have been studied in the literature in the re- cent years. The most updated results regarding these problems can be found in [17]. In the on-line version of QTSP we imagine that requests are given over time in a metric space and a server (the **traveling** **salesman**) has to decide which requests to serve and in what order to serve them, without yet knowing the whole sequence of requests, with the aim of fulfilling the quota by **traveling** the minimum possible amount of time. As it is common in the evaluation of on-line algorithms [14], the performance of the algorithm (cost of serving the requests needed to fulfill the quota), is matched against the performance of an optimum off-line server, that is a **traveling** **salesman** that knows all the requests ahead of time and decides which requests to serve and in what order, in order to fulfill the assigned quota. Clearly the off-line server cannot serve a request before its release time. The ratio between the former and the latter is called competitive ratio of the on-line algorithm (see [10]).

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Q assigned a new, lower balue.) Repeat the procedure: again work up the branch, discarding terminal nodes with bounds equal or greater than Zg until the first one smaller. is found; agai[r]

1 Motivation
The **traveling** **salesman** problem (TSP) involves finding a Hamiltonian cycle of minimum weight in a given undirected graph G = (V, E) associated with a weight function w : E → Z. It has been widely investigated by the opera- tional research (OR) community for more than half a century, because it is an important optimization problem with many industrial applications. Its simple structure has enabled the development of general techniques, such as cutting planes, variable fixing, Lagrangian relaxation, and heuristics. These techniques are the key to the success of dedicated solvers (e.g., Concorde [3]), and they can be adapted to a range of optimization problems. Some have even been integrated into general MIP solvers, leading to great improvements in OR.

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[12] Marco F Duarte and Yonina C Eldar, “Structured compressed sensing: From theory to applications,” Signal Processing, IEEE Transactions on, vol. 59, no. 9, pp. 4053–4085, 2011.
[13] WanSoo T Rhee and Michel Talagrand, “A sharp deviation inequality for the stochastic **traveling** **salesman** problem,” The Annals of Probability, vol. 17, no. 1, pp. 1–8, 1989.

categories, it does share some similarities with the last 2. On the one hand, the local interac- tions between neighboring regions with a constant communication delay relates to the second approach; on the other hand, the movement of the **traveling** wave arises naturally as in the last model from the communication between separately oscillating levels. Importantly though, in our model none of the units is an oscillator or pacemaker per se, but oscillations arise from bidirectional communication with temporal delays (see [ 58 ] for a similar approach to oscilla- tions’ generation). This property marks a decisive difference with all 3 previous models. Lastly, in our implementation, the wave’s direction does not depend on a gradient of time delays or a gradient of frequencies; instead, the same model can produce both FW or BW waves depend- ing on the signals provided: FW waves originate from perceptual inputs, and BW waves origi- nate from top-down activity. Regarding the anatomical origin of the **traveling** waves, it is possible to envision the first level of the model as a thalamic region (e.g., LGN) connected to the cortex or the entire hierarchy as reflecting successive cortical regions (e.g., V1, V2, etc.). In the brain, these regions would also interact with each other through indirect cortico-cortical connections, as well as via the thalamus (e.g., pulvinar); however, for simplicity, these second- ary connection pathways were omitted in the model.

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