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81
Lasserre hierarchy, higher eigenvalues, and approximation schemes for graph partitioning and quadratic integer programming with PSD objectives
 In Proceedings of 52nd Annual Symposium on Foundations of Computer Science (FOCS
, 2011
"... We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and ..."
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Cited by 25 (3 self)
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We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and Small Set expansion, as well as the Unique Games problem. These problems are notorious for the existence of huge gaps between the known algorithmic results and NPhardness results. Our algorithm is based on rounding semidefinite programs from the Lasserre hierarchy, and the analysis uses bounds for lowrank approximations of a matrix in Frobenius norm using columns of the matrix. For all the above graph problems, we give an algorithm running in time nO(r/ε 2) with approximation ratio 1+εmin{1,λr} , where λr is the r’th smallest eigenvalue of the normalized graph Laplacian L. In the case of graph bisection and small set expansion, the number of vertices in the cut is within lowerorder terms of the stipulated bound. Our results imply (1 + O(ε)) factor approximation in time nO(r
SPECTRAL ALGORITHMS FOR UNIQUE Games
"... We give a new algorithm for Unique Games which is based on purely spectral techniques, in contrast to previous work in the area, which relies heavily on semidefinite programming (SDP). Given a highly satisfiable instance of Unique Games, our algorithm is able to recover a good assignment. The appro ..."
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Cited by 15 (1 self)
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We give a new algorithm for Unique Games which is based on purely spectral techniques, in contrast to previous work in the area, which relies heavily on semidefinite programming (SDP). Given a highly satisfiable instance of Unique Games, our algorithm is able to recover a good assignment. The approximation guarantee depends only on the completeness of the game, and not on the alphabet size, while the running time depends on spectral properties of the LabelExtended graph associated with the instance of Unique Games. We further show that on input the integrality gap instance of Khot and Vishnoi, our algorithm runs in quasipolynomial time and decides that the instance if highly unsatisfiable. Notably, when run on this instance, the standard SDP relaxation of Unique Games fails. As a special case, we also rederive a polynomial time algorithm for Unique Games on expander constraint graphs. The main ingredient of our algorithm is a technique to effectively use the full spectrum of the underlying graph instead of just the second eigenvalue, which is of independent interest. The question of how to take advantage of the full spectrum of a graph in the design of algorithms has been often studied, but no significant progress was made prior to this work.
Reductions between Expansion Problems
, 2010
"... The SmallSet Expansion Hypothesis (Raghavendra, Steurer, STOC 2010) is a natural hardness assumption concerning the problem of approximating the edge expansion of small sets in graphs. This hardness assumption is closely connected to the Unique Games Conjecture (Khot, STOC 2002). In particular, the ..."
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Cited by 14 (1 self)
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The SmallSet Expansion Hypothesis (Raghavendra, Steurer, STOC 2010) is a natural hardness assumption concerning the problem of approximating the edge expansion of small sets in graphs. This hardness assumption is closely connected to the Unique Games Conjecture (Khot, STOC 2002). In particular, the SmallSet Expansion Hypothesis implies the Unique Games Conjecture (Raghavendra,
Randomly supported independence and resistance
 In 38th Annual ACM Symposium on Theory of Computation
, 2009
"... Abstract. We prove that for any positive integers q and k, there is a constant cq,k such that a uniformly random set of cq,knk log n vectors in [q] n with high probability supports a balanced kwise independent distribution. In the case of k ≤ 2 a more elaborate argument gives the stronger bound cq, ..."
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Cited by 14 (4 self)
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Abstract. We prove that for any positive integers q and k, there is a constant cq,k such that a uniformly random set of cq,knk log n vectors in [q] n with high probability supports a balanced kwise independent distribution. In the case of k ≤ 2 a more elaborate argument gives the stronger bound cq,knk. Using a recent result by Austrin and Mossel this shows that a predicate on t bits, chosen at random among predicates accepting cq,2t2 input vectors, is, assuming the Unique Games Conjecture, likely to be approximation resistant. These results are close to tight: we show that there are other constants, c ′ q,k, such that a randomly selected set of cardinality c ′ q,knk points is unlikely to support a balanced kwise independent distribution and, for some c> 0, a random predicate accepting ct2 / log t input vectors is nontrivially approximable with high probability. In a different application of the result of Austrin and Mossel we prove that, again assuming the Unique Games Conjecture, any predicate on t Boolean inputs accepting at least (32/33) · 2t inputs is approximation resistant. The results extend from balanced distributions to arbitrary product distributions.
Many Sparse Cuts via Higher Eigenvalues
, 2011
"... Cheeger’s fundamental inequality states that any edgeweighted graph has a vertex subset S such that its expansion (a.k.a. conductance) is bounded as follows: φ(S) def w(S, S̄) min{w(S), w(S̄)} 6 2 λ2, where w is the total edge weight of a subset or a cut and λ2 is the second smallest eigenvalue of ..."
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Cited by 12 (1 self)
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Cheeger’s fundamental inequality states that any edgeweighted graph has a vertex subset S such that its expansion (a.k.a. conductance) is bounded as follows: φ(S) def w(S, S̄) min{w(S), w(S̄)} 6 2 λ2, where w is the total edge weight of a subset or a cut and λ2 is the second smallest eigenvalue of the normalized Laplacian of the graph. Here we prove the following natural generalization: for any integer k ∈ [n], there exist ck disjoint subsets S1,..., Sck, such that max i φ(Si) 6 C λk log k where λi is the i th smallest eigenvalue of the normalized Laplacian and c < 1, C> 0 are suitable absolute constants. Our proof is via a polynomialtime algorithm to find such subsets, consisting of a spectral projection and a randomized rounding. As a consequence, we get the same upper bound for the small set expansion problem, namely for any k, there is a subset S whose weight is at most a O(1/k) fraction of the total weight and φ(S) 6 C λk log k. Both results are the best possible up to constant factors. The underlying algorithmic problem, namely finding k subsets such that the maximum expansion is minimized, besides extending sparse cuts to more than one subset, appears to be a natural clustering problem in its own right.
New Tools for Graph Coloring
"... How to color 3 colorable graphs with few colors is a problem of longstanding interest. The best polynomialtime algorithm uses n0.2072 colors. There are no indications that coloring using say O(log n) colors is hard. It has been suggested that lift and project based SDP relaxations could be used to ..."
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Cited by 10 (0 self)
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How to color 3 colorable graphs with few colors is a problem of longstanding interest. The best polynomialtime algorithm uses n0.2072 colors. There are no indications that coloring using say O(log n) colors is hard. It has been suggested that lift and project based SDP relaxations could be used to design algorithms that use nɛ colors for arbitrarily small ɛ> 0. We explore this possibility in this paper and find some cause for optimism. While the case of general graphs is till open, we can analyse the Lasserre relaxation for two interesting families of graphs. For graphs with low threshold rank (a class of graphs identified in the recent paper of Arora, Barak and Steurer on the unique games problem), Lasserre relaxations can be used to find an independent set of size Ω(n) (i.e., progress towards a coloring with O(log n) colors) in nO(D) time, where D is the threshold rank of the graph. This algorithm is inspired by recent work of Barak, Raghavendra, and Steurer on using Lasserre Hierarchy for unique games. The algorithm can also be used to show that known integrality gap instances for SDP relaxations like strict vector chromatic number cannot survive a few rounds of Lasserre lifting, which also seems reason for optimism. For distance transitive graphs of diameter ∆, we can show how to color them using O(log n) colors in n2O(∆) time. This family is interesting because the family of graphs of diameter O(1/ɛ) is easily seen to be complete for coloring with nɛ colors. The distancetransitive property implies that the graph “looks” the same in all neighborhoods.
Improved cheeger’s inequality: analysis of spectral partitioning algorithms through higher order spectral gap
 In 45th annual ACM symposium on Symposium on theory of computing, STOC ’13
, 2013
"... Let φ(G) be the minimum conductance of an undirected graph G, and let 0 = λ1 ≤ λ2 ≤... ≤ λn ≤ 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k ≥ 2, φ(G) = O(k) λ2√ λk and this performance guarantee is achieved by the spectral partitioning algorit ..."
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Cited by 10 (0 self)
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Let φ(G) be the minimum conductance of an undirected graph G, and let 0 = λ1 ≤ λ2 ≤... ≤ λn ≤ 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k ≥ 2, φ(G) = O(k) λ2√ λk and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger’s inequality, and the bound is optimal up to a constant factor for any k. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if λk is a constant for some constant k. This provides some theoretical justification to its empirical performance in image segmentation and clustering problems. We extend the analysis to spectral algorithms for other graph partitioning problems, including multiway partition, balanced separator, and maximum cut.
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