Hypergraph Samplers: Typical and Worst Case Behavior

TL;DR

This paper investigates the effectiveness of using hypergraph-based sampling for error reduction in randomized algorithms, revealing fundamental limits and showing that, in most cases, hypergraph sampling performs nearly as well as IID sampling.

Contribution

The paper establishes lower bounds on hypergraph size for error reduction and demonstrates that, under certain conditions, hypergraph sampling is nearly as effective as IID sampling for most algorithms.

Findings

Lower bounds on hypergraph edges for error reduction.

Hypergraph sampling nearly matches IID sampling in most cases.

Implications for dispersers and vertex-expanders.

Abstract

We study the utility and limitations of using $k$-uniform hypergraphs $H = ([n], E)$ ($n \ge \mathrm{poly}(k)$) in the context of error reduction for randomized algorithms for decision problems with one- or two-sided error. Our error reduction idea is sampling a uniformly random hyperedge of $H$, and repeating the algorithm $k$ times using the hyperedge vertices as seeds. This is a general paradigm, which captures every pseudorandom method generating $k$ seeds without repetition. We show two results which imply a gap between the typical and the worst-case behavior of using $H$ for error-reduction. First, in the context of one-sided error reduction, if using a random hyperedge of $H$ decreases the error probability from $p$ to $p^k + \epsilon$, then $H$ cannot have too few edges, i.e., $|E| = \Omega(n k^{-1} \epsilon^{-1})$. Thus, the number of random bits needed for reducing the error from $p$ to $p^k + \epsilon$ cannot be reduced below $\lg n+\lg(\epsilon^{-1})-\lg k+O(1)$. This is also true for hypergraphs of average uniformity $k$. Our result implies new lower bounds for dispersers and vertex-expanders. Second, if the vertex degrees are reasonably distributed, we show that in a $(1-o(1))$-fraction of the cases, choosing $k$ pseudorandom seeds using $H$ will reduce the error probability to at most $o(1)$ above the error probability of using $k$ IID seeds, for both algorithms with one- or two-sided error. Thus, despite our lower bound, for a $(1-o(1))$-fraction of randomized algorithms (and inputs) for decision problems, the advantage of using IID samples over samples obtained from a uniformly random edge of a reasonable hypergraph is negligible. A similar statement holds true for randomized algorithms with two-sided error.