Aditya Bhaskara

Polynomial integrality gaps for strong SDP relaxations of Densest k - subgraph

The Densest k-subgraph problem (i.e. find a size k subgraph with maximum number of edges), is one of the notorious problems in approximation algorithms. There is a significant gap between known upper and lower bounds for Densest k-subgraph: the current best algorithm gives an a O(n1/4) approximation, while even showing a small constant factor hardness requires significantly stronger assumptions than P ≠ NP. In addition to interest in designing better algorithms, a number of recent results have exploited the conjectured hardness of Densest k-subgraph and its variants. Thus, understanding the approximability of Densest k-subgraph is an important challenge. In this work, we give evidence for the hardness of approximating Densest k-subgraph within polynomial factors. Specifically, we expose the limitations of strong semidefinite programs from SDP hierarchies in solving Densest k-subgraph. Our results include: • A lower bound of Ω(n1/4/log3 n) on the integrality gap for Ω(log n/log log n) rounds of the Sherali-Adams relaxation for Densest k-subgraph. This also holds for the relaxation obtained from Sherali-Adams with an added SDP constraint. Our gap instances are in fact Erdos-Renyi random graphs. • For every e > 0, a lower bound of n2/53−e on the integrality gap of nΩ(e) rounds of the Lasserre SDP relaxation for Densest k-subgraph, and an nΩe(1) gap for n1−e rounds. Our construction proceeds via a reduction from random instances of a certain Max-CSP over large domains. In the absence of inapproximability results for Densest k-subgraph, our results show that beating a factor of nΩ(1) is a barrier for even the most powerful SDPs, and in fact even beating the best known n1/4 factor is a barrier for current techniques. Our results indicate that approximating Densest k-subgraph within a polynomial factor might be a harder problem than Unique Games or Small Set Expansion, since these problems were recently shown to be solvable using neΩ(1) rounds of the Lasserre hierarchy, where e is the completeness parameter in Unique Games and Small Set Expansion.

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.

Optimizing Display Advertising in Online Social Networks

Advertising is a significant source of revenue for most online social networks. Conventional online advertising methods need to be customized for online social networks in order to address their distinct characteristics. Recent experimental studies have shown that providing social cues along with ads, e.g. information about friends liking the ad or clicking on an ad, leads to higher click rates. In other words, the probability of a user clicking an ad is a function of the set of friends that have clicked the ad. In this work, we propose formal probabilistic models to capture this phenomenon, and study the algorithmic problem that then arises. Our work is in the context of display advertising where a contract is signed to show an ad to a pre-determined number of users. The problem we study is the following: given a certain number of impressions, what is the optimal display strategy, i.e. the optimal order and the subset of users to show the ad to, so as to maximize the expected number of clicks? Unlike previous models of influence maximization, we show that this optimization problem is hard to approximate in general, and that it is related to finding dense subgraphs of a given size. In light of the hardness result, we propose several heuristic algorithms including a two-stage algorithm inspired by influence-and-exploit strategies in viral marketing. We evaluate the performance of these heuristics on real data sets, and observe that our two-stage heuristic significantly outperforms the natural baselines.

Centrality of Trees for Capacitated k -Center

We consider the capacitated k-center problem. In this problem we are given a finite set of locations in a metric space and each location has an associated non-negative integer capacity. The goal is to choose (open) k locations (called centers) and assign each location to an open center to minimize the maximum, over all locations, of the distance of the location to its assigned center. The number of locations assigned to a center cannot exceed the center’s capacity. The uncapacitated k-center problem has a simple tight 2-approximation from the 80’s. In contrast, the first constant factor approximation for the capacitated problem was obtained only recently by Cygan, Hajiaghayi and Khuller who gave an intricate LP-rounding algorithm that achieves an approximation guarantee in the hundreds. In this paper we give a simple algorithm with a clean analysis and prove an approximation guarantee of 9. It uses the standard LP relaxation and comes close to settling the integrality gap (after necessary preprocessing), which is narrowed down to either 7,8 or 9. The algorithm proceeds by first reducing to special tree instances, and then uses our best-possible algorithm to solve such instances. Our concept of tree instances is versatile and applies to natural variants of the capacitated k-center problem for which we also obtain improved algorithms. Finally, we give evidence to show that more powerful preprocessing could lead to better algorithms, by giving an approximation algorithm that beats the integrality gap for instances where all non-zero capacities are the same.

On quadratic programming with a ratio objective

Quadratic Programming (QP) is the well-studied problem of maximizing over {−1,1} values the quadratic form ∑i≠jaijxixj. QP captures many known combinatorial optimization problems, and assuming the Unique Games conjecture, Semidefinite Programming (SDP) techniques give optimal approximation algorithms. We extend this body of work by initiating the study of Quadratic Programming problems where the variables take values in the domain {−1,0,1}. The specific problem we study is

Non-Negative Sparse Regression and Column Subset Selection with L1 Error

\begin{aligned} \textsf{QP-Ratio} &: \mbox{\ \ } \max_{\{-1,0,1\}^n} \frac{\sum_{i \not = j} a_{ij} x_i x_j}{\sum x_i^2} \end{aligned}

Provable Bounds for Learning Some Deep Representations

This is a natural relative of several well studied problems (in fact Trevisan introduced a normalized variant as a stepping stone towards a spectral algorithm for Max Cut Gain). Quadratic ratio problems are good testbeds for both algorithms and complexity because the techniques used for quadratic problems for the {−1,1} and {0,1} domains do not seem to carry over to the {−1,0,1} domain. We give approximation algorithms and evidence for the hardness of approximating these problems. We consider an SDP relaxation obtained by adding constraints to the natural eigenvalue (or SDP) relaxation for this problem. Using this, we obtain an

Smoothed analysis of tensor decompositions

\tilde{O}(n^{1/3})