The Lovasz Local Lemma due to Erdos and Lovasz is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events, each of which asserts that a random variable does not deviate much from its mean. As applications, we consider two classes of NP-hard integer programs: minimax and covering integer programs. A key technique, randomized rounding of linear relaxations, was developed by Raghavan and Thompson to derive good approximation algorithms for such problems. We use our extension of the Local Lemma to prove that randomized rounding produces, with non-zero probability, much better feasible solutions than known before, if the constraint matrices of these integer programs are column-sparse (e.g., routing using short paths, problems on hypergraphs with small dimension degree). This complements certain well-known results from discrepancy theory. We also generalize the method of pessimistic estimators due to Raghavan, to obtain constructive (algorithmic) versions of our results for covering integer programs.
A key corollary of our results is that for families of instances of CIPs, we get a good ( @math or @math ) integrality gap if @math grows at least as fast as @math . Bounds on the result of a greedy algorithm for CIPs relative to the optimal solution, are known @cite_29 @cite_7 . Our bound improves that of @cite_29 and is incomparable with @cite_7 ; for any given @math , @math , and the unit vector @math , our bound improves on @cite_7 if @math is more than a certain threshold. As it stands, randomized rounding produces such improved solutions for several CIPs only with a very low, sometimes exponentially small, probability. Thus, it does not imply a randomized algorithm, often. To this end, we generalize Raghavan's method of pessimistic estimators to derive an algorithmic (polynomial-time) version of our results for CIPs, in .
@cite_29 @cite_7
1818081266 2949393806 2125674105 2950890626 2007385254 2097646889 2952621790 2963986064 2000931246 2947175317 2001663593 2115831572 2604884452 179407972 2093259407 1556276033 2151239055 2008724760 1818659651 2950958145
Random instances of constraint satisfaction problems (CSPs) appear to be hard for all known algorithms when the number of constraints per variable lies in a certain interval. Contributing to the general understanding of the structure of the solution space of a CSP in the satisfiable regime, we formulate a set of technical conditions on a large family of random CSPs and prove bounds on three most interesting thresholds for the density of such an ensemble: namely, the satisfiability threshold, the threshold for clustering of the solution space, and the threshold for an appropriate reconstruction problem on the CSPs. The bounds become asymptoticlally tight as the number of degrees of freedom in each clause diverges. The families are general enough to include commonly studied problems such as random instances of Not-All-Equal SAT, k-XOR formulae, hypergraph 2-coloring, and graph k-coloring. An important new ingredient is a condition involving the Fourier expansion of clauses, which characterizes the class of ... We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with 2-variable constraints (2-CSP's). More concretely, we show for every 2-CSP instance I a rounding algorithm for r rounds of the Lasserre SDP hierarchy for I that obtains an integral solution that is at most worse than the relaxation's value (normalized to lie in [0,1]), as long as r > k ( ) ( ) ;, where k is the alphabet size of I, @math , and @math denotes the number of eigenvalues larger than @math in the normalized adjacency matrix of the constraint graph of @math . In the case that @math is a instance, the threshold @math is only a polynomial in @math , and is independent of the alphabet size. Also in this case, we can give a non-trivial bound on the number of rounds for instance. In particular our result yields an SDP-hierarchy based algorithm that matches the performance of the recent subexponential algorithm of Arora, Barak and Steurer (FOCS 2010) in the worst case, but runs faster on a natural family of instances, thus further restricting the set of possible hard instances for Khot's Unique Games Conjecture. Our algorithm actually requires less than the @math constraints specified by the @math level of the Lasserre hierarchy, and in some cases @math rounds of our program can be evaluated in time @math . The Lova´sz local lemma due to Erdods and Lova´sz (Infinite and Finite Sets, Colloq. Math. Soc. J. Bolyai 11, 1975, pp. 609-627) is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events, each of which asserts that a random variable does not deviate much from its mean. As applications, we consider two classes of NP-hard integer programs: minimax and covering integer programs. A key technique, randomized rounding of linear relaxations, was developed by Raghavan and Thompson (Combinatorica, 7 (1987), pp. 365-374) to derive good approximation algorithms for such problems. We use our extension of the local lemma to prove that randomized rounding produces, with nonzero probability, much better feasible solutions than known before, if the constraint matrices of these integer programs are column-sparse (e.g., routing using short paths, problems on hypergraphs with small dimension degree). This complements certain well-known results from discrepancy theory. We also generalize the method of pessimistic estimators due to Raghavan (J. Comput. System Sci., 37 (1988), pp. 130-143), to obtain constructive (algorithmic) versions of our results for covering integer programs. Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with @math variables and @math clauses, there is a value of @math beyond which the CSP will be unsatisfiable with high probability. Strong refutation is the problem of certifying that no variable assignment satisfies more than a constant fraction of clauses; this is the natural algorithmic problem in the unsatisfiable regime (when @math ). Intuitively, strong refutation should become easier as the clause density @math grows, because the contradictions introduced by the random clauses become more locally apparent. For CSPs such as @math -SAT and @math -XOR, there is a long-standing gap between the clause density at which efficient strong refutation algorithms are known, @math , and the clause density at which instances become unsatisfiable with high probability, @math . In this paper, we give spectral and sum-of-squares algorithms for strongly refuting random @math -XOR instances with clause density @math in time @math or in @math rounds of the sum-of-squares hierarchy, for any @math and any integer @math . Our algorithms provide a smooth transition between the clause density at which polynomial-time algorithms are known at @math , and brute-force refutation at the satisfiability threshold when @math . We also leverage our @math -XOR results to obtain strong refutation algorithms for SAT (or any other Boolean CSP) at similar clause densities. Our algorithms match the known sum-of-squares lower bounds due to Grigoriev and Schonebeck, up to logarithmic factors. Additionally, we extend our techniques to give new results for certifying upper bounds on the injective tensor norm of random tensors. We initiate a study of when the value of mathematical relaxations such as linear and semi-definite programs for constraint satisfaction problems (CSPs) is approximately preserved when restricting the instance to a sub-instance induced by a small random subsample of the variables. Let C be a family of CSPs such as 3SAT, Max-Cut, etc., and let Π be a mathematical program that is a relaxation for C, in the sense that for every instance P ∈ C, Π(P) is a number in [0, 1] upper bounding the maximum fraction of satisfiable constraints of P. Loosely speaking, we say that subsampling holds for C and Π if for every sufficiently dense instance P ∈ C and every e > 0, if we let P' be the instance obtained by restricting P to a sufficiently large constant number of variables, then Π(P') ∈ (1 ± e)Π(P). We say that weak subsampling holds if the above guarantee is replaced with Π(P') = 1 − θ(γ) whenever Π(P) = 1 − γ, where θ hides only absolute constants. We obtain both positive and negative results, showing that: 1. Subsampling holds for the BasicLP and BasicSDP programs. BasicSDP is a variant of the semi-definite program considered by Raghavendra (2008), who showed it gives an optimal approximation factor for every constraint-satisfaction problem under the unique games conjecture. BasicLP is the linear programming analog of BasicSDP. 2. For tighter versions of BasicSDP obtained by adding additional constraints from the Lasserre hierarchy, weak subsampling holds for CSPs of unique games type. 3. There are non-unique CSPs for which even weak subsampling fails for the above tighter semi-definite programs. Also there are unique CSPs for which (even weak) subsampling fails for the Sherali-Adams linear programming hierarchy. As a corollary of our weak subsampling for strong semi-definite programs, we obtain a polynomial-time algorithm to certify that random geometric graphs (of the type considered by Feige and Schechtman, 2002) of max-cut value 1 − γ have a cut value at most 1 − γ 10. More generally, our results give an approach to obtaining average-case algorithms for CSPs using semi-definite programming hierarchies. By the breakthrough work of Hastad [J ACM 48(4) (2001), 798–859], several constraint satisfaction problems are now known to have the following approximation resistance property: Satisfying more clauses than what picking a random assignment would achieve is NP-hard. This is the case for example for Max E3-Sat, Max E3-Lin, and Max E4-Set Splitting. A notable exception to this extreme hardness is constraint satisfaction over two variables (2-CSP); as a corollary of the celebrated Goemans-Williamson algorithm [J ACM 42(6) (1995), 1115–1145], we know that every Boolean 2-CSP has a nontrivial approximation algorithm whose performance ratio is better than that obtained by picking a random assignment to the variables. An intriguing question then is whether this is also the case for 2-CSPs over larger, non-Boolean domains. This question is still open, and is equivalent to whether the generalization of Max 2-SAT to domains of size d, can be approximated to a factor better than (1 − 1 d2). In an attempt to make progress towards this question, in this paper we prove, first, that a slight restriction of this problem, namely, a generalization of linear inequations with two variables per constraint, is not approximation resistant, and, second, that the Not-All-Equal Sat problem over domain size d with three variables per constraint, is approximation resistant, for every d ≥ 3. In the Boolean case, Not-All-Equal Sat with three variables per constraint is equivalent to Max 2-SAT and thus has a nontrivial approximation algorithm; for larger domain sizes, Max 2-SAT can be reduced to Not-All-Equal Sat with three variables per constraint. Our approximation algorithm implies that a wide class of 2-CSPs called regular 2-CSPs can all be approximated beyond their random assignment threshold. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004 Let @math be a nontrivial @math -ary predicate. Consider a random instance of the constraint satisfaction problem @math on @math variables with @math constraints, each being @math applied to @math randomly chosen literals. Provided the constraint density satisfies @math , such an instance is unsatisfiable with high probability. The problem is to efficiently find a proof of unsatisfiability. We show that whenever the predicate @math supports a @math - probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree @math (which runs in time @math ) refute a random instance of @math . In particular, the polynomial-time SOS algorithm requires @math constraints to refute random instances of CSP @math when @math supports a @math -wise uniform distribution on its satisfying assignments. Together with recent work of [LRS15], our result also implies that polynomial-size semidefinite programming relaxation for refutation requires at least @math constraints. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random CSPs. For every constraint predicate @math , they give a three-way hardness tradeoff between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent algorithmic results of [AOW15] and [RRS16], this full three-way tradeoff is , up to lower-order factors. We consider positive covering integer programs, which generalize set cover and which have attracted a long line of research developing (randomized) approximation algorithms. Srinivasan (2006) gave a rounding algorithm based on the FKG inequality for systems which are "column-sparse." This algorithm may return an integer solution in which the variables get assigned large (integral) values; Kolliopoulos & Young (2005) modified this algorithm to limit the solution size, at the cost of a worse approximation ratio. We develop a new rounding scheme based on the Partial Resampling variant of the Lovasz Local Lemma developed by Harris & Srinivasan (2013). This achieves an approximation ratio of 1 + ln([EQUATION]), where amin is the minimum covering constraint and Δ1 is the maximum e1-norm of any column of the covering matrix (whose entries are scaled to lie in [0, 1]); we also show nearly-matching inapproximability and integrality-gap lower bounds. Our approach improves asymptotically, in several different ways, over known results. First, it replaces Δ0, the maximum number of nonzeroes in any column (from the result of Srinivasan) by Δ1 which is always - and can be much - smaller than Δ0; this is the first such result in this context. Second, our algorithm automatically handles multi-criteria programs; we achieve improved approximation ratios compared to the algorithm of Srinivasan, and give, for the first time when the number of objective functions is large, polynomial-time algorithms with good multi-criteria approximations. We also significantly improve upon the upper-bounds of Kolliopoulos & Young when the integer variables are required to be within (1 + e) of some given upper-bounds, and show nearly-matching inapproximability. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 Reducing hidden bias in the data and ensuring fairness in algorithmic data analysis has recently received significant attention. We complement several recent papers in this line of research by introducing a general method to reduce bias in the data through random projections in a fair'' subspace. We apply this method to densest subgraph and @math -means. For densest subgraph, our approach based on fair projections allows to recover both theoretically and empirically an almost optimal, fair, dense subgraph hidden in the input data. We also show that, under the small set expansion hypothesis, approximating this problem beyond a factor of @math is NP-hard and we show a polynomial time algorithm with a matching approximation bound. We further apply our method to @math -means. In a previous paper, [NIPS 2017] showed that problems such as @math -means can be approximated up to a constant factor while ensuring that none of two protected class (e.g., gender, ethnicity) is disparately impacted. We show that fair projections generalize the concept of fairlet introduced by to any number of protected attributes and improve empirically the quality of the resulting clustering. We also present the first constant-factor approximation for an arbitrary number of protected attributes thus settling an open problem recently addressed in several works. par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 . For many random constraint satisfaction problems, by now there exist asymptotically tight estimates of the largest constraint density for which solutions exist. At the same time, for many of these problems, all known polynomial-time algorithms stop finding solutions at much smaller densities. For example, it is well-known that it is easy to color a random graph using twice as many colors as its chromatic number. Indeed, some of the simplest possible coloring algorithms achieve this goal. Given the simplicity of those algorithms, one would expect room for improvement. Yet, to date, no algorithm is known that uses (2 - epsiv)chi colors, in spite of efforts by numerous researchers over the years. In view of the remarkable resilience of this factor of 2 against every algorithm hurled at it, we find it natural to inquire into its origin. We do so by analyzing the evolution of the set of k-colorings of a random graph, viewed as a subset of 1,...,k n, as edges are added. We prove that the factor of 2 corresponds in a precise mathematical sense to a phase transition in the geometry of this set. Roughly speaking, we prove that the set of k-colorings looks like a giant ball for k ges 2chi, but like an error-correcting code for k les (2 - epsiv)chi. We also prove that an analogous phase transition occurs both in random k-SAT and in random hypergraph 2-coloring. And that for each of these three problems, the location of the transition corresponds to the point where all known polynomial-time algorithms fail. To prove our results we develop a general technique that allows us to establish rigorously much of the celebrated 1-step replica-symmetry-breaking hypothesis of statistical physics for random CSPs. We consider the problem of provably optimal exploration in reinforcement learning for finite horizon MDPs. We show that an optimistic modification to value iteration achieves a regret bound of @math where @math is the time horizon, @math the number of states, @math the number of actions and @math the number of time-steps. This result improves over the best previous known bound @math achieved by the UCRL2 algorithm of , 2010. The key significance of our new results is that when @math and @math , it leads to a regret of @math that matches the established lower bound of @math up to a logarithmic factor. Our analysis contains two key insights. We use careful application of concentration inequalities to the optimal value function as a whole, rather than to the transitions probabilities (to improve scaling in @math ), and we define Bernstein-based "exploration bonuses" that use the empirical variance of the estimated values at the next states (to improve scaling in @math ). We report on a novel technique called spatial coupling and its application in the analysis of random constraint satisfaction problems (CSP). Spatial coupling was invented as an engineering construction in the area of error correcting codes where it has resulted in efficient capacity-achieving codes for a wide range of channels. However, this technique is not limited to problems in communications, and can be applied in the much broader context of graphical models. We describe here a general methodology for applying spatial coupling to random constraint satisfaction problems and obtain lower bounds for their (rough) satisfiability threshold. The main idea is to construct a distribution of geometrically structured random K-SAT instances - namely the spatially coupled ensemble - which has the same (rough) satisfiability threshold, and is at the same time algorithmically easier to solve. Then by running well-known algorithms on the spatially coupled ensemble we obtain a lower bound on the (rough) satisfiability threshold of the original ensemble. The method is versatile because one can choose the CSP, there is a certain amount of freedom in the construction of the spatially coupled ensemble, and also in the choice of the algorithm. In this work we focus on random K-SAT but we have also checked that the method is successful for Coloring, NAE-SAT and XOR-SAT. We choose Unit Clause propagation for the algorithm which is analyzed over the spatially coupled instances. For K = 3, for instance, our lower bound is equal to 3.67 which is better than the current bounds in the literature. Similarly, for graph 3-colorability we get a bound of 2.22 which is also better than the current bounds in the literature. In a covering integer program (CIP), we seek an n-vector x of nonnegative integers, which minimizes cT ċ x, subject to Ax ≥ b, where all entries of A, b, c are nonnegative. In their most general form, CIPs include also multiplicity constraints of the type x ≤ d, i.e., arbitrarily large integers are not acceptable in the solution. The multiplicity constraints incur a dichotomy with respect to approximation between (0,1)-CIPs whose matrix A contains only zeros and ones and the general case. Let m denote the number of rows of A. The well known O(log m) cost approximation with respect to the optimum of the linear relaxation is valid for general CIPs, but multiplicity constraints can be dealt with effectively only in the (0,1) case. In the general case, existing algorithms that match the integrality gap for the cost objective violate the multiplicity constraints by a multiplicative O(log m) factor. We make progress by defining column-restricted CIPs, a strict superclass of (0,1)-CIPs, and showing how to find for them integral solutions of cost O(log m) times the LP optimum while violating the multiplicity constraints by a multiplicative O(1) factor. In the conclusion of his monumental paper on optimal inapproximability results, Hastad [13] suggested that Fourier analysis of Dictator (Long Code) Tests may not be universally applicable in the study of CSPs. His main open question was to determine if the technique could resolve the approximability of satisfiable 3-bit constraint satisfaction problems. In particular, he asked if the "Not Two" (NTW) predicate is non-approximable beyond the random assignment threshold of 5 8 on satisfiable instances. Around the same time, Zwick [30] showed that all satisfiable 3-CSPs are 5 8-approximable and conjectured that the 5 8 is optimal. In this work we show that Fourier analysis techniques can produce a Dictator Test based on NTW with completeness 1 and soundness 5 8. Our test's analysis uses the Bonami-Gross-Beckner hypercontractive inequality. We also show a soundness lower bound of 5 8 for all 3-query Dictator Tests with perfect completeness. This lower bound for Property Testing is proved in part via a semidefinite programming algorithm of Zwick [30]. Our work precisely determines the 3-query "Dictatorship Testing gap". Although this represents progress on Zwick's conjecture, current PCP "outer verifier" technology is insufficient to convert our Dictator Test into an NP-hardness-of-approximation result. Given an arc-weighted directed graph G = (V, A, spl lscr ) and a pair of nodes s, t, we seek to find an s-t walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the orienteering problem. Our main result is a quasi-polynomial time algorithm that yields an O(log OPT) approximation for this problem when f is a given submodular set function. We then extend it to the case when a node v is counted as visited only if the walk reaches v in its time window [R(v), D(v)]. We apply the algorithm to obtain several new results. First, we obtain an O(log OPT) approximation for a generalization of the orienteering problem in which the profit for visiting each node may vary arbitrarily with time. This captures the time window problem considered earlier for which, even in undirected graphs, the best approximation ratio known [Bansal, (2004)] is O(log sup 2 OPT). The second application is an O(log sup 2 k) approximation for the k-TSP problem in directed graphs (satisfying asymmetric triangle inequality). This is the first non-trivial approximation algorithm for this problem. The third application is an O(log sup 2 k) approximation (in quasi-poly time) for the group Steiner problem in undirected graphs where k is the number of groups. This improves earlier ratios (Garg, ) by a logarithmic factor and almost matches the inapproximability threshold on trees (Halperin and Krauthgamer, 2003). This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than spl Omega (log sup 1- spl epsi OPT), even in undirected graphs. Even though our algorithm runs in quasi-poly time, we believe that the implications for the approximability of several basic optimization problems are interesting. In this paper we study a fundamental open problem in the area of probabilistic checkable proofs: What is the smallest s such that NP ⊆ naPCP1,s[O(log n),3]? In the language of hardness of approximation, this problem is equivalent to determining the smallest s such that getting an s-approximation for satisfiable 3-bit constraint satisfaction problems ("3-CSPs") is NP-hard. The previous best upper bound and lower bound for s are 20 27+µ by Khot and Saket [KS06], and 5 8 (assuming NP subseteq BPP) by Zwick [Zwi98]. In this paper we close the gap assuming Khot's d-to-1 Conjecture. Formally, we prove that if Khot's d-to-1 Conjecture holds for any finite constant integer d, then NP naPCP1,5 8+ µ[O(log n),3] for any constant µ > 0. Our conditional result also solves Hastad's open question [Has01] on determining the inapproximability of satisfiable Max-NTW ("Not Two") instances and confirms Zwick's conjecture [Zwi98] that the 5 8-approximation algorithm for satisfiable 3-CSPs is optimal. We prove a structure theorem for the feasible solutions of the Arora-Rao-Vazirani SDP relaxation on low threshold rank graphs and on small-set expanders. We show that if G is a graph of bounded threshold rank or a small-set expander, then an optimal solution of the Arora-Rao-Vazirani relaxation (or of any stronger version of it) can be almost entirely covered by a small number of balls of bounded radius. Then, we show that, if k is the number of balls, a solution of this form can be rounded with an approximation factor of O(sqrt log k ) in the case of the Arora-Rao-Vazirani relaxation, and with a constant-factor approximation in the case of the k-th round of the Sherali-Adams hierarchy starting at the Arora-Rao-Vazirani relaxation. The structure theorem and the rounding scheme combine to prove the following result, where G=(V,E) is a graph of expansion (G), is the k-th smallest eigenvalue of the normalized Laplacian of G, and (G) = disjoint S_1,...,S_k 1 > log^ 2.5 k phi(G) or k (G) >> log k sqrt log n loglog n (G), then the Arora-Rao-Vazirani relaxation can be rounded in polynomial time with an approximation ratio O(sqrt log k ). Stronger approximation guarantees are achievable in time exponential in k via relaxations in the Lasserre hierarchy. Guruswami and Sinop [GS13] and Arora, Ge and Sinop [AGS13] prove that 1+eps approximation is achievable in time 2^ O(k) poly(n) if either > (G) poly(eps), or if SSE_ n k > sqrt log k log n (G) poly(eps), where SSE_s is the minimal expansion of sets of size at most s. We consider the problem of selecting the best subset of exactly @math columns from an @math matrix @math . We present and analyze a novel two-stage algorithm that runs in @math time and returns as output an @math matrix @math consisting of exactly @math columns of @math . In the first (randomized) stage, the algorithm randomly selects @math columns according to a judiciously-chosen probability distribution that depends on information in the top- @math right singular subspace of @math . In the second (deterministic) stage, the algorithm applies a deterministic column-selection procedure to select and return exactly @math columns from the set of columns selected in the first stage. Let @math be the @math matrix containing those @math columns, let @math denote the projection matrix onto the span of those columns, and let @math denote the best rank- @math approximation to the matrix @math . Then, we prove that, with probability at least 0.8, @math This Frobenius norm bound is only a factor of @math worse than the best previously existing existential result and is roughly @math better than the best previous algorithmic result for the Frobenius norm version of this Column Subset Selection Problem (CSSP). We also prove that, with probability at least 0.8, @math This spectral norm bound is not directly comparable to the best previously existing bounds for the spectral norm version of this CSSP. Our bound depends on @math , whereas previous results depend on @math ; if these two quantities are comparable, then our bound is asymptotically worse by a @math factor.