Results 1  10
of
15
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,
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.
Finding Small Sparse Cuts by Random Walk
"... Abstract. We study the problem of finding a small sparse cut in an undirected graph. Given an undirected graph G = (V,E) and a parameter k ≤ E, the small sparsest cut problem is to find a set S ⊆ V with minimum conductance among all sets with volume at most k. Using ideas developed in local graph ..."
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Cited by 5 (0 self)
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Abstract. We study the problem of finding a small sparse cut in an undirected graph. Given an undirected graph G = (V,E) and a parameter k ≤ E, the small sparsest cut problem is to find a set S ⊆ V with minimum conductance among all sets with volume at most k. Using ideas developed in local graph partitioning algorithms, we obtain the following bicriteria approximation algorithms for the small sparsest cut problem: – If there is a set U ⊆ V with conductance φ and vol(U) ≤ k, then there isapolynomial timealgorithm tofindaset S with conductance O ( √ φ/ǫ) and vol(S) ≤ k 1+ǫ for any ǫ> 1/k. – If there is a set U ⊆ V with conductance φ and vol(U) ≤ k, then there isapolynomial timealgorithm tofindaset S with conductance O ( √ φlogk/ǫ) and vol(S) ≤ (1+ǫ)k for any ǫ> 2logk/k. These algorithms can be implemented locally using truncated random walk, with running time almost linear to k. 1
Hypergraph Markov Operators, Eigenvalues and Approximation Algorithms
, 2014
"... The celebrated Cheeger’s Inequality [AM85, Alo86] establishes a bound on the expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on studying the eigenvalues and eigenvectors of the adjacency matrix (and other related matrices) of graphs. It ha ..."
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Cited by 4 (2 self)
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The celebrated Cheeger’s Inequality [AM85, Alo86] establishes a bound on the expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on studying the eigenvalues and eigenvectors of the adjacency matrix (and other related matrices) of graphs. It has remained open to define a suitable spectral model for hypergraphs whose spectra can be used to estimate various combinatorial properties of the hypergraph. In this paper we introduce a new hypergraph Laplacian operator (generalizing the Laplacian matrix of graphs) and study its spectra. We prove a Cheegertype inequality for hypergraphs, relating the second smallest eigenvalue of this operator to the expansion of the hypergraph. We bound other hypergraph expansion parameters via higher eigenvalues of this operator. We give bounds on the diameter of the hypergraph as a function of the second smallest eigenvalue of the Laplacian operator. The Markov process underlying the Laplacian operator can be viewed as a dispersion process on the vertices of the hypergraph that can be used to model rumour spreading in networks, brownian motion, etc., and might be of independent interest. We bound the Mixingtime of this process as a function of the second smallest eigenvalue of the Laplacian operator. All these results are generalizations of the corresponding results for
Approximation algorithms for hypergraph small set expansion and small set vertex expansion
 CoRR
"... The expansion of a hypergraph, a natural extension of the notion of expansion in graphs, is defined as the minimum over all cuts in the hypergraph of the ratio of the number of the hyperedges cut to the size of the smaller side of the cut. We study the Hypergraph Small Set Expansion problem, which ..."
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Cited by 3 (1 self)
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The expansion of a hypergraph, a natural extension of the notion of expansion in graphs, is defined as the minimum over all cuts in the hypergraph of the ratio of the number of the hyperedges cut to the size of the smaller side of the cut. We study the Hypergraph Small Set Expansion problem, which, for a parameter δ ∈ (0, 1/2], asks to compute the cut having the least expansion while having at most δ fraction of the vertices on the smaller side of the cut. We present two algorithms. Our first algorithm gives an Õ(δ−1 log n) approximation. The second algorithm finds a set with expansion Õ(δ−1( dmaxr−1 log r φ ∗ + φ∗)) in a r–uniform hypergraph with maximum degree dmax (where φ ∗ is the expansion of the optimal solution). Using these results, we also obtain algorithms for the Small Set Vertex Expansion problem: we get an Õ(δ−1 log n) approximation algorithm and an algorithm that finds a set with vertex expansion O δ−1 φV log dmax + δ
Approximation algorithms for semirandom graph partitioning problems
 CORR
, 2012
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