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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 15 (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,
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
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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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
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
NonUniform Graph Partitioning
"... We consider the problem of NONUNIFORM GRAPH PARTITIONING, where the input is an edgeweighted undirected graph G = (V, E) and k capacities n1,..., nk, and the goal is to find a partition {S1, S2,..., Sk} of V satisfying Sj  ≤ nj for all 1 ≤ j ≤ k, that minimizes the total weight of edges crossi ..."
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We consider the problem of NONUNIFORM GRAPH PARTITIONING, where the input is an edgeweighted undirected graph G = (V, E) and k capacities n1,..., nk, and the goal is to find a partition {S1, S2,..., Sk} of V satisfying Sj  ≤ nj for all 1 ≤ j ≤ k, that minimizes the total weight of edges crossing between different parts. This natural graph partitioning problem arises in practical scenarios, and generalizes wellstudied balanced partitioning problems such as MINIMUM BISECTION, MINIMUM BALANCED CUT, and MINIMUM kPARTITIONING. Unlike these problems, NONUNIFORM GRAPH PARTITIONING seems to be resistant to many of the known partitioning techniques, such as spreading metrics, recursive partitioning, and Räcke’s tree decomposition, because k can be a function of n and the capacities could be of different magnitudes. We present a bicriteria approximation algorithm for NONUNIFORM GRAPH PARTITIONING that approximates the objective within O(log n) factor while deviating from the required capacities by at most a constant factor. Our approach is to apply stoppingtime based concentration results to a simple randomized rounding of a configuration LP. These concentration bounds are needed as the commonly used techniques of bounded differences and bounded conditioned variances do not suffice. 1
NEW ROUNDING TECHNIQUES FOR THE DESIGN AND ANALYSIS OF APPROXIMATION ALGORITHMS
, 2014
"... We study two of the most central classical optimization problems, namely the Traveling Salesman problems and Graph Partitioning problems and develop new approximation algorithms for them. We introduce several new techniques for rounding a fractional solution of a continuous relaxation of these probl ..."
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We study two of the most central classical optimization problems, namely the Traveling Salesman problems and Graph Partitioning problems and develop new approximation algorithms for them. We introduce several new techniques for rounding a fractional solution of a continuous relaxation of these problems into near optimal integral solutions. The two most notable of those are the maximum entropy rounding by sampling method and a novel use of higher eigenvectors of graphs.