Transformations and Exact Node Routing Solutions by Column Generation

This chapter examines arc routing problems from the “dual” perspective of node routing. It first attempts to explain and respond to a question from a “typical” engineering and applied mathematics graduate who was exposed to classical node routing problems such as the transportation and the traveling salesman problems, and ventured a little into the vast literature on problems and solutions for the different variants of vehicle routing (node routing) problems. Why shouldn’t (or why should) any arc routing problem be viewed as some version of a node routing problem, pending an appropriate transformation of the corresponding graph? The first part of this chapter will examine this question addressing the issue of when such a transformation is necessary and the complementary question of when, or for what arc routing problems, from computational point of view, a transformation to node routing is inappropriate. In addition, the first part will attempt to provide a partial account of the different transformation schemes proposed over the years for arc routing problems into node routing setting. For an excellent write-up of exact solution methodologies for “hard” arc routing problems addressed without transformation to a node routing setting the reader is directed to chapters in this book by Eglese and Letchford, Benavent, Corberán, and Sanchis, and Johnson.