Delay-constrained shortest paths: approximation algorithms and second-order cone models

Routing real-time traffic with maximum packet delay in contemporary telecommunication networks requires not only choosing a path, but also reserving transmission capacity along its arcs, as the delay is a nonlinear function of both components. The problem is known to be solvable in polynomial time under quite restrictive assumptions, i.e., Equal Rate Allocations (all arcs are reserved the same capacity) and identical reservation costs, whereas the general problem is NP-hard. We first extend the approaches to the ERA version to a pseudo-polynomial Dynamic Programming one for integer arc costs, and a FPTAS for the case of general arc costs. We then show that the general problem can be formulated as a mixed-integer Second-Order Cone (SOCP) program, and therefore solved with off-the-shelf technology. We compare two formulations: one based on standard big-M constraints, and one where Perspective Reformulation techniques are used to tighten the continuous relaxation. Extensive computational experiments on both real-world networks and randomly-generated realistic ones show that the ERA approach is fast and provides an effective heuristic for the general problem whenever it manages to find a solution at all, but it fails for a significant fraction of the instances that the SOCP models can solve. We therefore propose a three-pronged approach that combines the fast running time of the ERA algorithm and the effectiveness of the SOCP models, and show that it is capable of solving realistic-sized instances with high accuracy at different levels of network load in a time compatible with real-time usage in an operating environment

Static and Dynamic Routing Under Disjoint Dominant Extreme Demands

This paper considers the special case of the robust network design problem where the dominant extreme points of the demand polyhedron have a disjoint support. In this case static and dynamic routing lead to the same optimal solution, both for the splittable and the unspittable case. As a consequence, the robust network design problem with (splittable) dynamic routing is polynomially solvable, whereas it is coNP-Hard in the general case. This result applies to particular instances of the single-source Hose model

Robust optimisation of green wireless LANs under rate uncertainty and user mobility

We present a robust optimisation approach to energy savings in wireless local area networks, that incorporates both link capacity fluctuations and user mobility under Bertsimas and Sim's robust optimization paradigm. Preliminary computational results are discussed

A branch-and-Benders-cut method for nonlinear power design in green wireless local area networks

We consider a problem arising in the design of green wireless local area networks. Decisions on powering-on a set of access points (APs), via the assignment of one power level (PL) to each opened AP, and decisions on the assignment of the user terminals (UTs) to the opened APs, have to be taken simultaneously. The PL assigned to an AP affects, in a nonlinear way, the capacity of the connections between the AP and the UTs that are assigned to it. The objective is to minimize the overall power consumption of the APs, which has two components: location/capacity dimensioning costs of the APs; assignment costs that depend on the total demands assigned to the APs. We develop a branch-and-Benders-cut (BBC) method where, in a non-standard fashion, the master problem includes the variables of the Benders subproblem, but relaxes their integrality. The BBC method has been tested on a large set of instances, and compared to a Benders decomposition algorithm on a subset of instances without assignment costs, where the two approaches can be compared. The computational results show the superiority of BBC in terms of solution quality, scalability and robustness

A strongly polynomial algorithm for the Uniform Balanced Network Flow Problem

AbstractSeveral balanced optimization problems have been analysed in the literature. Here, the balanced network flow problem in the uniform case is studied, and it is shown that it can be solved by the Newton's approach inO(n2 log3 n) max-flow computations. The key of the proof is an extension of Radzik's analysis of Newton's method for linear fractional combinatorial optimization problems

Balanced paths in acyclic networks: tractable cases and related approaches

Given a weighted acyclic network G and two nodes s and t in G, we consider the problem of computing k balanced paths from s to t, that is, k paths such that the difference in cost between the longest and the shortest path is minimized.The problem has several variants. We show that, whereas the general problem is solvable in pseudopolynomial time, both the arc-disjoint and the node-disjoint variants (i.e., the variants where the k paths are required to be arc-disjoint and node-disjoint, respectively) are strongly NP-Hard. We then address some significant special cases of such variants, and propose exact as well as approximate algorithms for their solution. The proposed approaches are also able to solve versions of the problem in which k origin-destination pairs are provided, and a set of k paths linking the origin-destination pairs has to be computed in such a way to minimize the difference in cost between the longest and the shortest path in the set

Pattern Generation Policies to Cope with Robustness in Home Care

We consider the Robust Home Care problem, where caregiver-to-patient assignment, scheduling of patient requests and caregiver routing must be taken jointly in a given planning horizon, and patient demand is subject to uncertainty. We propose four alternative policies to fix scheduling decisions and experiment their impact when used as a building block of a decomposition approach. Preliminary experiments on large size instances show that such policies allow to efficiently compute robust solutions of good quality in terms of balancing caregivers' workload and in terms of number of satisfied uncertain requests

Robust portfolio asset allocation and risk measures

Many financial optimization problems involve future values of security prices, interest rates and exchange rates which are not known in advance, but can only be forecast or estimated. Several methodologies have therefore, been proposed to handle the uncertainty in financial optimization problems. One such methodology is Robust Statistics, which addresses the problem of making estimates of the uncertain parameters that are insensitive to small variations. A different way to achieve robustness is provided by Robust Optimization which, given optimization problems with uncertain parameters, looks for solutions that will achieve good objective function values for the realization of these parameters in given uncertainty sets. Robust Optimization thus offers a vehicle to incorporate an estimation of uncertain parameters into the decision making process. This is true, for example, in portfolio asset allocation. Starting with the robust counterparts of the classical mean-variance and minimumvariance portfolio optimization problems, in this paper we review several mathematical models, and related algorithmic approaches, that have recently been proposed to address uncertainty in portfolio asset allocation, focusing on Robust Optimization methodology. We also give an overview of some of the computational results that have been obtained with the described approaches. In addition we analyse the relationship between the concepts of robustness and convex risk measures