Undirected Rural Postman Problem Research Articles

A Divide and Conquer Strategy for Sweeping Coverage Path Planning

One of the main challenges faced by floor treatment service robots is to compute optimal paths that completely cover a set of target areas. Short paths are of noticeable importance because their length is directly proportional to the energy used by the robot. Such a problem is known to be NP-hard; therefore, efficient algorithms are needed. In particular, computation efficiency is important for mobile robots with limited onboard computation capability. The general problem is known as coverage path planning (CPP). The CPP has several variants for single regions and for disjoint regions. In this research, we are investigating the solutions for disjoint target regions (rooms) that have fixed connection points (doors). In particular, we have found effective simplifications for the cases of rooms with one and two doors, while the challenging case of an unbounded number of rooms can be solved by approximation. As a result, this work presents a divide and conquer strategy (DnCS) to address the problem of finding a path for a sweeping robot that needs to sweep a set of disjoint rooms that are connected by fixed doors and corridors. The strategy divides the problem into computing the sweeping paths for the target rooms and then merges those paths into one solution by optimising the room visiting order. In this strategy, a geometrical approach for room coverage and an undirected rural postman problem optimisation are strategically combined to solve the coverage of the entire area of interest. The strategy has been tested in several synthetic maps and a real scenario showing short computation times and complete coverage.

The target visitation arc routing problem

This paper studies the target visitation arc routing problem on an undirected graph. This problem combines the well-known undirected rural postman problem and the linear ordering problem. In this problem, there is a set of required edges partitioned into targets, which must be traversed and there are pairwise preferences for the order in which some targets are serviced, which generates a revenue if the preference is satisfied. The aim is to find a tour that traverses all required edges at least once, and offers a compromise between the revenue generated by the order in which targets are serviced, and the routing cost of the tour. A linear integer programming formulation including some families of valid inequalities is proposed. Despite the difficulty of the problem, the model can be used to solve to optimality around 62% of the test instances.

A Nozzle Path Planner for 3-D Printing Applications

Additive manufacturing technologies have been widely applied in both household and industrial applications. The fabrication time of a three-dimensional (3-D) printed object can be shortened by optimizing its segments printing order. The computational times required by existing nozzle path planning algorithms can increase rapidly with the number of printing segments. In this article, the nozzle path planning problem is formulated as an undirected rural postman problem and a computationally efficient heuristic search algorithm is proposed to find fast routes and mitigate overheads in printing processes. Both simulation and experimental results concur that the proposed algorithm can significantly speed up printing processes and outperform its counterparts in 3-D printing applications.

An ACO-Based Tool-Path Optimizer for 3-D Printing Applications

Layered additive manufacturing, also known as three-dimensional (3-D) printing, has revolutionized transitional manufacturing processes. Fabrication of 3-D models with complex structures is now feasible with 3-D printing technologies. By performing careful tool-path optimization, the printing process can be speeded up, while the visual quality of printed objects can be improved simultaneously. The optimization process can be perceived as an undirected rural postman problem (URPP) with multiple constraints. In this paper, a tool-path optimizer is proposed, which further optimizes solutions generated from a slicer software to alleviate visual artifacts in 3-D printing and shortens print time. The proposed optimizer is based on a modified ant colony optimization (ACO), which exploits unique properties in 3-D printing. Experiment results verify that the proposed optimizer can deliver significant improvements in computational time, print time, and visual quality of printed objects over optimizers based on conventional URPP and ACO solvers.

Multi-depot rural postman problems

This paper studies multi-depot rural postman problems on an undirected graph. These problems extend the well-known undirected rural postman problem to the case where there are several depots instead of just one. Linear integer programming formulations that only use binary variables are proposed for the problem that minimizes the overall routing costs and for the model that minimizes the length of the longest route. An exact branch-and-cut algorithm is presented for each considered model, where violated constraints of both types are separated in polynomial time. Despite the difficulty of the problems, the numerical results from a series of computational experiments with various types of instances illustrate a quite good behavior of the algorithms. When the overall routing costs are minimized, over 43 % of the instances were optimally solved at the root node, and 95 % were solved at termination, most of them with a small additional computational effort. When the length of the longest route is minimized, over 25 % of the instances were optimally solved at the root node, and 99 % were solved at termination.

Undirected postman problems with zigzagging option: A cutting-plane approach

The paper devises a new model and associated cutting-plane and branch-and-cut approaches for a variant of the undirected Chinese and rural postman problem where some of the edges offer the flexibility of either being serviced twice by two separate traversals or by a single zigzag traversal. The kernel of the proposed cutting-plane algorithm is a separation procedure for generalized blossom inequalities. We show that the currently best known separation procedure of Letchford, Reinelt and Theis [A faster exact separation algorithm for blossom inequalities. In: Nemhauser G, Bienstock D, editors, Integer programming and combinatorial optimization, vol. 3064. Berlin: Springer; 2004. (chapter 10)] is applicable and leads to a highly efficient solution approach which can handle large-scale problem instances.

On the general routing polytope

The General Routing Problem ( GRP) consists of finding a minimum length closed walk in an edge-weighted undirected graph, subject to containing certain sets of required nodes and edges. It is related to the Rural Postman Problem and the Graphical Traveling Salesman Problem. We examine the 0/1-polytope associated with the GRP introduced by Ghiani and Laporte [A branch-and-cut algorithm for the Undirected Rural Postman Problem, Math. Program. Ser. A 87 (3) (2000) 467–481]. We show that whenever it is not full-dimensional, the set of equations and facets can be characterized, and the polytope is isomorphic to the full-dimensional polytope associated with another GRP instance which can be obtained in polynomial time. We also offer a node-lifting method. Both results are applied to prove the facet-defining property of some classes of valid inequalities. As a tool, we study more general polyhedra associated to the GRP.

A constructive heuristic for the Undirected Rural Postman Problem

This paper describes a constructive heuristic for the well-known Undirected Rural Postman Problem. At each iteration, the procedure inserts a connected component of the required edges and performs a local postoptimization. Computational results on a set of benchmark instances with up to 350 vertices show that the proposed procedure is competitive with the classical Frederickson procedure. Its use is recommended when a high-quality solution is needed in a short amount of time (e.g., in laser plotter applications).

On the Undirected Rural Postman Problem: Tight Bounds Based on a New Formulation

The Rural Postman Problem (RPP) is a classic “edge-routing” problem. A mathematical programming formulation of the RPP that differs fundamentally from those in the literature was introduced, but not tested computationally, by Garfinkel and Webb (1999). A rudimentary algorithm that yields lower bounds via cutting planes and upper bounds via heuristics is developed and tested for a variation of that formulation. Computational results are encouraging, especially in terms of the relatively small number of added inequalities needed to get good lower bounds, and the fact that the vast majority of these have efficient, exact separation procedures. Note that a first algorithm based on this new formulation is computationally competitive, allowing the possibility of far more efficient and complex future realizations.

A branch-and-cut algorithm for the Undirected Rural Postman Problem

The well-known Undirected Rural Postman Problem is considered and a binary linear problem using new dominance relations is presented. Polyhedral properties are investigated and a branch-and-cut algorithm is developed. Extensive computational results indicate that the algorithm is capable of solving much larger instances than previously reported.

A branch-and-cut algorithm for the Undirected Rural Postman Problem

A branch-and-cut algorithm for the Undirected Rural Postman Problem

Improvement Procedures for the Undirected Rural Postman Problem

This article describes new construction and postoptimization heuristics for the Undirected Rural Postman Problem. Extensive computational tests indicate that some combinations of these heuristics consistently produce optimal or high-quality solutions.

Arc Routing Problems, Part II: The Rural Postman Problem

This is the second half of a two-part survey on arc routing problems. The first part appeared in the March–April 1995 issue of this journal. Here, the rural postman problem (RPP) is reviewed. The paper is organized as follows: applications, the undirected RPP, the directed RPP, the stacker crane problem, and the capacitated arc routing problem.