How to Pay, Come What May: Approximation Algorithms for Demand-Robust Covering Problems

Abstract

Robust optimization has traditionally focused on uncertainty in data and costs in optimization problems to formulate models whose solutions will be optimal in the worstcase among the various uncertain scenarios in the model. While these approaches may be thought of defining data-or cost-robust problems, we formulate a new "demand-robust" model motivated by recent work on two-stage stochastic optimization problems. We propose this in the framework of general covering problems and prove a general structural lemma about special types of first-stage solutions for such problems: there exists a first-stage solution that is a minimal feasible solution for the union of the demands for some subset of the scenarios and its objective function value is no more than twice the optimal. We then provide approximation algorithms for a variety of standard discrete covering problems in this setting, including minimum cut, minimum multi-cut, shortest paths, Steiner trees, vertex cover and un-capacitated facility location. While many of our results draw from rounding approaches recently developed for stochastic programming problems, we also show new applications of old metric rounding techniques for cut problems in this demand-robust setting.

Stochastic Combinatorial Optimization with Controllable Risk Aversion Level

“…Consequently, we can use the following flow-based integer programming formulation to find that near-optimal solution (cf. [11,6]): y 0,e (t) ∀ v ∈ {r, t}, ∀ t ∈ S A , ∀ A (26) y 0,e (t) ≤ x e , y A,e (t) ≤ r A,e ∀ e ∈ E, ∀ t ∈ S A , ∀ A (27)…”

Stochastic Combinatorial Optimization with Controllable Risk Aversion Level

“…P for each x ∈ X (i.e. P(ω) = 0 implies that Q x (ω) = 0), there will be a corresponding weighing function f x given precisely by (6). Thus, in this context, we see that f x is simply the Radon-Nikodym derivative of Q x w.r.t.…”

“…Consequently, our result extends the class of problems that can be efficiently treated by the SAA method. Finally, by combining with techniques developed in earlier work [18,10,23,6], we obtain the first approximation algorithms for a large class of 2-stage stochastic combinatorial optimization problems under the risk-adjusted setting.…”

“…On another front, motivated by robustness concerns, Dhamdhere et al [6] have considered the case where Φ is the max operator and developed approximation algorithms for various 2-stage stochastic combinatorial optimization problems with recourse under that setting. Unfortunately, their techniques only work under the more restrictive scenario model 1 .…”

“…It should be noted, however, that the above robust optimization framework has some limitations. First, it is clear that the approaches used by Dhamdhere et al [6] and Feige et al [8] for specifying the underlying probability distribution are less general than the black-box model. Secondly, since the worst-case scenario may occur with an exponentially small probability, it seems unlikely that sampling techniques can be applied to efficiently solve the aforementioned robust optimization problems.…”