Ravishankar Krishnaswamy

Online Primal-Dual for Non-linear Optimization with Applications to Speed Scaling

We give a principled method to design online algorithms (for potentially non-linear problems) using a mathematical programming formulation of the problem, and also to analyze the competitiveness of the resulting algorithm using the dual program. This method can be viewed as an extension of the online primal-dual method for linear programming problems, to nonlinear programs. We show the application of this method to two online speed-scaling problems: one involving scheduling jobs on a speed scalable processor so as to minimize energy plus an arbitrary sum scheduling objective, and one involving routing virtual circuit connection requests in a network of speed scalable routers so as to minimize the aggregate power or energy used by the routers. This analysis shows that competitive algorithms exist for problems that had resisted analysis using the dominant potential function approach in the speed-scaling literature, and provides alternate cleaner analysis for other known results. This gives us another tool in the design and analysis of primal-dual algorithms for online problems.

Scalably scheduling power-heterogeneous processors

We show that a natural online algorithm for scheduling jobs on a heterogeneous multiprocessor, with arbitrary power functions, is scalable for the objective function of weighted flow plus energy.

Better Scalable Algorithms for Broadcast Scheduling

In the classical <i>broadcast scheduling problem</i>, there are <i>n</i> pages stored at a server, and requests for these pages arrive over time. Whenever a page is broadcast, it satisfies all outstanding requests for that page. The objective is to minimize average <i>flow time</i> of the requests. For any ε > 0, we give a (1+ε)-speed <i>O</i>(1/ε³)-competitive online algorithm for broadcast scheduling. This improves over the recent breakthrough result of Im and Moseley [2010], where they obtained a (1+ε)-speed <i>O</i>(1/ε¹¹)-competitive algorithm. Our algorithm and analysis are considerably simpler than Im and Moseley [2010]. More importantly, our techniques also extend to the general setting of <i>nonuniform page sizes</i> and <i>dependent requests</i>. This is the first scalable algorithm for broadcast scheduling with varying size pages and resolves the main open question from Im and Moseley [2010].

Relax, No Need to Round: Integrality of Clustering Formulations

We study exact recovery conditions for convex relaxations of point cloud clustering problems, focusing on two of the most common optimization problems for unsupervised clustering: k-means and k-median clustering. Motivations for focusing on convex relaxations are: (a) they come with a certificate of optimality, and (b) they are generic tools which are relatively parameter-free, not tailored to specific assumptions over the input. More precisely, we consider the distributional setting where there are k clusters in Rm and data from each cluster consists of n points sampled from a symmetric distribution within a ball of unit radius. We ask: what is the minimal separation distance between cluster centers needed for convex relaxations to exactly recover these k clusters as the optimal integral solution? For the k-median linear programming relaxation we show a tight bound: exact recovery is obtained given arbitrarily small pairwise separation ε > O between the balls. In other words, the pairwise center separation is δ > 2+ε. Under the same distributional model, the k-means LP relaxation fails to recover such clusters at separation as large as δ = 4. Yet, if we enforce PSD constraints on the k-means LP, we get exact cluster recovery at separation as low as δ > min{2 + √2k/m}, 2+√2 + 2/m} + ε. In contrast, common heuristics such as Lloyds algorithm (a.k.a. the k means algorithm) can fail to recover clusters in this setting; even with arbitrarily large cluster separation, k-means++ with overseeding by any constant factor fails with high probability at exact cluster recovery. To complement the theoretical analysis, we provide an experimental study of the recovery guarantees for these various methods, and discuss several open problems which these experiments suggest.

Scheduling jobs with varying parallelizability to reduce variance

We give a (2+ε)-speed <i>O</i>(1)-competitive algorithm for scheduling jobs with arbitrary speed-up curves for the l₂ norm of flow. We give a similar result for the broadcast setting with varying page sizes.

Better scalable algorithms for broadcast scheduling

In the classic broadcast scheduling problem, there are n pages stored at a server, and requests for these pages arrive over time. Whenever a page is broadcast, it satisfies all outstanding requests for that page. The objective is to minimize the average flowtime of the requests. In this paper, for any e > 0, we give a (1 + e)-speed O(1/e3)-competitive online algorithm for the broadcast scheduling problem, even when page sizes are not identical. This improves over the recent breakthrough result of Im and Moseley [18], where they obtained a (1 + e)-speed O(1/e11)-competitive algorithm for the setting when all pages have the same size. This is the first scalable algorithm for broadcast scheduling with varying size pages, and resolves the main open question from [18]. Furthermore, our algorithm and analysis are considerably simpler than [18], and also extend to the general setting of dependent requests.

Network-wide deployment of intrusion detection and prevention systems

Traditional efforts for scaling network intrusion detection (NIDS) and intrusion prevention systems (NIPS) have largely focused on a single-vantage-point view. In this paper, we explore an alternative design that exploits spatial, network-wide opportunities for distributing NIDS and NIPS functions. For the NIDS case, we design a linear programming formulation to assign detection responsibilities to nodes while ensuring that no node is overloaded. We describe a prototype NIDS implementation adapted from the Bro system to analyze traffic per these assignments, and demonstrate the advantages that this approach achieves. For NIPS, we show how to maximally leverage specialized hardware (e.g., TCAMs) to reduce the footprint of unwanted traffic on the network. Such hardware constraints make the optimization problem NP-hard, and we provide practical approximation algorithms based on randomized rounding.

Unconditional differentially private mechanisms for linear queries

We investigate the problem of designing differentially private mechanisms for a set of d linear queries over a database, while adding as little error as possible. Hardt and Talwar [HT10] related this problem to geometric properties of a convex body defined by the set of queries and gave a O(log3 d)-approximation to the minimum l22 error, assuming a conjecture from convex geometry called the Slicing or Hyperplane conjecture. In this work we give a mechanism that works unconditionally, and also gives an improved O(log2 d) approximation to the expected l22 error. We remove the dependence on the Slicing conjecture by using a result of Klartag [Kla06] that shows that any convex body is close to one for which the conjecture holds; our main contribution is in making this result constructive by using recent techniques of Dadush, Peikert and Vempala [DPV10]. The improvement in approximation ratio relies on a stronger lower bound we derive on the optimum. This new lower bound goes beyond the packing argument that has traditionally been used in Differential Privacy and allows us to add the packing lower bounds obtained from orthogonal subspaces. We are able to achieve this via a symmetrization argument which argues that there always exists a near optimal differentially private mechanism which adds noise that is independent of the input database! We believe this result should be of independent interest, and also discuss some interesting consequences.

Online and stochastic survivable network design

Consider the edge-connectivity survivable network design problem: given a graph G = (V,E) with edge-costs, and edge-connectivity requirements rij for every pair of vertices i,j, find an (approximately) minimum-cost network that provides the required connectivity. While this problem is known to admit good approximation algorithms in the offline case, no algorithms were known for this problem in the online setting. In this paper, we give a randomized O(rmax log3 n) competitive online algorithm for this edge-connectivity network design problem, where rmax = maxij rij. Our algorithms use the standard embeddings of graphs into random subtrees (i.e., into singly connected subgraphs) as an intermediate step to get algorithms for higher connectivity. Our results for the online problem give us approximation algorithms that admit strict cost-shares with the same strictness value. This, in turn, implies approximation algorithms for (a) the rent-or-buy version and (b) the (two-stage) stochastic version of the edge-connected network design problem with independent arrivals. For these two problems, if we are in the case when the underlying graph is complete and the edge-costs are metric (i.e., satisfy the triangle inequality), we improve our results to give O(1)-strict cost shares, which gives constant-factor rent-or-buy and stochastic algorithms for these instances.

Nonclairvoyantly scheduling power-heterogeneous processors

We show that a natural nonclairvoyant online algorithm for scheduling jobs on a power-heterogeneous multiprocessor is bounded-speed bounded-competitive for the objective of flow plus energy.