Hypergraph Samplers: Typical and Worst Case Behavior
Vedat Levi Alev, Uriya A. First
TL;DR
This work analyzes how sampling seeds from a $k$-uniform hypergraph impacts error reduction in randomized decision algorithms. It introduces and leverages the notions of $(bcepsilon,p)$-samplers and $(bcepsilon,p)$-confiners to compare hypergraph-based seed selection with IID seeds, and then derives both typical-case and worst-case results. The authors show that for most density-$p$ subsets, the hypergraph sampler behaves almost identically to the ideal complete hypergraph, with strong concentration bounds, while in worst-case scenarios, confinement cannot be pushed below $p^k$ by much; they further quantify this gap and connect to dispersers and vertex expanders. The work culminates in precise, fine-grained bounds via the function $f_{k,r}(p)$ and a detailed analytic framework, shedding light on when sparse hypergraph samplers are near-optimal and where intrinsic limits arise. Overall, the results have implications for designing pseudorandom seed strategies with limited randomness and for understanding the trade-offs in dispersers and vertex-expanders in error-reduction contexts.
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 $p^k + ε$, then $H$ cannot have too few edges, i.e., $|E| = Ω(n k^{-1} ε^{-1})$. Thus, the number of random bits needed for reducing the error from $p$ to $p^k + ε$ cannot be reduced below $\lg n+\lg(ε^{-1})-\lg k+O(1)$. 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.