Robust constrained shortest path problems under budgeted uncertainty

Abstract

We study the robust constrained shortest path problem under resource uncertainty. After proving that the problem is N P-hard in the strong sense for arbitrary uncertainty sets, we focus on budgeted uncertainty sets introduced by Bertsimas and Sim (2003) and their extension to variable uncertainty by Poss (2013). We apply classical techniques to show that the problem with capacity constraint can be solved in pseudo-polynomial time. However, we prove that the problem with time windows is N P-hard in the strong sense when Γ is not fixed, using a reduction from the independent set problem. We introduce then new robust labels that yield dynamic programming algorithms for the problems with time windows and capacity constraints. The running times of these algorithms are pseudo-polynomial when Γ is fixed, exponential otherwise. We present numerical results for the problem with time windows which show the effectiveness of the label-setting algorithm based on the new robust labels. Our numerical results also highlight the reduction in price of robustness obtained when using variable budgeted uncertainty instead of classical budgeted uncertainty.

Robust Combinatorial Optimization with Locally Budgeted Uncertainty

“…Advantages of this set include its intuitive description for a decision maker, and that robust counterparts remain efficiently solvable for nominal problems that can be solved efficiently, even though the budgeted uncertainty set has an exponential number of extreme points. These benefits have lead to a substantial amount of research into robust optimization problems with budgeted uncertainty sets, see, e.g., [3,7,10,11,14] and many more. But there are also limitations to this approach, which has lead to the development of alternative uncertainty sets.…”

Some Black-box Reductions for Objective-robust Discrete Optimization Problems Based on their LP-Relaxations

“…We note also that some of these results can be derived from our reduction in Section 2. Finally, it is worth noting that there is a number of results on special problems, such as SHORTESTPATH [3], MINCOSTFLOW [8], MACHINESCHEDULING [12], VEHICLEROUTING [1], two-stage robust optimization [15,19], mostly under a class of budget uncertainty. In general, this seems to be a growing area of research, see, e.g., the theses by Poss [25] and Ilyina [22].…”

Robust Combinatorial Optimization with Locally Budgeted Uncertainty

“…Advantages of this set include its intuitive description for a decision maker, and that robust counterparts remain efficiently solvable for nominal problems that can be solved efficiently, even though the budgeted uncertainty set has an exponential number of extreme points. These benefits have lead to a substantial amount of research into robust optimization problems with budgeted uncertainty sets, see, e.g., [3,7,10,11,18] and many more. But there are also limitations to this approach, which has lead to the development of alternative uncertainty sets.…”