We consider the traveling salesman problem when the cities are points in R^d for some fixed d and distances are computed according to geometric distances, determined by some norm. We show that for any polyhedral norm, the problem of finding a tour of maximum length can be solved in polynomial time. If arithmetic operations are assumed to take unit time, our algorithms run in time O(n log n), where f is the number of facets of the polyhedron determining the polyhedral norm. Thus for example we have O(n log n) algorithms for the cases of points in the plane under the Rectilinear and Sup norms. This is in contrast to the fact that finding a minimum length tour in each case is NP-hard. Our approach can be extended to the more general case of quasi-norms with not necessarily symmetric unit ball, where we get a complexity of O(n log n).

maximum flow, networks, parametric flow networks, graphs, optimization, selection, sequencing A natural extension of the maximum flow problem is the parametric maximum flow problem, in which some of the arc capacities in the network are functions of a single parameter λ. Previous approaches to the problem compute the maximum flow for a given sequence of parameter values sequentially taking advantage of the solution at the previous parameter value to speed up the computation at the next. In this paper, we present a new Simultaneous Parametric Maximum Flow (SPMF) algorithm that finds the maximum flow and a minimum cut of an important class of parametric networks for all values of parameter λ simultaneously. Instead of working with the original parametric network, a new non-parametric network is derived from the original and the SPMF gives a particular state of the flows in the derived network, from which the nested minimum-cuts under all λ-values are derived in a single scan of the vertices in a sorted order. SPMF simultaneously discovers all breakpoints of λ where the maximum flow as a step-function of λ jumps. The maximum flows at these λ-values are calculated in O(m) time from the minimum-cuts; m is the number of arcs. Generalization beyond bipartite networks is also shown.

Abstract. The parametric maximum flow problem is an extension of the classical maximum flow problem in which the capacities of certain arcs are not fixed but are functions of a single parameter. Gallo et al. [6] showed that certain versions of the push-relabel algorithm for ordinary maximum flow can be extended to the parametric problem while only increasing the worst-case time bound by a constant factor. Recently Zhang et al. [14,13] proposed a novel, simple balancing algorithm for the parametric problem on bipartite networks. They claimed good performance for their algorithm on networks arising from a real-world application. We describe the results of an experimental study comparing the performance of the balancing algorithm, the GGT algorithm, and a simplified version of the GGT algorithm, on networks related to those of the application of Zhang et al. as well as networks designed to be hard for the balancing algorithm. Our implementation of the balancing algorithm beats both versions of the GGT algorithm on networks related to the application, thus supporting the observations of Zhang et al. On the other hand, the GGT algorithm is more robust; it beats the balancing algorithm on some natural networks, and by asymptotically increasing amount on networks designed to be hard for the balancing algorithm. 1

This paper gives output sensitive parallel algorithms whose performance

We study linear programming models that contain transportation constraints in their formulation. Typically, these models have a multi-stage nature and the transportation constraints together with the associated ow variables are used to achieve consistency between consecutive stages. We describe how to reformulate these models by projecting out the ow variables. The reformulation can be more desirable since it has fewer variables and can be solved faster. We apply these ideas to reformulate two well-known workforce stang and scheduling problems: the shift scheduling problem and the tour scheduling problem. We also present computational results. 1

this paper, we define the minimax flow problem and design an O(k \Delta M(n;m)) time optimal algorithm for a special case of the problem in which the weights on arcs are either 0 or 1, where n is the number of vertices, m is the number of arcs, k (where 1 k m) is the number of arcs with nonzero weights, and M(n;m) is the best time bound for finding a maximum flow in a network.

We present a preflow-push algorithm for determining the vertex connectivity of an n-node, m-edge graph with positive vertex capacities. We give a deterministic algorithm that computes (u) = min v 6=u (u; v) in time O(mn log n), where (u; v) is the capacity of the minimum vertex cut between u and v. This leads to a deterministic algorithm for computing in time O(knm log n) and a Monte-Carlo algorithm with expected time O(nm log 2 n), where k is the number of nodes in a minimum vertex cut. 1 Introduction Vertex connectivity and edge connectivity problems and algorithms are closely related. Often, however, running times for known vertex connectivity algorithms are asymptotically worse than corresponding edge connectivity algorithms. In this paper, we show that an elegant and powerful edge-connectivity algorithm recently discovered by Hao and Orlin can be modified to work in the context of vertex connectivity without an asymptotic loss in performance guarantees. In particular, Hao a...

parametric flow networks, graphs, optimization, selection, sequencing A new algorithm, SPMF simple, for finding the complete chain of solutions of the product selection model is presented in this report. λ-directed simple residual path is identified to the only kind of residual path necessary for the new algorithm. By augmenting the right amount of flows along λ-directed simple residual paths, the new algorithm is monotone convergent.

In a sustained development scenario it is often the case that an investment is to be made over time in facilities that generate benefits. The benefits result from joint synergies between the facilities expressed as positive utilities specific to some subsets of facilities. As incremental budgets to finance fixed facility costs become available over time, additional facilities can be opened. The question is which facilities should be opened in order to guarantee that the overall benefit return over time is on the highest possible trajectory. This problem is common in situations such as ramping up a communication or transportation network where the facilities are hubs or service stations, or when introducing new technologies such as alternative fuels for cars and the facilities are fueling stations, or when expanding the production capacity with new machines, or when facilities are functions in a developing organization that is forced to make choices of where to invest limited funding. An intuitive strategy frequently used to evolve the set of facilities is a greedy approach that picks additional facilities which provide a highest rate of return on each budget increment. Such strategy is shown to have adverse impact by locking the system into future suboptimal solutions with total benefit that can be arbitrarily small compared to the optimal evolution sequence. It is shown here that the degree of suboptimality of such greedy strategy is extreme in that it can lead to the loss of the majority of future benefits.

Abstract. We describe a two-level push-relabel algorithm for the maximum flow problem and compare it to the competing codes. The algorithm generalizes a practical algorithm for bipartite flows. Experiments show that the algorithm performs well on several problem families. 1