One More Weight is Enough: Toward the Optimal Traffic Engineering with OSPF

Traffic Engineering (TE) leverages information of network traffic to generate a routing scheme optimizing the traffic distribution so as to advance network performance. However, optimize the link weights for OSPF to the offered traffic is an known NP-hard problem. In this paper, motivated by the fairness concept of congestion control, we firstly propose a new generic objective function, where various interests of providers can be extracted with different parameter settings. And then, we model the optimal TE as the utility maximization of multi-commodity flows with the generic objective function and theoretically show that any given set of optimal routes corresponding to a particular objective function can be converted to shortest paths with respect to a set of positive link weights. This can be directly configured on OSPF-based protocols. On these bases, we employ the Network Entropy Maximization(NEM) framework and develop a new OSPF-based routing protocol, SPEF, to realize a flexible way to split traffic over shortest paths in a distributed fashion. Actually, comparing to OSPF, SPEF only needs one more weight for each link and provably achieves optimal TE. Numerical experiments have been done to compare SPEF with the current version of OSPF, showing the effectiveness of SPEF in terms of link utilization and network load distribution

Sensitivity Analysis for Shortest Path Problems and Maximum Capacity Path Problems in Undirected Graphs

This paper addresses sensitivity analysis questions concerning the shortest path problem and the maximum capacity path problem in an undirected network. For both problems, we determine the maximum and minimum weights that each edge can have so that a given path remains optimal. For both problems, we show how to determine these maximum and minimum values for all edges in O(m + K log K) time, where m is the number of edges in the network, and K is the number of edges on the given optimal path

Optimal multi-objective discrete decision making using a multidirectional modified Physarum solver

This paper will address a bio-inspired algorithm able to incrementally grow decision graphs in multiple directions for discrete multi-objective optimization. The algorithm takes inspiration from the slime mould Physarum Polycephalum, an amoeboid organism that in its plasmodium state extends and optimizes a net of veins looking for food. The algorithm is here used to solve multi-objective Traveling Salesman and Vehicle Routing Problems selected as representative examples of multi-objective discrete decision making problems. Simulations on selected test case showed that building decision sequences in two directions and adding a matching ability (multidirectional approach) is an advantageous choice if compared with the choice of building decision sequences in only one direction (unidirectional approach). The ability to evaluate decisions from multiple directions enhances the performance of the solver in the construction and selection of optimal decision sequences

A computational comparison of two simplicial decomposition approaches for the separable traffic assignment problems : RSDTA and RSDVI

Draft pel 4th Meeting del Euro Working Group on Transportation (Newcastle 9-11 setembre de 1.996)The class of simplicial decomposition methods has shown to constitute efficient tools for the solution of the variational inequality formulation of the general traffic assignment problem. The paper presents a particular implementation of such an algorithm, called RSDVI, and a restricted simplicial decomposition algorithm, developed adhoc for diagonal, separable, problems named RSDTA. Both computer codes are compared for large scale separable traffic assignment problems. Some meaningful figures are shown for general problems with several levels of asymmetry.Preprin

Exact solution of the evasive flow capturing problem

The Evasive Flow Capturing Problem is defined as the problem of locating a set of law enforcement facilities on the arcs of a road network to intercept unlawful vehicle flows traveling between origin-destination pairs, who in turn deviate from their route to avoid any encounter with such facilities. Such deviations are bounded by a given tolerance. We first propose a bilevel program that, in contrast to previous studies, does not require a priori route generation. We then transform this bilevel model into a single-stage equivalent model using duality theory to yield a compact formulation. We finally reformulate the problem by describing the extreme rays of the polyhedral cone of the compact formulation and by projecting out the auxiliary variables, which leads to facet-defining inequalities and a cut formulation with an exponential number of constraints. We develop a branch-and-cut algorithm for the resulting model, as well as two separation algorithms to solve the cut formulation. Through extensive experiments on real and randomly generated networks, we demonstrate that our best model and algorithm accelerate the solution process by at least two orders of magnitude compared with the best published algorithm. Furthermore, our best model significantly increases the size of the instances that can be solved optimally

Exact solution of the evasive flow capturing problem

The Evasive Flow Capturing Problem is defined as the problem of locating a set of law enforcement facilities on the arcs of a road network to intercept unlawful vehicle flows traveling between origin-destination pairs, who in turn deviate from their route to avoid any encounter with such facilities. Such deviations are bounded by a given tolerance. We first propose a bilevel program that, in contrast to previous studies, does not require a priori route generation. We then transform this bilevel model into a single-stage equivalent model using duality theory to yield a compact formulation. We finally reformulate the problem by describing the extreme rays of the polyhedral cone of the compact formulation and by projecting out the auxiliary variables, which leads to facet-defining inequalities and a cut formulation with an exponential number of constraints. We develop a branch-and-cut algorithm for the resulting model, as well as two separation algorithms to solve the cut formulation. Through extensive experiments on real and randomly generated networks, we demonstrate that our best model and algorithm accelerate the solution process by at least two orders of magnitude compared with the best published algorithm. Furthermore, our best model significantly increases the size of the instances that can be solved optimally.</p

Complexity of determining exact tolerances for min-max combinatorial optimization problems

Suppose that we are given an instance of a combinatorial optimization problemwith min-max objective along with an optimal solution for it. Let the cost of asingle element be varied. We refer to the range of values of the element’s costfor which the given optimal solution remains optimal as its exact tolerance. Inthis paper we examine the problem of determining the exact tolerance of eachelement in combinatorial optimization problems with min-max objectives. Weshow that under very weak assumptions, the exact tolerance of each elementcan be determined in polynomial time if and only if the original optimizationproblem can be solved in polynomial time

Quadratically-Regularized Optimal Transport on Graphs

Optimal transportation provides a means of lifting distances between points on a geometric domain to distances between signals over the domain, expressed as probability distributions. On a graph, transportation problems can be used to express challenging tasks involving matching supply to demand with minimal shipment expense; in discrete language, these become minimum-cost network flow problems. Regularization typically is needed to ensure uniqueness for the linear ground distance case and to improve optimization convergence; state-of-the-art techniques employ entropic regularization on the transportation matrix. In this paper, we explore a quadratic alternative to entropic regularization for transport over a graph. We theoretically analyze the behavior of quadratically-regularized graph transport, characterizing how regularization affects the structure of flows in the regime of small but nonzero regularization. We further exploit elegant second-order structure in the dual of this problem to derive an easily-implemented Newton-type optimization algorithm.Comment: 27 page