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80
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 Program-ming 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 Program-ming 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 prob-lem. These problems are notorious for the existence of huge gaps between the known algorith-mic results and NP-hardness results. Our algorithm is based on rounding semidefinite pro-grams from the Lasserre hierarchy, and the analysis uses bounds for low-rank 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 ap-proximation 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 lower-order 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 Label-Extended 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 quasi-polynomial 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 re-derive 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 Small-Set 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 Small-Set 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 Small-Set 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 k-wise 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 k-wise 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 k-wise independent distribution and, for some c> 0, a random predicate accepting ct2 / log t input vectors is non-trivially 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 edge-weighted 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 edge-weighted 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 polynomial-time 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.
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 parti-tioning problems, including multi-way partition, balanced separator, and maximum cut.
New Tools for Graph Coloring
"... How to color 3 colorable graphs with few colors is a problem of longstanding interest. The best polynomial-time 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 polynomial-time 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 distance-transitive property implies that the graph “looks” the same in all neighborhoods.
