Hypergraph Samplers: Typical and Worst Case Behavior
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910004838137856 |
|---|---|
| author | Alev, Vedat Levi First, Uriya A. |
| author_facet | Alev, Vedat Levi First, Uriya A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20039 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hypergraph Samplers: Typical and Worst Case Behavior Alev, Vedat Levi First, Uriya A. Data Structures and Algorithms Computational Complexity Combinatorics 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. |
| title | Hypergraph Samplers: Typical and Worst Case Behavior |
| topic | Data Structures and Algorithms Computational Complexity Combinatorics |
| url | https://arxiv.org/abs/2601.20039 |