Aravindan Vijayaraghavan

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.

Approximation algorithms for semi-random partitioning problems

In this paper, we propose and study a new semi-random model for graph partitioning problems. We believe that it captures many properties of real-world instances. The model is more flexible than the semi-random model of Feige and Kilian and planted random model of Bui, Chaudhuri, Leighton and Sipser. We develop a general framework for solving semi-random instances and apply it to several problems of interest. We present constant factor bi-criteria approximation algorithms for semi-random instances of the Balanced Cut, Multicut, Min Uncut, Sparsest Cut and Small Set Expansion problems. We also show how to almost recover the optimal solution if the instance satisfies an additional expanding condition. Our algorithms work in a wider range of parameters than most algorithms for previously studied random and semi-random models. Additionally, we study a new planted algebraic expander model and develop constant factor bi-criteria approximation algorithms for graph partitioning problems in this model.

Approximation Algorithms and Hardness of the k -Route Cut Problem

We study the <i>k</i>-route cut problem: given an undirected edge-weighted graph <i>G</i> = (<i>V</i>, <i>E</i>), a collection {(<i>s</i>₁, <i>t</i>₁), (<i>s</i>₂, <i>t</i>₂), …, (<i>s_r</i>, <i>t_r</i>)} of source-sink pairs, and an integer connectivity requirement <i>k</i>, the goal is to find a minimum-weight subset <i>E</i>′ of edges to remove, such that the connectivity of every pair (<i>s_i</i>, <i>t_i</i>) falls below <i>k</i>. Specifically, in the edge-connectivity version, EC-kRC, the requirement is that there are at most (<i>k</i> − 1) edge-disjoint paths connecting <i>s_i</i> to <i>t_i</i> in <i>G</i>∖<i>E</i>′, while in the vertex-connectivity version, VC-kRC, the same requirement is for vertex-disjoint paths. Prior to our work, poly-logarithmic approximation algorithms have been known for the special case where <i>k</i> ⩽ 3, but no non-trivial approximation algorithms were known for any value <i>k</i> > 3, except in the single-source setting. We show an <i>O</i>(<i>k</i>log ^3/2<i>r</i>)-approximation algorithm for EC-kRC with uniform edge weights, and several polylogarithmic bi-criteria approximation algorithms for EC-kRC and VC-kRC, where the connectivity requirement <i>k</i> is violated by a constant factor. We complement these upper bounds by proving that VC-kRC is hard to approximate to within a factor of <i>k</i>^ε for some fixed ε > 0. We then turn to study a simpler version of VC-kRC, where only one source-sink pair is present. We give a simple bi-criteria approximation algorithm for this case, and show evidence that even this restricted version of the problem may be hard to approximate. For example, we prove that the single source-sink pair version of VC-kRC has no constant-factor approximation, assuming Feige’s Random κ-AND assumption.

Constant factor approximation for balanced cut in the PIE model

We propose and study a new semi-random semi-adversarial model for Balanced Cut, a planted model with permutation invariant random edges (PIE). Our model is much more general than planted models considered previously. Consider a set of vertices V partitioned into two clusters L and R of equal size. Let G be an arbitrary graph on V with no edges between L and R. Let Erandom be a set of edges sampled from an arbitrary permutation-invariant distribution (a distribution that is invariant under permutation of vertices in L and in R). Then we say that G+Erandom is a graph with permutation-invariant random edges. We present an approximation algorithm for the Balanced Cut problem that finds a balanced cut of cost O(|Erandom|)+n polylog(n) in this model. In the regime when |Erandom| = Ω(n polylog(n)), this is a constant factor approximation with respect to the cost of the planted cut.

Sorting noisy data with partial information

In this paper, we propose two semi-random models for the Minimum Feedback Arc Set Problem and present approximation algorithms for them. In the first model, which we call the Random Edge Flipping model, an instance is generated as follows. We start with an arbitrary acyclic directed graph and then randomly flip its edges (the adversary may later un-flip some of them). In the second model, which we call the Random Backward Edge model, again we start with an arbitrary acyclic graph but now add new random backward edges (the adversary may delete some of them). For the first model, we give an approximation algorithm that finds a solution of cost (1+ δ) OPT + n polylog n, where OPT is the cost of the optimal solution. For the second model, we give an approximation algorithm that finds a solution of cost O(planted) + n polylog n, where planted is the cost of the planted solution. Additionally, we present an approximation algorithm for semi-random instances of Minimum Directed Balanced Cut.

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

Detecting high log-densities: an O ( n ¼ ) approximation for densest k -subgraph

\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}

Uniqueness of Tensor Decompositions with Applications to Polynomial Identifiability.

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