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23
Optimal ColumnBased LowRank Matrix Reconstruction
"... We prove that for any realvalued matrix X ∈ R m×n, and positive integers r � k, there is a subset of r columns of √ X such that projecting X onto their span gives a r+1approximation to best rankk approximation of X r−k+1 in Frobenius norm. We show that the tradeoff we achieve between the number ..."
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We prove that for any realvalued matrix X ∈ R m×n, and positive integers r � k, there is a subset of r columns of √ X such that projecting X onto their span gives a r+1approximation to best rankk approximation of X r−k+1 in Frobenius norm. We show that the tradeoff we achieve between the number of columns and the approximation ratio is optimal up to lower order terms. Furthermore, there is a deterministic algorithm to find such a subset of columns that runs in O(rnm ω log m) arithmetic operations where ω is the exponent of matrix multiplication. We also give a faster randomized algorithm that runs in O(rnm 2) arithmetic operations. 1
Approximation Limits of Linear Programs (Beyond Hierarchies)
, 2013
"... We develop a framework for proving approximation limits of polynomialsize linear programs from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any linear program as opposed to only programs generate ..."
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Cited by 19 (7 self)
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We develop a framework for proving approximation limits of polynomialsize linear programs from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any linear program as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n1/2−ɛ)approximations for CLIQUE require linear programs of size 2nΩ(ɛ). This lower bound applies to linear programs using a certain encoding of CLIQUE as a linear optimization problem. Moreover, we establish a similar result for approximations of semidefinite programs by linear programs. Our main technical ingredient is a quantitative improvement of Razborov’s rectangle corruption lemma (1992) for the high error regime, which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix.
Bypassing UGC from some Optimal Geometric Inapproximability Results
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 177
, 2010
"... The Unique Games conjecture (UGC) has emerged in recent years as the starting point for several optimal inapproximability results. While for none of these results a reverse reduction to Unique Games is known, the assumption of bijective projections in the Label Cover instance seems critical in these ..."
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Cited by 8 (2 self)
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The Unique Games conjecture (UGC) has emerged in recent years as the starting point for several optimal inapproximability results. While for none of these results a reverse reduction to Unique Games is known, the assumption of bijective projections in the Label Cover instance seems critical in these proofs. In this work we bypass the UGC assumption in inapproximability results for two geometric problems, obtaining a tight NPhardness result in each case. The first problem known as the Lp Subspace Approximation is a generalization of the classic least squares regression problem. Here, the input consists of a set of points S = {a1,..., am} ⊆ R n and a parameter k (possibly depending on n). The goal is to find a subspace H of R n of dimension k that minimizes the sum of the p th powers of the distances to the points. For p = 2, k = n − 1, this reduces to the least squares regression problem, while for p = ∞, k = 0 it reduces to the problem of finding a ball of minimum radius enclosing all the points. We show that for any fixed p (2 < p < ∞) it is NPhard to approximate this problem to within a factor of γp − ɛ for constant ɛ> 0, where γp is the pth moment of a standard Gaussian variable. This matches the factor γp approximation algorithm obtained by Deshpande, Tulsiani and Vishnoi
Constant Factor Lasserre Integrality Gaps for Graph Partitioning Problems
"... Partitioning the vertices of a graph into two roughly equal parts while minimizing the number of edges crossing the cut is a fundamental problem (called Balanced Separator) that arises in many settings. For this problem, and variants such as the Uniform Sparsest Cut problem where the goal is to mini ..."
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Partitioning the vertices of a graph into two roughly equal parts while minimizing the number of edges crossing the cut is a fundamental problem (called Balanced Separator) that arises in many settings. For this problem, and variants such as the Uniform Sparsest Cut problem where the goal is to minimize the fraction of pairs on opposite sides of the cut that are connected by an edge, there are large gaps between the known approximation algorithms and nonapproximability results. While no constant factor approximation algorithms are known, even APXhardness is not known either. In this work we prove that for balanced separator and uniform sparsest cut, semidefinite programs from the Lasserre hierarchy (which are the most powerful relaxations studied in the literature) have an integrality gap bounded away from 1, even for Ω(n) levels of the hierarchy. This complements recent algorithmic results in [GS11] which used the Lasserre hierarchy to give an approximation scheme for these problems (with runtime depending on the spectrum of the graph). Along the way, we make an observation that simplifies the task of lifting “polynomial constraints ” (such as the global balance constraint in balanced separator) to higher levels of the Lasserre hierarchy. We also obtain a similar result for Max Cut and prove that even linear number of levels of the Lasserre hierarchy have an integrality gap exceeding 18/17 − o(1), though in this case there are known NPhardness results with this gap.
How to Sell Hyperedges: The Hypermatching Assignment Problem
, 2013
"... We are given a set of clients with budget constraints and a set of indivisible items. Each client is willing to buy one or more bundles of (at most) k items each (bundles can be seen as hyperedges in a khypergraph). If client i gets a bundle e, she pays bi,e and yields a net profit wi,e. The Hyperm ..."
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Cited by 3 (2 self)
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We are given a set of clients with budget constraints and a set of indivisible items. Each client is willing to buy one or more bundles of (at most) k items each (bundles can be seen as hyperedges in a khypergraph). If client i gets a bundle e, she pays bi,e and yields a net profit wi,e. The Hypermatching Assignment Problem (HAP) is to assign a set of pairwise disjoint bundles to clients so as to maximize the total profit while respecting the budgets. This problem has various applications in production planning and budgetconstrained auctions and generalizes wellstudied problems in combinatorial optimization: for example the weighted (unweighted) khypergraph matching problem is the special case of HAP with one client having unbounded budget and general (unit) profits; the Generalized Assignment Problem (GAP) is the special case of HAP with k = 1. Let ε> 0 denote an arbitrarily small constant. In this paper we obtain the following main results: • We give a randomized (k + 1 + ) approximation algorithm for HAP, which is based on rounding the 1round Lasserre strengthening of a novel LP. This is one of a few approximation results based on Lasserre hierarchies and our approach might be of independent interest. We remark that for weighted khypergraph matching no LP nor SDP relaxation is known to have integrality gap better than k − 1 + 1/k for general k [Chan and Lau, SODA’10]. • For the relevant special case that one wants to maximize the total revenue (i.e., bi,e = wi,e), we present a local search based (k + O( k))/2 approximation algorithm for k = O(1). This almost matches the best known (k + 1 + )/2 approximation ratio by Berman [SWAT’00] for
The lasserre hierarchy in almost diagonal form
 CoRR
"... The Lasserre hierarchy is a systematic procedure for constructing a sequence of increasingly tight relaxations that capture the convex formulations used in the best available approximation algorithms for a wide variety of optimization problems. Despite the increasing interest, there are very few t ..."
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Cited by 2 (1 self)
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The Lasserre hierarchy is a systematic procedure for constructing a sequence of increasingly tight relaxations that capture the convex formulations used in the best available approximation algorithms for a wide variety of optimization problems. Despite the increasing interest, there are very few techniques for analyzing Lasserre integrality gaps. Satisfying the positive semidefinite requirement is one of the major hurdles to constructing Lasserre gap examples. We present a novel characterization of the Lasserre hierarchy based on moment matrices that differ from diagonal ones by matrices of rank one (almost diagonal form). We provide a modular recipe to obtain positive semidefinite feasibility conditions by iteratively diagonalizing rank one matrices. Using this, we prove strong lower bounds on integrality gaps of Lasserre hierarchy for two basic capacitated covering problems. For the minknapsack problem, we show that the integrality gap remains arbitrarily large even at level n − 1 of Lasserre hierarchy. For the minsum of tardy jobs scheduling problem, we show that the integrality gap is unbounded at level Ω( n) (even when the objective function is integrated as a constraint). These bounds are interesting on their own, since both problems admit FPTAS. 1
Optimization over Polynomials: Selected Topics
"... Minimizing a polynomial function over a region defined by polynomial inequalities models broad classes of hard problems from combinatorics, geometry and optimization. New algorithmic approaches have emerged recently for computing the global minimum, by combining tools from real algebra (sums of s ..."
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Minimizing a polynomial function over a region defined by polynomial inequalities models broad classes of hard problems from combinatorics, geometry and optimization. New algorithmic approaches have emerged recently for computing the global minimum, by combining tools from real algebra (sums of squares of polynomials) and functional analysis (moments of measures) with semidefinite optimization. Sums of squares are used to certify positive polynomials, combining an old idea of Hilbert with the recent algorithmic insight that they can be checked efficiently with semidefinite optimization. The dual approach revisits the classical moment problem and leads to algorithmic methods for checking optimality of semidefinite relaxations and extracting global minimizers. We review some selected features of this general methodology, illustrate how it applies to some combinatorial graph problems, and discuss links with other relaxation methods.