Zigzag inequalities: a new class of facet-inducing inequalities for Arc Routing Problems

In this paper we introduce a new class of facet-inducing inequalities for the Windy Rural Postman Problem and the Windy General Routing Problem. These inequalities are called Zigzag inequalities because they cut off fractional solutions containing a zigzag associated with variables with 0.5 value. Two different types of inequalities, the Odd Zigzag and the Even Zigzag inequalities, are presented. Finally, their application to other known Arc Routing Problems is discussed.     

A note on the Undirected Rural Postman Problem polytope

This note refers to the article by G. Ghiani and G. Laporte ``A branch-and-cut algorithm for the Undirected Rural Postman Problem', Math. Program. 87 (2000). We show that some conditions for the facet-defining property of the basic non-trivial inequalities are not sufficient and that the Rural Postman Problem polytope is more complex even when focusing on canonical inequalities only.     

New results for the Directed Profitable Rural Postman Problem

We study a generalization of the Directed Rural Postman Problem where not all arcs requiring a service have to be visited provided that a penalty cost is paid if a service arc is not crossed. The problem, known as Directed Profitable Rural Postman Problem, looks for a tour visiting the selected set of service arcs while minimizing both traveling and penalty costs. We add different valid inequalities to a known mathematical formulation of the problem and develop a branch-and-cut algorithm that introduces connectivity constraints both in a “lazy” and in a standard way. We also propose a matheuristic followed by an improvement heuristic (final refinement). The matheuristic exploits information provided by a problem relaxation to select promising service arcs used to solve optimally Directed Rural Postman problems. The ex-post refinement tries to improve the solution provided by the matheuristic using a branch-and-cut algorithm. The method gets a quick convergence through the introduction of connectivity cuts that are not guaranteed to be valid inequalities, and thus may exclude integer feasible solutions.     

中国投递员问题综述

§1 引言 中国投递员问题(Chinese Postman Problem)是1960年我们从生产实际中提出的一个数学问题,它是从下述实际问题中抽象出来的:“一个投递员应该怎样选择一条路线,才能既把所有由他负责送的信都送到,而所走的路程又最短。” 在我们开始研究中国投递员问题以前,国外有人研究过所谓旅行售货员问题     

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.     

Lower bounds and heuristics for the Windy Rural Postman Problem

In this paper we present several heuristic algorithms and a cutting-plane algorithm for the Windy Rural Postman Problem. This problem contains several important Arc Routing Problems as special cases and has very interesting real-life applications. Extensive computational experiments over different sets of instances are also presented.     

The Hierarchical Chinese Postman Problem: The slightest disorder makes it hard,yet disconnectedness is manageable

The Hierarchical Chinese Postman Problem is finding a shortest traversal of all edges of a graph respecting precedence constraints given by a partial order on classes of edges. We show that the special case with connected classes is NP-hard even on orders decomposable into a chain and an incomparable class. For the case with linearly ordered (possibly disconnected) classes, we get 5/3-approximations and fixed-parameter algorithms by transferring results from the Rural Postman Problem.     

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. Received: December 1, 1997 / Accepted: October 13, 1999?Published online January 27, 2000     

关于E0的Steiner邮路问题

给定图G＝(V，E，ω)，E0真包含于E是是一个指定通过的边子集，本文讨论了关于E0的Steiner邮路问题的特殊情况，即由E0导出的子图仅有两个连通分支．我们分别考虑了三种不同的情形，并给出了子闭迹消去算法和带限制的最短链算法，前者是一个基于整数规划的精确算法，而后者是一个近似算法．     

The directed profitable location Rural Postman Problem

In this paper we introduce an extension of the well known Rural Postman Problem, which combines arc routing with profits and facility location. Profitable arcs must be selected, facilities located at both end-points of the selected arcs, and a tour identified so as to maximize the difference between the profit collected along the arcs and the cost of traversing the arcs and installing the facilities. We analyze properties of the problem, present a mathematical programming formulation and a branch-and-cut algorithm. In an extensive computational experience the algorithm could solve instances with up to 140 vertices and 190 arcs and up to 50 vertices and 203 arcs.     

New inequalities for the General Routing Problem

A large new class of valid inequalities is introduced for the General Routing Problem which properly contains the class of ‘K-C constraints’. These are also valid for the Rural Postman Problem. A separation algorithm is given for a subset of these inequalities which runs in polynomial time.     

Solving the Prize-collecting Rural Postman Problem

In this work we present an algorithm for solving the Prize-collecting Rural Postman Problem. This problem was recently defined and is a generalization of other arc routing problems like, for instance, the Rural Postman Problem. The main difference is that there are no required edges. Instead, there is a profit function on the edges that must be taken into account only the first time that an edge is traversed.     

A branch and cut approach to the cardinality constrained circuit problem

The Cardinality Constrained Circuit Problem (CCCP) is the problem of finding a minimum cost circuit in a graph where the circuit is constrained to have at most k edges. The CCCP is NP-Hard. We present classes of facet-inducing inequalities for the convex hull of feasible circuits, and a branch-and-cut solution approach using these inequalities. Received: April 1998 / Accepted: October 2000?Published online October 26, 2001     

The general routing problem polyhedron: Facets from the RPP and GTSP polyhedra

In this paper we study the polyhedron associated with the General Routing Problem (GRP). This problem, first introduced by Orloff in 1974, is a generalization of both the Rural Postman Problem (RPP) and the Graphical Traveling Salesman Problem (GTSP) and, thus, is NP -hard. We describe a formulation of the problem such that from every non-trivial facet-inducing inequality for the RPP and GTSP polyhedra, we obtain facet-inducing inequalities for the GRP polyhedron. We describe a new family of facet-inducing inequalities for the GRP, the honeycomb constraints, which seem to be very useful for solving GRP and RPP instances. Finally, new classes of facets obtained by composition of facet-inducing inequalities are presented.     

A branch-and-cut algorithm for a generalization of the Uncapacitated Facility Location Problem

Summary We introduce a generalization of the well-know Uncapacitated Facility Location Problem, in which clients can be served not only by single facilities but also by sets of facilitities. The problem, calledGaneralized Uncapacitated Facility Lacition Problem (GUFLP), was inspired by the Index Selection Problem in physical database design. We for mulate GUFLP as a Set Packing Problem, showing that our model contains all the clique inequalities (in polynomial number). Moreover, we describe and exact separation procedure for odd-hole inequalities, based on the particular structure of the problem. These results are used within a branch-and-cut algorithm for the exact solution of GUFLP. Computational results on two different classes of test problems are given.     

Two-Connected Networks with Rings of Bounded Cardinality

We study the problem of designing at minimum cost a two-connected network such that each edge belongs to a cycle using at most K edges. This problem is a particular case of the two-connected networks with bounded meshes problem studied by Fortz, Labbé and Maffioli (Operations Research, vol. 48, no. 6, pp. 866–877, 2000).In this paper, we compute a lower bound on the number of edges in a feasible solution, we show that the problem is strongly NP-complete for any fixed K, and we derive a new class of facet defining inequalities. Numerical results obtained with a branch-and-cut algorithm using these inequalities show their effectiveness for solving the problem.     

Branch-and-cut algorithms for the split delivery vehicle routing problem

In this paper we present two exact branch-and-cut algorithms for the Split Delivery Vehicle Routing Problem (SDVRP) based on two relaxed formulations that provide lower bounds to the optimum. Procedures to obtain feasible solutions to the SDVRP from a feasible solution to the relaxed formulations are presented. Computational results are presented for 4 classes of benchmark instances. The new approach is able to prove the optimality of 17 new instances. In particular, the branch-and-cut algorithm based on the first relaxed formulation is able to solve most of the instances with up to 50 customers and two instances with 75 and 100 customers.     

A branch-and-cut algorithm for capacitated network design problems

We present a branch-and-cut algorithm to solve capacitated network design problems. Given a capacitated network and point-to-point traffic demands, the objective is to install more capacity on the edges of the network and route traffic simultaneously, so that the overall cost is minimized. We study a mixed-integer programming formulation of the problem and identify some new facet defining inequalities. These inequalities, together with other known combinatorial and mixed-integer rounding inequalities, are used as cutting planes. To choose the branching variable, we use a new rule called “knapsack branching”. We also report on our computational experience using real-life data. Received April 29, 1997 / Revised version received January 9, 1999? Published online June 28, 1999