Searcharxiv⌕ Search

arXiv subjects

Anupam Gupta

Publications and source records attributed to Anupam Gupta.

158 records · Page 9Linked to original sources
arXiv2011

Iterative Constructions and Private Data Release

In this paper we study the problem of approximately releasing the cut function of a graph while preserving differential privacy, and give new algorithms (and new analyses of existing algorithms) in both the interactive and non-interactive settings. Our algorithms in the interactive setting are achieved by revisiting the problem of releasing differentially private, approximate answers to a large number of queries on a database. We show that several algorithms for this problem fall into the same basic framework, and are based on the existence of objects which we call iterative database construction algorithms. We give a new generic framework in which new (efficient) IDC algorithms give rise to new (efficient) interactive private query release mechanisms. Our modular analysis simplifies and tightens the analysis of previous algorithms, leading to improved bounds. We then give a new IDC algorithm (and therefore a new private, interactive query release mechanism) based on the Frieze/Kannan low-rank matrix decomposition. This new release mechanism gives an improvement on prior work in a range of parameters where the size of the database is comparable to the size of the data universe (such as releasing all cut queries on dense graphs). We also give a non-interactive algorithm for efficiently releasing private synthetic data for graph cuts with error O(|V|^{1.5}). Our algorithm is based on randomized response and a non-private implementation of the SDP-based, constant-factor approximation algorithm for cut-norm due to Alon and Naor. Finally, we give a reduction based on the IDC framework showing that an efficient, private algorithm for computing sufficiently accurate rank-1 matrix approximations would lead to an improved efficient algorithm for releasing private synthetic data for graph cuts. We leave finding such an algorithm as our main open problem.

cs.DS↗
arXiv2011

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?".

cs.DS↗
arXiv2011

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.

cs.DS↗
arXiv2011

Approximation Algorithms for Correlated Knapsacks and Non-Martingale Bandits

In the stochastic knapsack problem, we are given a knapsack of size B, and a set of jobs whose sizes and rewards are drawn from a known probability distribution. However, we know the actual size and reward only when the job completes. How should we schedule jobs to maximize the expected total reward? We know O(1)-approximations when we assume that (i) rewards and sizes are independent random variables, and (ii) we cannot prematurely cancel jobs. What can we say when either or both of these assumptions are changed? The stochastic knapsack problem is of interest in its own right, but techniques developed for it are applicable to other stochastic packing problems. Indeed, ideas for this problem have been useful for budgeted learning problems, where one is given several arms which evolve in a specified stochastic fashion with each pull, and the goal is to pull the arms a total of B times to maximize the reward obtained. Much recent work on this problem focus on the case when the evolution of the arms follows a martingale, i.e., when the expected reward from the future is the same as the reward at the current state. What can we say when the rewards do not form a martingale? In this paper, we give constant-factor approximation algorithms for the stochastic knapsack problem with correlations and/or cancellations, and also for budgeted learning problems where the martingale condition is not satisfied. Indeed, we can show that previously proposed LP relaxations have large integrality gaps. We propose new time-indexed LP relaxations, and convert the fractional solutions into distributions over strategies, and then use the LP values and the time ordering information from these strategies to devise a randomized adaptive scheduling algorithm. We hope our LP formulation and decomposition methods may provide a new way to address other correlated bandit problems with more general contexts.

cs.DS↗
arXiv2010

Forest Density Estimation

We study graph estimation and density estimation in high dimensions, using a family of density estimators based on forest structured undirected graphical models. For density estimation, we do not assume the true distribution corresponds to a forest; rather, we form kernel density estimates of the bivariate and univariate marginals, and apply Kruskal's algorithm to estimate the optimal forest on held out data. We prove an oracle inequality on the excess risk of the resulting estimator relative to the risk of the best forest. For graph estimation, we consider the problem of estimating forests with restricted tree sizes. We prove that finding a maximum weight spanning forest with restricted tree size is NP-hard, and develop an approximation algorithm for this problem. Viewing the tree size as a complexity parameter, we then select a forest using data splitting, and prove bounds on excess risk and structure selection consistency of the procedure. Experiments with simulated data and microarray data indicate that the methods are a practical alternative to Gaussian graphical models.

stat.ML↗
arXiv2010

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.

cs.DS↗
arXiv2010

When LP is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings

Consider a random graph model where each possible edge $e$ is present independently with some probability $p_e$. Given these probabilities, we want to build a large/heavy matching in the randomly generated graph. However, the only way we can find out whether an edge is present or not is to query it, and if the edge is indeed present in the graph, we are forced to add it to our matching. Further, each vertex $i$ is allowed to be queried at most $t_i$ times. How should we adaptively query the edges to maximize the expected weight of the matching? We consider several matching problems in this general framework (some of which arise in kidney exchanges and online dating, and others arise in modeling online advertisements); we give LP-rounding based constant-factor approximation algorithms for these problems. Our main results are the following: We give a 4 approximation for weighted stochastic matching on general graphs, and a 3 approximation on bipartite graphs. This answers an open question from [Chen etal ICALP 09]. Combining our LP-rounding algorithm with the natural greedy algorithm, we give an improved 3.46 approximation for unweighted stochastic matching on general graphs. We introduce a generalization of the stochastic online matching problem [Feldman etal FOCS 09] that also models preference-uncertainty and timeouts of buyers, and give a constant factor approximation algorithm.

cs.DS↗
arXiv2009

Differentially Private Combinatorial Optimization

Consider the following problem: given a metric space, some of whose points are "clients", open a set of at most $k$ facilities to minimize the average distance from the clients to these facilities. This is just the well-studied $k$-median problem, for which many approximation algorithms and hardness results are known. Note that the objective function encourages opening facilities in areas where there are many clients, and given a solution, it is often possible to get a good idea of where the clients are located. However, this poses the following quandary: what if the identity of the clients is sensitive information that we would like to keep private? Is it even possible to design good algorithms for this problem that preserve the privacy of the clients? In this paper, we initiate a systematic study of algorithms for discrete optimization problems in the framework of differential privacy (which formalizes the idea of protecting the privacy of individual input elements). We show that many such problems indeed have good approximation algorithms that preserve differential privacy; this is even in cases where it is impossible to preserve cryptographic definitions of privacy while computing any non-trivial approximation to even the_value_ of an optimal solution, let alone the entire solution. Apart from the $k$-median problem, we study the problems of vertex and set cover, min-cut, facility location, Steiner tree, and the recently introduced submodular maximization problem, "Combinatorial Public Projects" (CPP).

cs.DS↗
arXiv2009

Scheduling with Outliers

In classical scheduling problems, we are given jobs and machines, and have to schedule all the jobs to minimize some objective function. What if each job has a specified profit, and we are no longer required to process all jobs -- we can schedule any subset of jobs whose total profit is at least a (hard) target profit requirement, while still approximately minimizing the objective function? We refer to this class of problems as scheduling with outliers. This model was initiated by Charikar and Khuller (SODA'06) on the minimum max-response time in broadcast scheduling. We consider three other well-studied scheduling objectives: the generalized assignment problem, average weighted completion time, and average flow time, and provide LP-based approximation algorithms for them. For the minimum average flow time problem on identical machines, we give a logarithmic approximation algorithm for the case of unit profits based on rounding an LP relaxation; we also show a matching integrality gap. For the average weighted completion time problem on unrelated machines, we give a constant factor approximation. The algorithm is based on randomized rounding of the time-indexed LP relaxation strengthened by the knapsack-cover inequalities. For the generalized assignment problem with outliers, we give a simple reduction to GAP without outliers to obtain an algorithm whose makespan is within 3 times the optimum makespan, and whose cost is at most (1 + ε) times the optimal cost.

cs.DS↗
arXiv2008

Simpler Analyses of Local Search Algorithms for Facility Location

We study local search algorithms for metric instances of facility location problems: the uncapacitated facility location problem (UFL), as well as uncapacitated versions of the $k$-median, $k$-center and $k$-means problems. All these problems admit natural local search heuristics: for example, in the UFL problem the natural moves are to open a new facility, close an existing facility, and to swap a closed facility for an open one; in $k$-medians, we are allowed only swap moves. The local-search algorithm for $k$-median was analyzed by Arya et al. (SIAM J. Comput. 33(3):544-562, 2004), who used a clever ``coupling'' argument to show that local optima had cost at most constant times the global optimum. They also used this argument to show that the local search algorithm for UFL was 3-approximation; their techniques have since been applied to other facility location problems. In this paper, we give a proof of the $k$-median result which avoids this coupling argument. These arguments can be used in other settings where the Arya et al. arguments have been used. We also show that for the problem of opening $k$ facilities $F$ to minimize the objective function $Φ_p(F) = \big(\sum_{j \in V} d(j, F)^p\big)^{1/p}$, the natural swap-based local-search algorithm is a $Θ(p)$-approximation. This implies constant-factor approximations for $k$-medians (when $p=1$), and $k$-means (when $p = 2$), and an $O(\log n)$-approximation algorithm for the $k$-center problem (which is essentially $p = \log n$).

cs.DS↗
arXiv2007

How to Complete a Doubling Metric

In recent years, considerable advances have been made in the study of properties of metric spaces in terms of their doubling dimension. This line of research has not only enhanced our understanding of finite metrics, but has also resulted in many algorithmic applications. However, we still do not understand the interaction between various graph-theoretic (topological) properties of graphs, and the doubling (geometric) properties of the shortest-path metrics induced by them. For instance, the following natural question suggests itself: \emph{given a finite doubling metric $(V,d)$, is there always an \underline{unweighted} graph $(V',E')$ with $V\subseteq V'$ such that the shortest path metric $d'$ on $V'$ is still doubling, and which agrees with $d$ on $V$.} This is often useful, given that unweighted graphs are often easier to reason about. We show that for any metric space $(V,d)$, there is an \emph{unweighted} graph $(V',E')$ with shortest-path metric $d':V'\times V' \to \R_{\geq 0}$ such that -- for all $x,y \in V$, the distances $d(x,y) \leq d'(x,y) \leq (1+\eps) \cdot d(x,y)$, and -- the doubling dimension for $d'$ is not much more than that of $d$, where this change depends only on $\e$ and not on the size of the graph. We show a similar result when both $(V,d)$ and $(V',E')$ are restricted to be trees: this gives a simpler proof that doubling trees embed into constant dimensional Euclidean space with constant distortion. We also show that our results are tight in terms of the tradeoff between distortion and dimension blowup.

cs.DM↗
arXiv2007

Dial a Ride from k-forest

The k-forest problem is a common generalization of both the k-MST and the dense-$k$-subgraph problems. Formally, given a metric space on $n$ vertices $V$, with $m$ demand pairs $\subseteq V \times V$ and a ``target'' $k\le m$, the goal is to find a minimum cost subgraph that connects at least $k$ demand pairs. In this paper, we give an $O(\min\{\sqrt{n},\sqrt{k}\})$-approximation algorithm for $k$-forest, improving on the previous best ratio of $O(n^{2/3}\log n)$ by Segev & Segev. We then apply our algorithm for k-forest to obtain approximation algorithms for several Dial-a-Ride problems. The basic Dial-a-Ride problem is the following: given an $n$ point metric space with $m$ objects each with its own source and destination, and a vehicle capable of carrying at most $k$ objects at any time, find the minimum length tour that uses this vehicle to move each object from its source to destination. We prove that an $α$-approximation algorithm for the $k$-forest problem implies an $O(α\cdot\log^2n)$-approximation algorithm for Dial-a-Ride. Using our results for $k$-forest, we get an $O(\min\{\sqrt{n},\sqrt{k}\}\cdot\log^2 n)$- approximation algorithm for Dial-a-Ride. The only previous result known for Dial-a-Ride was an $O(\sqrt{k}\log n)$-approximation by Charikar & Raghavachari; our results give a different proof of a similar approximation guarantee--in fact, when the vehicle capacity $k$ is large, we give a slight improvement on their results.

cs.DS↗