Postman Problem Research Articles

The Capacitated Chinese Postman Problem: Lower Bounds and Solvable Cases

SYNOPTIC ABSTRACTThe Capacitated Chinese Postman Problem (CCPP) is an important arc routing problem in which demands occurring on arcs of a network have to be serviced by vehicles of fixed capacity at minimal total cost. In this paper, we investigate the behavior of a matching-based lower bound for this problem and introduce a new bounding technique. We then turn to special classes of the CCPP that one can solve directly when suitable restrictions apply to the problem data and the network structure.

An Eulerian exposition

AbstractAn overview of Eulerian graphs is presented. In particular, characterizations of Eulerian graphs and digraphs as well as algorithms for constructing Eulerian circuits are discussed. A solution to the Chinese postman problem is followed by a study of subgraphs and supergraphs of Eulerian graphs. After an introduction to randomly Eulerian graphs and digraphs, we conclude with a summary of a variety of results involving enumeration.

Binary group and Chinese postman polyhedra

A new proof of the characterization of the Chinese postman polyhedra is given. In developing this proof, a theorem of Gomory about homomorphic lifting of facets for group polyhedra is generalized to subproblems. Some results for the Chinese postman problem are generalized to binary group problems. In addition, a connection is made between Fulkerson's blocking polyhedra and a blocking pair of binary group problems. A connection is also developed between minors and lifting of facets for group problems.

Computational experiments with algorithms for a class of routing problems

The vehicle routing problem, a generalization of the infamous traveling salesman problem, is a well-known distribution management problem that has been the focus of much research attention. On the other hand, generalizations of arc routing problems, such as the Chinese postman problem, have been comparatively neglected. In a recent paper, we studied a class of capacitated arc routing problems from primarily a theoretical point of view. In this paper, we focus on the development and testing of algorithms for solving the capacitated Chinese postman problem. Extensive computational results are presented and analyzed.

The mixed postman problem

The mixed postman problem is formulated as an integer linear program. A one to one correspondence is established between the extreme points of the linear programming polyhedron and prime assigned Euler networks. A prime assigned Euler network gives rise to a set of length equivalent prime postman tours. Some prime postman tour is optimal. Thus, it is sufficient to search the extreme points of the polyhedron to find an optimal postman tour.

The Chinese Postman Problem for Mixed Networks

The Chinese postman problem is to find a least cost way to traverse each arc of a network at least once and to return to the vertex from which you started. Diverse problems such as the routing of road crews, police patrol scheduling, garbage collection and the programming of computer map printers can be modelled as Chinese postman problems. This paper surveys available solution techniques for the Chinese postman problem for totally undirected networks (when all streets are two-way streets) and for totally directed networks (when all streets are one-way streets). A known solution technique for networks with both directed and undirected arcs (both one-way and two-way streets) in which the degree of each vertex is an even number is also reviewed. A solution technique for these mixed networks in which some vertices have odd degree is presented. This technique is based on the before mentioned technique and requires the solution of a minimum cost flow problem on a network that is an extension of the original network. Some of the additional arcs in this network have gain factors (i.e., the flow leaving these arcs equals the flow entering times the gain factor) and the flows are required to be integer valued.

Approximation Algorithms for Some Routing Problems

Several polynomial time approximation algorithms for some $NP$-complete routing problems are presented, and the worst-case ratios of the cost of the obtained route to that of an optimal are determined. A mixed-strategy heuristic with a bound of 9/5 is presented for the stacker-crane problem (a modified traveling salesman problem). A tour-splitting heuristic is given for k-person variants of the traveling salesman problem, the Chinese postman problem, and the stacker-crane problem, for which a minimax solution is sought. This heuristic has a bound of $e + 1 - 1/k$, where e is the bound for the corresponding 1-person algorithm.

An algorithm for the solution of the postman's problem

The postman's problem may be defined as follows: given an interconnected network of streets, the postman must cover every street of the network so that his path is as short as possible. This paper describes an algorithm, which produces solutions to real planar street plans within feasible limits of both computer time and storage.

On the complexity of edge traversing

It is shown that the Chinese Postman Problem, although tractable in the totally directed and the totally undirected cases, is NP-complete in the mixed case. A simpler version of the same problem is shown algorithmically equivalent to the max-flow problem with unit edge capacities.

Survey of Deterministic Networks

Abstract The state-of-the-art of deterministic networks is surveyed with a discussion of shortest path, transportation, assignment, transshipment, maximum flow, minimum spanning tree, Chinese postman, Euler path and multicommodity flow models. The computational complexity of network models is discussed. The survey concentrates on models and algorithms that can be used to solve large-scale problems.

Survey of Deterministic Networks

Abstract The state-of-the-art of deterministic networks is surveyed with a discussion of shortest path, transportation, assignment, transshipment, maximum flow, minimum spanning tree, Chinese postman, Euler path and multicommodity flow models. The computational complexity of network models is discussed. The survey concentrates on models and algorithms that can be used to solve large-scale problems.

A fundamental problem in vehicle routing

AbstractAn important but difficult combinatorial problem, in general, is to find the optimal route for a single vehicle on a given network. This paper defines a problem type, called the General Routing Problem, and gives an algorithm for its solution. The classical Traveling Salesman Problem and the Chinese Postman Problem are shown to be special limiting cases of the General Routing Problem. The algorithm provides a unified approach to both node and arc oriented routing problems, and exploits special properties of most real transportation networks such as sparsity of the associated adjacency matrix, and the tendency for arc symmetry at many nodes. For node oriented routing problems, this approach tends to produce large reduction in effective problem size.

The optimum traversal of a graph

For a given graph (network) having costs [ c ij ] associated with its links, the present paper examines the problem of finding a cycle which traverses every link of the graph at least once, and which incurs the minimum cost of traversal. This problem (called the graph traversal problem, or the Chinese postman problem [9]) can be formulated in ways analogous to those used for the well-known travelling salesman problem, and using this apparent similarity, Bellman and Cooke [1] have produced a dynamic programming formulation. This method of solution of the graph traversal problem requires computational times which increase exponentially with the number of links in the graph. Approximately the same rate of increase of computational effort with problem size would result by any other method adapting a travelling salesman algorithm to the present problem. This paper describes an efficient algorithm for the optimal solution of the graph traversal problem based on the matching method of Edmonds [5, 6]. The computational time requirements of this algorithm increase as a low order (2 or 3) power of the number of links in the graph. Computational results are given for graphs of up to 50 vertices and 125 links. The paper then discusses a generalised version of the graph traversal problem, where not one but a number of cycles are required to traverse the graph. In this case each link has (in addition to its cost) a quantity q ij associated with it, and the sum of the quantities of the links in any one cycle must be less than a given amount representing the cycle capacity. A heuristic algorithm for the solution of this problem is given. The algorithm is based on the optimal algorithm for the single-cycle graph traversal problem and is shown to produce near-optimal results. There is a large number of possible applications where graph traversal problems arise. These applications include: the spraying of roads with salt-grit to prevent ice formation, the inspection of electric power lines, gas, or oil pipelines for faults, the delivery of letter post, etc.

Matching, Euler tours and the Chinese postman

The solution of the Chinese postman problem using matching theory is given. The convex hull of integer solutions is described as a linear programming polyhedron. This polyhedron is used to show that a good algorithm gives an optimum solution. The algorithm is a specialization of the more generalb-matching blossom algorithm. Algorithms for finding Euler tours and related problems are also discussed.

Routing for Public Service Vehicles

City services requiring the routing of vehicles which must cover every street in a complex highway network are investigated. An overview of the problems arising in planning such services as trash collection, snow plowing, and street cleaning, as well as the formulation of the general theoretical problem, known in literature as the Chinese postman problem, are presented. A fairly complete analysis is made of the trash collection and snow plowing problems for the city of Cambridge, MA, and differences between the real world problems and the theoretical mold are explored. A brief description of several existing algorithms to solve vehicle routing problems is given. A literature review and description of available methods is presented. A sample routing problem for the city of Cambridge is worked using an algorithm.