Traveling Salesman Tour Research Articles

Bi-sided facility location problems: an efficient algorithm for k-centre, k-median, and travelling salesman problems

This study introduces a general framework, called Bi-sided facility location, for a wide range of problems in the area of combined facility location and routing problems such as locating test centres and designing the network of supermarkets. It is based on a multi-objective optimisation model to enhance the service quality which the clients received, called client-side, and enhance the interconnection quality and eligibility among the centres, called center-side. Well-known problems such as k-median and k-centre for the client-side and the travelling salesman problem for the centre-side are taken into account in this paper. After discussing the complexity of this kind of combination, we propose a heuristic approximation algorithm to find approximation Pareto-optimal solutions for the problem. The algorithm is an efficient local search utilising geometric objects such as the Voronoi diagram and Delaunay triangulation as well as algorithms for computing approximation travelling salesman tour. In addition to the comprehensive theoretical analysis of the proposed models and algorithm, we apply the algorithm to different instances and benchmarks, and compare it with NSGA-II based on set coverage and spacing metrics. The results confirm the efficiency of the algorithm in terms of running time and providing a diverse set of efficient trade-off solutions. Highlights Introducing a general bi-side location model considering centres and clients’ utilities Discussing and proving the NP-hardness of the model in the general framework Considering two instances; k-centre and k-median for client-side and TSP for centre-side Proposing an efficient geometric-based algorithm for solving the problems Implementing, testing, and comparing the proposed algorithm on several benchmarks

Adjustment Factors for Analytic Approximations of Traveling Salesman Tour Lengths

Adjustment Factors for Analytic Approximations of Traveling Salesman Tour Lengths

Adjustment Factors for Analytic Approximations of Traveling Salesman Tour Lengths

Adjustment Factors for Analytic Approximations of Traveling Salesman Tour Lengths

An Analytic Latency Model for a Next-Hop Data-Ferrying Swarm on Random Geometric Graphs

Next-hop data-ferrying is a data-driven approach to ferrying data between graph components of a disconnected network. In contrast to pre-planned routing methods like the Traveling Salesman tour, only the next-hop is planned, allowing the ferry to be reactive to the data flows within the network. When multiple ferries are used, explicit coordination between them is difficult due to the data-driven approach of the algorithm, and the distributed and disconnected nature of the problem. However, coordinated behavior can still be produced through swarm intelligent means and while this is a useful pragmatic result, mathematical models describing system properties are difficult to develop. This generally results in algorithm correctness being demonstrated through simulation. In this article, we describe the development of a mathematical model of such a next-hop ferrying swarm. We show that the closed-form expression for user data latency is a close match to simulation results for Random Geometric Graphs under the assumption of a zero expected degree. In non-zero degree graphs, a near constant offset between the model and simulation is observed.

An improved lower bound for the Traveling Salesman constant

Let X1,X2,…,Xn be independent uniform random variables on [0,1]2. Let L(X1,…,Xn) be the length of the shortest Traveling Salesman tour through these points. Beardwood et al (1959) showed that there exists a constant β such that limn→∞L(X1,…,Xn)n=βalmost surely. It was shown that β≥0.625. Building upon an approach proposed by Steinerberger (2015), we improve the lower bound to β≥0.6277.

Wasserstein Distance and the Distributionally Robust TSP

One of the most common strategies used in large-scale logistics problems is districting, in which one divides a service region into smaller pieces. This is particularly useful when one does not know the true locations of demand, but has only limited information such as a probability distribution or a small set of sample data. In “Wasserstein Distance and the Distributionally Robust TSP,” Carlsson et al. show how to partition a region so as to balance the traveling salesman tours of a set of unknown points, using a distributionally robust method defined by a bound on the Wasserstein distance to a set of landmark points.

Aggregating courier deliveries

AbstractWe consider the problem of efficiently scheduling deliveries by an uncapacitated courier from a central location under online arrivals. We consider both adversary‐controlled and Poisson arrival processes. In the adversarial setting we provide a randomized (3βΔ/2δ − 1)‐competitive algorithm, where β is the approximation ratio of the traveling salesman problem, δ is the minimum distance between the central location and any customer, and Δ is the length of the optimal traveling salesman tour overall customer locations and the central location. We provide instances showing that this analysis is tight. We also prove a 1 + 0.271Δ/δ lower‐bound on the competitive ratio of any algorithm in this setting. In the Poisson setting, we relax our assumption of deterministic travel times by assuming that travel times are distributed with a mean equal to the excursion length. We prove that optimal policies in this setting follow a threshold structure and describe this structure. For the half‐line metric space we bound the performance of the randomized algorithm in the Poisson setting, and show through numerical experiments that the performance of the algorithm is often much better than this bound.

Simple cubic graphs with no short traveling salesman tour

Let tsp(G) denote the length of a shortest traveling salesman tour in a graph G. We prove that for any ε>0, there exists a simple 2-connected planar cubic graph G1 such that tsp(G1)≥(1.25−ε)⋅|V(G1)|, a simple 2-connected bipartite cubic graph G2 such that tsp(G2)≥(1.2−ε)⋅|V(G2)|, and a simple 3-connected cubic graph G3 such that tsp(G3)≥(1.125−ε)⋅|V(G3)|.

Multi-depot traveling salesmen location problems on networks with special structure

Suppose that on any day, calls for service are generated by nodes of a transportation network independently with known probabilities. The calls are allocated to p service units that visit the allocated customers on shortest closed tours. That is, each service unit performs an optimal traveling salesman tour on its set of allocated calls (customers). We need to find optimal locations for the service units to minimize the expected travel distance. For this problem on a path, we present an $$O(n^4)$$ algorithm, where n is the number of nodes of the network. We also present an $$O(n^5)$$ algorithm for the problem on a tree with $$p=2$$.

Cubic TSP: A 1.3-Approximation

We prove that every simple bridgeless cubic graph with $n \ge 8$ vertices has a traveling salesman tour of length at most $1.3\cdot n - 2$, which can be constructed in polynomial time.MSC codestraveling salesman problemgraphic TSPcubic graphsMSC codes05C3868R1090C27

Maximum ATSP with Weights Zero and One via Half-Edges

We present a fast combinatorial $3/4$-approximation algorithm for the maximum asymmetric TSP with weights zero and one. The approximation factor of this algorithm matches the currently best one given by Blaser in 2004 and based on linear programming. Our algorithm first computes a maximum size matching and a maximum weight cycle cover without certain cycles of length two but possibly with {\em half-edges} - a half-edge of a given edge $e$ is informally speaking a half of $e$ that contains one of the endpoints of $e$. Then from the computed matching and cycle cover it extracts a set of paths, whose weight is large enough to be able to construct a traveling salesman tour with the claimed guarantee.

A minimum spanning tree based heuristic for the travelling salesman tour

This paper presents a heuristic to find the travelling salesman tour (TST) in a connected network. The approach first identifies a node and two associated arcs that are desirable for inclusion in the required TST. If we let this node be denoted by \(p\) and two selected arcs emanating from this node be denoted by \(\left( {p,q} \right)\,{\text{and}}\,\left( {p,k} \right),\) then we find a path joining the two nodes \(q\,{\text{and}}\, k\) passing through all the remaining nodes of the given network. A sum of these lengths, i.e. length of the links \(\left( {p,q} \right)\,{\text{and}}\,\left( {p,k} \right)\) along with the length of the path that joins the nodes \(q\,{\text{and}}\, k\) passing through all the remaining nodes will result in a feasible TST, hence gives an upper bound on the TST. A simple procedure is outlined to identify: (1) the node \(p\), (2) the two corresponding links \(\left( {p,q} \right)\,{\text{and}}\,\left( {p,k} \right),\) and (3) the path joining the nodes \(q\,{\text{and}}\, k\) passing through all the remaining nodes. The approach is based on the minimum spanning tree; hence the complexity of the TST is reduced. The network in the present context has been assumed to be a connected network with at least two arcs emanating from each node.

Multiregion Inspection by Combining Clustered Traveling Salesman Tours With Sampling-Based Motion Planning

This paper develops an efficient approach to generate a collision-free and dynamically feasible trajectory that enables a robotic vehicle to inspect the entire workspace or a subset consisting of one or several regions. The approach makes it possible to specify constraints on the order in which the regions should be inspected by using colors to ensure that regions with the same color are inspected as a group. A key aspect is the transformation of the multiregion inspection into a clustered traveling salesman problem (CTSP). This is achieved by generating several inspection points on the medial axis of each region to increase the visibility and by grouping the inspection points into clusters. We also develop a fast branch-and-bound CTSP solver to find low-cost clustered tours. These tours guide sampling-based motion planning, as it expands a motion tree in search for a collision-free and dynamically feasible trajectory to carry out the multiregion inspection. The approach is evaluated in simulation using a snake-like robot model to carry out inspection tasks in complex environments. Comparisons to related work show significant speedups.

Curvature-constrained traveling salesman tours for aerial surveillance in scenarios with obstacles

The curvature-constrained traveling salesman problem with obstacles deals with finding a minimum length tour which includes a set of landmarks and avoids obstacles, for a kinematically constrained vehicle. Its great practical importance is mainly due to surveillance tasks of unmanned aerial vehicles. The problem constitutes a combination of the well-studied Dubins traveling salesman problem and the flight path planning problem. We present heuristic algorithms that are based on different strategies of extending a tour by inserting new landmarks. Each insert operation comprises the optimization of overflight directions for the given sequence of landmarks. Path finding between landmarks is done by a discrete routing model. It allows arbitrary flight directions and turn angles as well as maneuvers of different strengths, thus fully exploiting the flight capabilities of the aircraft. The performance of the algorithms is evaluated for agile and less agile aerial vehicles, using randomly generated scenarios with obstacles of different size and number.

Probabilistic analysis of power assignments

AbstractA fundamental problem for wireless ad hoc networks is the assignment of suitable transmission powers to the wireless devices such that the resulting communication graph is connected. The goal is to minimize the total transmit power in order to maximize the life‐time of the network. Our aim is a probabilistic analysis of this power assignment problem. We prove complete convergence for arbitrary combinations of the dimension d and the distance‐power gradient p. Furthermore, we prove that the expected approximation ratio of the simple spanning tree heuristic is strictly less than its worst‐case ratio of 2.Our main technical novelties are two‐fold: First, we find a way to deal with the unbounded degree that the communication network induced by the optimal power assignment can have. Minimum spanning trees and traveling salesman tours, for which strong concentration results are known in Euclidean space, have bounded degree, which is heavily exploited in their analysis. Second, we apply a recent generalization of Azuma‐Hoeffding's inequality to prove complete convergence for the case for both power assignments and minimum spanning trees (MSTs). As far as we are aware, complete convergence for p > d has not been proved yet for any Euclidean functional. © 2017 The Authors Random Structures & Algorithms Published by Wiley Periodicals, Inc., 51, 483–505, 2017

Avoiding 7-Circuits in 2-Factors of Cubic Graphs

Let $G$ be a cyclically $4$-edge-connected cubic graph with girth at least $7$ on $n$ vertices. We show that $G$ has a $2$-factor $F$ such that at least a linear amount of vertices is not in $7$-circuits of $F$. More precisely, there are at least $n/657$ vertices of $G$ that are not in $7$-circuits of $F$. If $G$ is cyclically $6$-edge-connected then the bound can be improved to $n/189$. As a corollary we obtain bounds on the oddness and on the length of the shortest travelling salesman tour in a cyclically $4$-edge-connected ($6$-edge-connected) cubic graph of girth at least $7$.

Euclidean Hub-and-Spoke Networks

The hub-and-spoke distribution paradigm has been a fundamental principle in geographic network design for more than 40 years. One of the primary advantages that such networks possess is their ability to exploit economies of scale in transportation by aggregating network flows through common sources. In this paper, we consider the problem of designing an optimal hub-and-spoke network in continuous Euclidean space: the “spokes” of the network are distributed uniformly over a service region, and our objective is to determine the optimal number of hub nodes and their locations. We consider seven different backbone network topologies for connecting the hub nodes, namely, the Steiner and minimum spanning trees, a travelling salesman tour, a star network, a capacitated vehicle routing tour, a complete bipartite graph, and a complete graph. We also perform an additional analysis on a multilevel network in which network flows move through multiple levels of transshipment before reaching the service region. We describe the asymptotically optimal (or near-optimal) configurations that minimize the total network costs as the demand in the region becomes large and give an approximation algorithm that solves our problem on a convex planar region for any values of the relevant input parameters.

Routing vehicles to minimize fuel consumption

We consider a generalization of the capacitated vehicle routing problem known as the cumulative vehicle routing problem in the literature. Cumulative VRPs are known to be a simple model for fuel consumption in VRPs. We examine four variants of the problem, and give constant factor approximation algorithms. Our results are based on a well-known heuristic of partitioning the traveling salesman tours and the use of the averaging argument.

On the asymptotic optimality of an algorithm for solving the maximum m-PSP in a multidimensional Euclidean space

An efficient algorithm A with a guaranteed error estimate is presented for solving the problem of finding several edge-disjoint Hamiltonian circuits (traveling salesman tours) of maximum weight in a complete weighted undirected graph in a multidimensional Euclidean space ℝk. The time complexity of the algorithm is O(n3). The number of traveling salesman tours for which the algorithm is asymptotically optimal is established.

Power assignment problems in wireless communication: Covering points by disks, reaching few receivers quickly, and energy-efficient travelling salesman tours

In this paper, we present approximation algorithms for a variety of problems occurring in the design of energy-efficient wireless communication networks. We first study the k-station network problem, where for a set S of stations and some constant k, one wants to assign transmission powers to at most k senders such that every station in S can receive a signal from at least one sender. We give a (1 + ϵ)-approximation algorithm for this problem. The second problem deals with energy-efficient networks, allowing bounded hop multicast operations, that is given a subset C of the stations S and a designated source node s ∈ S, we want to assign powers to the sending stations, such that every node in C can be reached by a transmission from s within k hops. For this problem, we provide an algorithm which runs in time linear in ∣ S∣. The last problem deals with a variant of the non-metric TSP problem where the edge costs correspond to the Euclidean distances to the power of some α ⩾ 1; this problem is motivated by data aggregation schemes in wireless sensor networks. We provide a simple constant approximation algorithm, which improves upon previous results when 2 ⩽ α ⩽ 2.7.