Abstract The general routing problem (GRP) is a fundamental ‐hard vehicle routing problem, first defined by Orloff in 1974. It contains as special cases the Chinese postman problem, the rural postman problem, the graphical TSP, and the Steiner TSP. We examine in detail a known constructive heuristic for the GRP, due to Christofides and others. We show how to speed it up, in both theory and practice, while obtaining solutions that are at least as good. Computational results show that, for large instances, our implementation is faster than the original by several orders of magnitude.

Abstract Arc routing problems (ARPs) are a special kind of vehicle routing problem, in which the demands are located on edges or arcs, instead of nodes. There is a huge literature on ARPs, and a variety of exact and heuristic algorithms are available. Recently, however, we encountered some real‐life ARPs with over 10 000 roads, which is much larger than those usually considered in the literature. For these problems, we develop fast upper‐ and lower‐bounding procedures. We also present extensive computational results.

Abstract Network flow problems form an important and much‐studied family of combinatorial optimization problems, with a huge array of practical applications. Two network flow problems in particular have received a great deal of attention: the maximum flow and minimum‐cost flow problems. We review the progress that has been made on exact solution algorithms for these two problems, with an emphasis on worst‐case running times.

Abstract Matchings and T‐joins are fundamental and much‐studied concepts in graph theory and combinatorial optimization. One important application of matchings and T‐joins is in the computation of strong lower bounds for arc routing problems (ARPs). An ARP is a special kind of vehicle routing problem, in which the demands are located along edges or arcs, rather than at nodes. We point out that the literature on applying matchings and T‐joins to ARPs does not fully exploit the structure of real‐life road networks. We propose some ways to exploit this structure. Computational results show significant running time improvements, without deteriorating the quality of the lower bounds.

It is well known that the standard (linear) knapsack problem can be solved exactly by dynamic programming in 𝒪(nc) time, where n is the number of items and c is the capacity of the knapsack. The quadratic knapsack problem, on the other hand, is NP-hard in the strong sense, which makes it unlikely that it can be solved in pseudo-polynomial time. We show, however, that the dynamic programming approach to the linear knapsack problem can be modified to yield a highly effective constructive heuristic for the quadratic version. In our experiments, the lower bounds obtained by our heuristic were consistently within a fraction of a percent of optimal. Moreover, the addition of a simple local search step enabled us to obtain the optimal solution of all instances considered.

Minimum linear arrangement is a classical basic combinatorial optimization problem from the 1960s that turns out to be extremely challenging in practice. In particular, for most of its benchmark instances, even the order of magnitude of the optimal solution value is unknown, as testified by the surveys on the problem that contain tables in which the best-known solution value often has one more digit than the best-known lower bound value. In this paper, we propose a linear programming-based approach to compute lower bounds on the optimum. This allows us, for the first time, to show that the best-known solutions are indeed not far from optimal for most of the benchmark instances.

Abstract In recent years, there has been an increased literature on so‐called generalized network design problems (GNDPs), such as the generalized minimum spanning tree problem and the generalized traveling salesman problem. In a GNDP, the node set of a graph is partitioned into “clusters,” and the feasible solutions must contain one node from each cluster. Up to now, the polyhedra associated with different GNDPs have been studied independently. The purpose of this article is to show that it is possible, to a certain extent, to derive polyhedral results for all GNDPs simultaneously. Along the way, we point out some interesting connections to other polyhedra, such as the quadratic semiassignment polytope and the boolean quadric polytope. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011

The famous Padberg–Rao separation algorithm for b-matching polyhedra can be implemented to run in $\mathcal{O}(|V|^2|E|\log(|V|^2/|E|))$ time in the uncapacitated case, and in $\mathcal{O}(|V||E|^2\log(|V|^2/|E|))$ time in the capacitated case. We give a new and simple algorithm for the capacitated case which can be implemented to run in $\mathcal{O}(|V|^2|E|\log(|V|^2/|E|))$ time.