Abstract
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 2 (0, 1/2], asks to compute the cut having the least expansion while having at most fraction of the vertices on the smaller side of the cut. We present two algorithms. Our first algorithm gives an ˜ O( 1 p logn) approximation. The second algorithm finds a set with expansion ˜ O( 1 ( p dmaxr 1 logr + )) in ar‐uniform hypergraph with maximum degreedmax (where is the expansion of the optimal solution). Using these results, we also obtain algorithms for the Small Set Vertex Expansion problem: we get an ˜ O( 1 p logn) approximation algorithm and an algorithm that finds a set with vertex expansion O 1 p V logdmax + 1 V (where V is the vertex expansion of the optimal solution). For = 1/2, Hypergraph Small Set Expansion is equivalent to the hypergraph expansion problem. In this case, our approximation factor of O( p logn) for expansion in hypergraphs matches the corresponding approximation factor for expansion in graphs due to Arora, Rao, and Vazirani.
Highlights
The expansion of a hypergraph, a natural extension of the notion of expansion in graphs, is defined as follows.Definition 1.1 (Hypergraph Expansion)
Given a hypergraph H = (V, E) on n vertices, we say that an edge e ∈ E is cut by a set S if e ∩ S = ∅ and e ∩ S = ∅ (i. e., some vertices in e lie in S and some vertices lie outside of S)
Hypergraph expansion and related hypergraph partitioning problems are of immense practical importance, having applications in parallel and distributed computing [6], VLSI circuit design and computer architecture [13], scientific computing [10] and other areas
Summary
The expansion of a hypergraph, a natural extension of the notion of expansion in graphs, is defined as follows. There is a randomized polynomial-time approximation algorithm for the Small-Set Vertex Expansion problem that, given a graph G = (V, E) and parameters ε ∈ (0, 1) and δ ∈ (0, 1/2)), finds a set S ⊂ V of size at most (1 + ε)δ n such that φV (S) ≤ Oε log n δ −1 log δ −1 log log δ −1 · φGV,δ . There is a randomized polynomial-time algorithm for the Small-Set Vertex Expansion problem that, given a graph G = (V, E) of maximum degree dmax and parameters ε ∈ (0, 1) and δ ∈ (0, 1/2), finds a set S ⊂ V of size at most (1 + ε)δ n such that φV (S) ≤ Oε φGV,δ log dmax · δ −1 log δ −1 log log δ −1 + δ −1φGV,δ.
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