Robust and MaxMin Optimization under Matroid and Knapsack Uncertainty Sets

Consider the following problem: given a set system (U,I) and an edge-weighted graph G = (U, E) on the same universe U, find the set A in I such that the Steiner tree cost with terminals A is as large as possible: "which set in I is the most difficult to connect up?" This is an example of a max-min problem: find the set A in I such that the value of some minimization (covering) problem is as large as possible. In this paper, we show that for certain covering problems which admit good deterministic online algorithms, we can give good algorithms for max-min optimization when the set system I is given by a p-system or q-knapsacks or both. This result is similar to results for constrained maximization of submodular functions. Although many natural covering problems are not even approximately submodular, we show that one can use properties of the online algorithm as a surrogate for submodularity. Moreover, we give stronger connections between max-min optimization and two-stage robust optimization, and hence give improved algorithms for robust versions of various covering problems, for cases where the uncertainty sets are given by p-systems and q-knapsacks.Comment: 17 pages. Preliminary version combining this paper and http://arxiv.org/abs/0912.1045 appeared in ICALP 201

Thresholded Covering Algorithms for Robust and Max-Min Optimization

The general problem of robust optimization is this: one of several possible scenarios will appear tomorrow, but things are more expensive tomorrow than they are today. What should you anticipatorily buy today, so that the worst-case cost (summed over both days) is minimized? Feige et al. and Khandekar et al. considered the k-robust model where the possible outcomes tomorrow are given by all demand-subsets of size k, and gave algorithms for the set cover problem, and the Steiner tree and facility location problems in this model, respectively. In this paper, we give the following simple and intuitive template for k-robust problems: "having built some anticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat". In this paper we show that this template gives us improved approximation algorithms for k-robust Steiner tree and set cover, and the first approximation algorithms for k-robust Steiner forest, minimum-cut and multicut. All our approximation ratios (except for multicut) are almost best possible. As a by-product of our techniques, we also get algorithms for max-min problems of the form: "given a covering problem instance, which k of the elements are costliest to cover?".Comment: 24 page

A parameterized view to the robust recoverable base problem of matroids under structural uncertainty

We study a robust recoverable version of the matroid base problem where the uncertainty is imposed on combinatorial structures rather than on weights as studied in the literature. We prove that the problem is NP-hard even when a given matroid is uniform or graphic. On the other hand, we prove that the problem is fixed-parameter tractable with respect to the number of scenarios

One Step at a Time: Optimizing SDN Upgrades in ISP Networks

© 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.Nowadays, there is a fast-paced shift from legacy telecommunication systems to novel Software Defined Network (SDN) architectures that can support on-the-fly network reconfiguration, therefore, empowering advanced traffic engineering mechanisms. Despite this momentum, migration to SDN cannot be realized at once especially in high-end cost networks of Internet Service Providers (ISPs). It is expected that ISPs will gradually upgrade their networks to SDN over a period that spans several years. In this paper, we study the SDN upgrading problem in an ISP network: which nodes to upgrade and when. We consider a general model that captures different migration costs and network topologies, and two plausible ISP objectives; first, the maximization of the traffic that traverses at least one SDN node, and second, the maximization of the number of dynamically selectable routing paths enabled by SDN nodes. We leverage the theory of submodular and supermodular functions to devise algorithms with provable approximation ratios for each objective. Using real-world network topologies and traffic matrices, we evaluate the performance of our algorithms and show up to 54% gains over state-of-the-art methods. Moreover, we describe the interplay between the two objectives; maximizing one may cause a factor of 2 loss to the other.EC/H2020/679158/EU/Resolving the Tussle in the Internet: Mapping, Architecture, and Policy Making/ResolutioNe

Optimizing Gradual SDN Upgrades in ISP Networks

Nowadays, there is a fast-paced shift from legacy telecommunication systems to novel software-defined network (SDN) architectures that can support on-the-fly network reconfiguration, therefore, empowering advanced traffic engineering mechanisms. Despite this momentum, migration to SDN cannot be realized at once especially in high-end networks of Internet service providers (ISPs). It is expected that ISPs will gradually upgrade their networks to SDN over a period that spans several years. In this paper, we study the SDN upgrading problem in an ISP network: which nodes to upgrade and when we consider a general model that captures different migration costs and network topologies, and two plausible ISP objectives: 1) the maximization of the traffic that traverses at least one SDN node, and 2) the maximization of the number of dynamically selectable routing paths enabled by SDN nodes. We leverage the theory of submodular and supermodular functions to devise algorithms with provable approximation ratios for each objective. Using real-world network topologies and traffic matrices, we evaluate the performance of our algorithms and show up to 54% gains over state-of-the-art methods. Moreover, we describe the interplay between the two objectives; maximizing one may cause a factor of 2 loss to the other. We also study the dual upgrading problem, i.e., minimizing the upgrading cost for the ISP while ensuring specific performance goals. Our analysis shows that our proposed algorithm can achieve up to 2.5 times lower cost to ensure performance goals over state-of-the-art methods.EC/H2020/679158/EU/Resolving the Tussle in the Internet: Mapping, Architecture, and Policy Making/ResolutioNe

Approximation Algorithms for Distributionally Robust Stochastic Optimization with Black-Box Distributions

Two-stage stochastic optimization is a framework for modeling uncertainty, where we have a probability distribution over possible realizations of the data, called scenarios, and decisions are taken in two stages: we make first-stage decisions knowing only the underlying distribution and before a scenario is realized, and may take additional second-stage recourse actions after a scenario is realized. The goal is typically to minimize the total expected cost. A criticism of this model is that the underlying probability distribution is itself often imprecise! To address this, a versatile approach that has been proposed is the {\em distributionally robust 2-stage model}: given a collection of probability distributions, our goal now is to minimize the maximum expected total cost with respect to a distribution in this collection. We provide a framework for designing approximation algorithms in such settings when the collection is a ball around a central distribution and the central distribution is accessed {\em only via a sampling black box}. We first show that one can utilize the {\em sample average approximation} (SAA) method to reduce the problem to the case where the central distribution has {\em polynomial-size} support. We then show how to approximately solve a fractional relaxation of the SAA (i.e., polynomial-scenario central-distribution) problem. By complementing this via LP-rounding algorithms that provide {\em local} (i.e., per-scenario) approximation guarantees, we obtain the {\em first} approximation algorithms for the distributionally robust versions of a variety of discrete-optimization problems including set cover, vertex cover, edge cover, facility location, and Steiner tree, with guarantees that are, except for set cover, within $O(1)$-factors of the guarantees known for the deterministic version of the problem

Target-based Optimization in Operations Management

Ph.DDOCTOR OF PHILOSOPH