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

Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms

Constrained submodular maximization problems have long been studied, with near-optimal results known under a variety of constraints when the submodular function is monotone. The case of non-monotone submodular maximization is less understood: the first approximation algorithms even for the unconstrainted setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC '09, APPROX '09) show how to approximately maximize non-monotone submodular functions when the constraints are given by the intersection of p matroid constraints; their algorithm is based on local-search procedures that consider p-swaps, and hence the running time may be n^Omega(p), implying their algorithm is polynomial-time only for constantly many matroids. In this paper, we give algorithms that work for p-independence systems (which generalize constraints given by the intersection of p matroids), where the running time is poly(n,p). Our algorithm essentially reduces the non-monotone maximization problem to multiple runs of the greedy algorithm previously used in the monotone case. Our idea of using existing algorithms for monotone functions to solve the non-monotone case also works for maximizing a submodular function with respect to a knapsack constraint: we get a simple greedy-based constant-factor approximation for this problem. With these simpler algorithms, we are able to adapt our approach to constrained non-monotone submodular maximization to the (online) secretary setting, where elements arrive one at a time in random order, and the algorithm must make irrevocable decisions about whether or not to select each element as it arrives. We give constant approximations in this secretary setting when the algorithm is constrained subject to a uniform matroid or a partition matroid, and give an O(log k) approximation when it is constrained by a general matroid of rank k.Comment: In the Proceedings of WINE 201

Back to the Source: an Online Approach for Sensor Placement and Source Localization

Source localization, the act of finding the originator of a disease or rumor in a network, has become an important problem in sociology and epidemiology. The localization is done using the infection state and time of infection of a few designated sensor nodes; however, maintaining sensors can be very costly in practice. We propose the first online approach to source localization: We deploy a priori only a small number of sensors (which reveal if they are reached by an infection) and then iteratively choose the best location to place new sensors in order to localize the source. This approach allows for source localization with a very small number of sensors; moreover, the source can be found while the epidemic is still ongoing. Our method applies to a general network topology and performs well even with random transmission delays

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

A General Framework for Sensor Placement in Source Localization

When an epidemic spreads in a given network of individuals or communities, can we detect its source using only the information provided by a small set of nodes? We propose a general framework that incorporates two dimensions. First, we can either rely exclusively on a set of selected nodes (i.e., sensors) which always reveal their state independently of any particular epidemic (these are called static), or we can add some sensors (called dynamic) as an epidemic spreads, depending on which additional information is required. Second, the method can either localizes the source after an epidemic has spread through the entire network (offline), or while the epidemic is ongoing (online). We empirically study the performance of offline and online localization both with and without dynamic sensors. Our analysis shows that, by using dynamic sensors, the number of sensors necessary to localize the source is reduced by up to a factor of 10 and that, even with high-variance transmission delays, the source can be localized by using fewer than 5% of the nodes as sensors

Tractable Policies in Dynamic Robust Optimization

In many sequential decision problems, uncertainty is revealed over time and we need to make decisions in the face of uncertainty. This is a fundamental problem arising in many applications such as facility location, resource allocation and capacity planning under demand uncertainty. Robust optimization is an approach to model uncertainty where we optimize over the worst-case realization of parameters within an uncertainty set. While computing an optimal solution in dynamic robust optimization is usually intractable, affine policies (or linear decision rules) are widely used as an approximate solution approach. However, there is a stark contrast between the observed good empirical performance and the bad worst-case theoretical performance bounds. In the first part of this thesis, we address this stark contrast between theory and practice. In particular, we introduce a probabilistic approach in Chapter 2 to analyze the performance of affine policies on randomly generated instances and show they are near-optimal with high probability under reasonable assumptions. In Chapter 3, we study these policies under important models of uncertainty such as budget of uncertainty sets and intersection of budgeted sets and show that affine policies give an optimal approximation matching the hardness of approximation. In the second part of the thesis and based on our analysis of affine policies, we design new tractable policies for dynamic robust optimization. In particular, in Chapter 4, we present a tractable framework to design piecewise affine policies that can be computed efficiently and improve over affine policies for many instances. In Chapter 5, we introduce extended affine policies and threshold policies and show that their performance guarantees are significantly better than previous policies. Finally, in Chapter 6, we study piecewise static policies and their limitations for solving some classes of dynamic robust optimization problems

Localizing the Source of an Epidemic Using Few Observations

Localizing the source of an epidemic is a crucial task in many contexts, including the detection of malicious users in social networks and the identification of patient zeros of disease outbreaks. The difficulty of this task lies in the strict limitations on the data available: In most cases, when an epidemic spreads, only few individuals, who we will call sensors, provide information about their state. Furthermore, as the spread of an epidemic usually depends on a large number of variables, accounting for all the possible spreading patterns that could explain the available data can easily result in prohibitive computational costs. Therefore, in the field of source localization, there are two central research directions: The design of practical and reliable algorithms for localizing the source despite the limited data, and the optimization of data collection, i.e., the identification of the most informative sensors. In this dissertation we contribute to both these directions. We consider network epidemics starting from an unknown source. The only information available is provided by a set of sensor nodes that reveal if and when they become infected. We study how many sensors are needed to guarantee the identification of the source. A set of sensors that guarantees the identification of the source is called a double resolving set (DRS); the minimum size of a DRS is called the double metric dimension (DMD). Computing the DMD is, in general, hard, hence estimating it with bounds is desirable. We focus on G(N,p) random networks for which we derive tight bounds for the DMD. We show that the DMD is a non-monotonic function of the parameter p, hence there are critical parameter ranges in which source localization is particularly difficult. Again building on the relationship between source localization and DRSs, we move to optimizing the choice of a fixed number K of sensors. First, we look at the case of trees where the uniqueness of paths makes the problem simpler. For this case, we design polynomial time algorithms for selecting K sensors that optimize certain metrics of interest. Next, turning to general networks, we show that the optimal sensor set depends on the distribution of the time it takes for an infected node u to infect a non-infected neighbor v, which we call the transmission delay from u to v. We consider both a low- and a high-variance regime for the transmission delays. We design algorithms for sensor placement in both cases, and we show that they yield an improvement of up to 50% over state-of-the-art methods. Finally, we propose a framework for source localization where some sensors (called dynamic sensors) can be added while the epidemic spreads and the localization progresses. We design an algorithm for joint source localization and dynamic sensor placement; This algorithm can handle two regimes: offline localization, where we localize the source after the epidemic spread, and online localization, where we localize the source while the epidemic is ongoing. We conduct an empirical study of offline and online localization and show that, by using dynamic sensors, the number of sensors we need to localize the source is up to 10 times less with respect to a strategy where all sensors are deployed a priori. We also study the resistance of our methods to high-variance transmission delays and show that, even in this setting, using dynamic sensors, the source can be localized with less than 5% of the nodes being sensors