Postman Problem Research Articles

Approximation algorithms for solving the constrained arc routing problem in mixed graphs

Given a mixed graph G with vertex set V, let E and A denote the sets of edges and arcs, respectively. We use Q+ and Z+ to denote the sets of positive rational numbers and positive integers, respectively. For any connected mixed graph G=(V,E∪A;w;l,u) with a length function w:E∪A→Q+ and two integer functions l,u:E∪A→Z+ satisfying l(e)⩽u(e) for each e∈E∪A, we are asked to determine a minimum length tour T traversing each e∈E∪A at least l(e) and at most u(e) times. This new constrained arc routing problem generalizes the mixed Chinese postman problem. Let n=|V| and m=|E∪A| denote the number of vertices and edges (including arcs), respectively. Using network flow techniques, we design a (1+1/l0)-approximation algorithm in time O(n2m3logn) to solve this constrained arc routing problem such that l(e)<u(e) holds for each edge e∈E,l(e)⩽u(e) holds for each arc e∈A and l0=min{l(e)|e∈E}. In addition, we present two optimal combinatorial algorithms in times O(n3) and O(nm2logn) to solve this problem for the cases A=∅ and E=∅, respectively.

Heurística para solução do problema da coleta de resíduos sólidos domiciliares (RSD) com base no problema do carteiro chinês capacitado com múltiplas viagens (PCCC-MV)

The fast growing world population, mainly in urban centers, increases the generation of domestic solid waste. Every day, tons of solid waste generated in households which are up to the government of each city to have strategies to collect, transport and dispose them. Considering that most of the heuristics and models proposed so far consider a trip per shift and ignores the idle time of the shift after returning the truck to the depot, the proposed heuristic of this paper gener-ates for each truck multiple trips per shift in order to collect more solid waste without increasing the fleet. Thus, this paper proposes a heuristic for solving the problem of collecting solid waste based on the Capacitate Chinese Postman Problem with Multiple Trips (CCPP-MV) where each truck can make more than one trip in a time shift. To validate the proposed heuristic, we performed a practical application at the municipality of Cariacica, Espirito Santo, which has 176 neighbor-hoods.

Graphs inducing totally balanced and submodular Chinese postman games

A Chinese postman (CP) game is induced by a weighted undirected, connected graph in which the edges are identified as players and a vertex is chosen as post-office location.Granot and Granot (2012) characterized graphs that give rise to CP games that are balanced. This note completes this line of research by characterizing graphs that give rise to CP games that are submodular (totally balanced, respectively).

Supereulerian graphs with small matching number and 2-connected hamiltonian claw-free graphs

Motivated by the Chinese Postman Problem, Boesch, Suffel, and Tindell [The spanning subgraphs of Eulerian graphs, J. Graph Theory 1 (1977), pp. 79–84] proposed the supereulerian graph problem which seeks the characterization of graphs with a spanning Eulerian subgraph. Pulleyblank [A note on graphs spanned by Eulerian graphs, J. Graph Theory 3 (1979), pp. 309–310] showed that the supereulerian problem, even within planar graphs, is NP-complete. In this paper, we settle an open problem raised by An and Xiong on characterization of supereulerian graphs with small matching numbers. A well-known theorem by Chvátal and Erdös [A note on Hamilton circuits, Discrete Math. 2 (1972), pp. 111–135] states that if G satisfies α(G)≤κ(G), then G is hamiltonian. Flandrin and Li in 1989 showed that every 3-connected claw-free graph G with α(G)≤2 κ(G) is hamiltonian. Our characterization is also applied to show that every 2-connected claw-free graph G with α(G)≤3 is hamiltonian, with only one well-characterized exceptional class.

The Tube Challenge

The Tube Challenge consists of visiting all stations of the London Underground in the least possible time. The competition started in 1959 and the current record was established in August 2013. This paper shows that under some simplifications, this problem and some of its variants can be cast as a Generalized Traveling Salesman Problem, as a Traveling Salesman Problem, as a Rural Postman Problem or as a Chinese Postman Problem defined on a directed graph.

An ILP-refined tabu search for the Directed Profitable Rural Postman Problem

In transportation services, the costs are highly dependent on the opportunity to serve neighboring customers. In this paper we study the problem faced by a shipper that has to serve a set of custom...

Efficient complete coverage of a known arbitrary environment with applications to aerial operations

The problem of coverage of known space arises in a multitude of domains, including search and rescue, mapping, and surveillance. In many of these applications, it is desirable or even necessary for the solution to guarantee both the complete coverage of the free space, as well as the efficiency of the generated trajectory in terms of distance traveled. A novel algorithm is introduced, based on the boustrophedon cellular decomposition technique, for computing an efficient complete coverage path for a known environment populated with arbitrary obstacles. This hierarchical approach first partitions the space to be covered into non-overlapping cells, then solves the Chinese postman problem to compute an Eulerian circuit traversing through these cells, and finally concatenates per-cell seed spreader motion patterns into a complete coverage path. Practical considerations of the coverage system are also explored for operations with a non-holonomic aerial vehicle. The effects of various system parameters are evaluated in controlled environments using a high-fidelity flight simulator, in addition to over 200 km of in-field flight sessions with a fixed-wing unmanned aerial vehicle.

The time-dependent rural postman problem: polyhedral results

The Chinese postman problem is a famous and classical problem in graph theory. This paper introduces a new variant of this problem, namely, the time-dependent rural postman problem, which is motivated by applications, involving scheduling with time-dependent processing time. We present an arc-path formulation for the problem with a constraint set that is divided into two parts. The first part is linear and has a strong combinatorial structure, which defines the polytope of the arc-path alternation sequence (APAS), while the second part is nonlinear and is closely related to time-dependent travel time. First, we investigate the facial structure of the APAS polytope and present several facet inequalities. Second, considering the travel and service time functions as piecewise functions, we linearize the nonlinear inequalities and propose several strong, valid inequalities. Finally, we propose a cutting plane algorithm and report numerical results on several randomly generated instances.

Path Planning for Mobile Mapping System Considering the Geometry of the GPS Satellite

In this paper, we propose a path planning method for a Mobile Mapping System (MMS) considering GPS geometry for the purpose of increasing GPS availability. We calculate Position Dilution of Precision (PDOP) using aerial laser survey data that provides geography and height of buildings near roads. PDOP is used for weighting road network data. By using the Chinese postman problem, we then calculate the optimum path that passes through all routes at least once. We describe currentMMS operation first and clear up problems with the current method. We then describe the proposed method in detail. Finally, we conduct evaluation tests and describe results in detail. The evaluation test verifies that the proposed method increases the GPS FIX ratio by 5% under the same GPS geometry and proves the effectiveness of the proposed method.

An optimal path planning strategy for multiple target search by a mobile robot

AbstractAlthough there is an obvious relationship between the Chinese Postman Problem (CPP) and the path‐planning problem for multiple target‐search tasks by a robot, no attempts have been made so far to connect the two. A novel map transformation framework (MTF) is proposed in this paper. With MTF, a feature map or a topological map can be converted into a standard topological map on which many graph‐search algorithms suitable for CPP can be employed to carry out path planning for multiple target search. Theoretical analysis proves that the route generated by this approach is optimal. Both simulation results and experiments indicate that the MTF‐based algorithm is better than several other target‐search algorithms. © 2012 Institute of Electrical Engineers of Japan. Published by John Wiley &amp; Sons, Inc.

Graphs Inducing Totally Balanced and Submodular Chinese Postman Games

A Chinese postman (CP) game is induced by a a weighted undirected, connected graph in which the edges are identified as players and a vertex is chosen as post-office location. Granot and Granot (2012) characterized graphs that give rise to CP games that are balanced. This note completes this line of research by characterizing graphs that give rise to CP games that are submodular (totally balanced, respectively).

On Games Arising from Multi-Depot Chinese Postman Problems

This paper introduces cooperative games arising from multi-depot Chinese postman problems and explores the properties of these games. A multi-depot Chinese postman problem (MDCP) is represented by a connected (di)graph G, a set of k depots that is a subset of the vertices of G, and a non-negative weight function on the edges of G. A solution to the MDCP is a minimum weight tour of the (di)graph that visits all edges (arcs) of the graph and that consists of a collection of subtours such that the subtours originate from different depots, and each subtour starts and ends at the same depot. A cooperative Chinese postman (CP) game is induced by a MDCP by associating every edge of the graph with a different player. This paper characterizes globally and locally k-CP balanced and submodular (di)graphs. A (di)graph G is called globally (locally) k-CP balanced (respectively submodular), if the induced CP game of the corresponding MDCP problem on G is balanced (respectively submodular) for any (some) choice of the locations of the k depots and every non-negative weight function.

Coordinated road-network search route planning by a team of UAVs

This paper presents a road-network search route planning algorithm by which multiple autonomous vehicles are able to efficiently visit every road identified in the map in the context of the Chinese postman problem. Since the typical Chinese postman algorithm can be applied solely to a connected road-network in which ground vehicles are involved, it is modified to be used for a general type of road map including unconnected roads as well as the operational and physical constraints of unmanned aerial vehicles (UAVs). For this, a multi-choice multi-dimensional knapsack problem is formulated to find an optimal solution minimising flight time and then solved via mixed integer linear programming. To deal with the dynamic constraints of the UAVs, the Dubins theory is used for path generation. In particular, a circular–circular–circular type of the Dubins path is exploited based on a differential geometry to guarantee that the vehicles follow the road precisely in a densely distributed road environment. Moreover, to overcome the computational burden of the multi-choice multi-dimensional knapsack algorithm, a nearest insertion and auction-based approximation algorithm is newly introduced. The properties and performance of the proposed algorithm are evaluated via numerical simulations operating on a real village map and randomly generated maps with different parameters.

GIS technology as an environment for testing an advanced mathematical model for optimization of road maintenance

In the paper, we consider a Chinese postman problem and show how integration of an advanced mathematical model with GIS technology can be beneficial providing a powerful tool for developing, testing and applying sophisticated heuristics for arc routing problems. We give details of an implementation of several recently developed heuristics for the Chinese postman problem with priority nodes that are integrated with GIS software. In the end, we show how a GIS visualization can be helpful in finding and presenting the results.

On graphs which can or cannot induce Chinese Postman games with a non-empty core

We study the Chinese postman (CP) cooperative game induced by a connected, weighted, undirected graph G, wherein players reside at the edges of G and a postman, starting from a post-office location (i.e., a vertex of G), needs to traverse all edges before returning to the post-office. We provide a complete characterization of all connected graphs for which there exists a positive edge-cost function such that the induced CP game has a non-empty core, and, consequently, we derive a complete characterization of all connected graphs for which there does not exist a positive edge-cost function which induces a CP game with a non-empty core. Membership in these classes of graphs can be verified in strongly polynomial time.

Uncertain Programming Model for Chinese Postman Problem with Uncertain Weights

IChinese postman problem is one of the classical combinatorial optimization problems with many applications. However, in application, some uncertain factors are frequently encountered. This paper employs uncertain programming to deal with Chinese postman problem with uncertain weight Within the framework of uncertainty theory, the concepts of expected shortest route, <TEX>${\alpha}$</TEX>-shortest route, and distribution shortest route are proposed. After that, expected shortest model, and <TEX>${\alpha}$</TEX>-shortest model are constructed. Taking advantage of properties of uncertainty theory, these models can be transf-ormed into their corresponding deterministic forms, which can be solved by classical algorithm..

A branch-and-cut algorithm for the maximum benefit Chinese postman problem

The Maximum Benefit Chinese Postman Problem (MBCPP) is an NP-hard problem that considers several benefits associated with each edge, one for each time the edge is traversed with a service. The objective is to find a closed walk with maximum benefit. We propose an IP formulation for the undirected MBCPP and, based on the description of its associated polyhedron, we propose a branch-and-cut algorithm and present computational results on instances with up to 1,000 vertices and 3,000 edges.

Test Planning in Digital Microfluidic Biochips Using Efficient Eulerization Techniques

Digital microfluidic technology is now being extensively used for implementing a lab-on-a-chip. Microfluidic biochips are often used for safety-critical applications, clinical diagnosis, and for genome analysis. Thus, devising effective and faster testing methodologies to warrant correct operations of these devices after manufacture and during bioassay operations, is very much needed. In this paper, we propose an Euler tour based technique to obtain the route plan of a test droplet for the purpose of structural testing of biochips. The method is applicable to various digital microfluidic biochip architectures, e.g., fully reconfigurable arrays, application specific biochips, pin-constrained irregular geometry biochips, and to defect-tolerant biochips. We show that in general, the optimal Eulerization and subsequent determination of an Euler tour in the graph model of a biochip can be abstracted in terms of the classical Chinese postman problem. The Euler tour can be identified by running the classical Hierholzer's algorithm, which relies on a simple cycle decomposition and splicing method. This improved Eulerization technique leads to an efficient test plan for the chip. This can also be used in phase-based test planning that yields savings in testing time. The method provides a unified approach towards structural testing and can be easily adopted to design a droplet routing procedure for functional testing of digital microfluidic biochips.

Solving Chinese Postman Problem on Time Varying Network with Timed Automata

基于时间自动机理论,提出了时间窗、时间依赖服务代价以及时间依赖旅行时间这3 类时变网络中国邮路问题的统一建模的语义模型和求解方法.首先,将中国邮路问题可行解条件和时变参数与时间自动机联系起来,建立了3 类问题的统一时间自动机系统(timed automata system,简称TAS)模型;然后,将时变网络中国邮路问题归结为TAS 模型上的一系列可达性判定问题,并利用形式化验证算法给出了有效的求解方法.由于TAS 模型中存在<em>O</em>(|<em>A</em>|+|<em>A_R</em>|+1)个时间自动机,限制了问题求解规模.为此,通过扩展时间自动机语义,提出了TAS 模型中的时间自动机合并策略,进而将TAS 模型转换为一个广义时间自动机(GTA)模型.基于GTA 模型,利用UPPAAL 工具对9 组、共54 个随机算例进行实验.实验结果表明,该方法在求解精度上明显优于运筹学领域的方法.

Operations research: a model-based approach

The book covers the standard models and techniques used in decision making in organizations. The main emphasis of the book is on modeling business-related scenarios and the generation of decision alternatives. Fully solved examples from many areas are used to illustrate the main concepts without getting bogged down in technical details. The book presents an approach to operations research that is heavily based on modeling and makes extensive use of sensitivity analyses. It is a result of many years of combined teaching experience of the authors. The second edition adds new material on multicriteria optimization, postman problems, Lagrangian relaxation, cutting planes, machine scheduling, and Markov chains. Support material is found on a free website and includes some algorithms, additional fully solved problems and slides for instructors.