Worst-Case and Average-Case Hardness of Hypercycle and Database Problems

TL;DR

The paper investigates the fine-grained complexity of hypergraph problems, focusing on worst-case hardness for hypercycle detection in $u$-uniform hypergraphs and average-case hardness for counting small subhypergraphs, with tight bounds and new algorithms. It develops a unified framework linking worst-case hardness to average-case hardness via low-degree polynomials and color-coding, and demonstrates worst-case-to-average-case reductions that extend to non-uniform hypergraphs and database queries. A core contribution is the pairing of weighted and unweighted hypercycle results: tight lower bounds for min-weight hypercycles, and near-tight upper/lower bounds for unweighted hypercycles, including fast matrix-multiplication–based algorithms for longer cycles. The reductions also yield average-case hardness for counting hypergraphs in Erdős-Rényi models and for self-join-free database queries, highlighting practical implications for query evaluation in real-world databases and CSP solving. Overall, the work advances the understanding of how cycle- and substructure-counting problems in hypergraphs behave under worst-case and average-case assumptions, with several open problems on longer cycles and color-coding counting techniques.

Abstract

In this paper we present tight lower-bounds and new upper-bounds for hypergraph and database problems. We give tight lower-bounds for finding minimum hypercycles. We give tight lower-bounds for a substantial regime of unweighted hypercycle. We also give a new faster algorithm for longer unweighted hypercycles. We give a worst-case to average-case reduction from detecting a subgraph of a hypergraph in the worst-case to counting subgraphs of hypergraphs in the average-case. We demonstrate two applications of this worst-case to average-case reduction, which result in average-case lower bounds for counting hypercycles in random hypergraphs and queries in average-case databases. Our tight upper and lower bounds for hypercycle detection in the worst-case have immediate implications for the average-case via our worst-case to average-case reductions.

Figures (7)

• Figure 1: The running time of minimum weight hyper-cycle in a weighted $u$-uniform hyper-graph. The exponent of the running time is indicated by the black line and the lower bound is matching, as indicated by the hatched red area. Note the running time is $O(n^u)$ for a $u$ length hyper-cycle and $O(n^{2u-1})$ for a $2u-1$ length hyper-cycle. Then for all hyper-cycles of length $k>2u-1$ the running time continues to be $O(n^{2u-1})$.
• Figure 2: The running time of unweighted hyper-cycle in a $u$-uniform hyper-graph. The exponent of the running time is indicated by the black line. The lower bound is matching for the hatched red area from cycles of length $u$ to $\gamma_3^{-1}(u)$ and the running time is $O(n^k)$ for a $k$-hypercycle. Then from $\gamma_3^{-1}(u)$ to $2u-1$ the running time is $\tilde{O}(n^{k-3+\omega})$ for $k$-hypercycle where $\omega$ is the matrix multiplication constant ($\omega < 2.3716$mmConstantOmega). The dotted line represents the running time the algorithm would achieve if $\omega=2$. There is an algorithm running in $\tilde{O}(n^{2u-1-(3-\omega)})$ time for all $k \geq 2u-1$. The shading stops after $\gamma_3^{-1}(u)$ because we don't know of any tight lower bounds past this point.
• Figure 3: An example of a small subgraph with hyperedges of multiple different sizes. This graph contains hyperedges $\{A,B\}, \{B,C\}, \{B, D\}, \{C, D\}, \{A,B,C\},$ and $\{B,C,D,E\}$. We depict the $3$-width edge as a red circle, the $4$-width edge as a blue circle, and the $2$-width edges as black lines.
• Figure 4: An example of two small graphs and corresponding $H$-partite graphs.
• Figure 5: The case of $k=7$ where we start with a $3$-uniform graph. The double circles indicate the 'farthest apart' three nodes can be. Note the purple and blue arcs indicate the furthest a uniformity $4$ hyperedge can go while including the top node, and that neither hyperedge includes all three double circled nodes. Further note that the red edge, which is a hyperedge of uniformity $5$, covers all three double circled nodes. This is what $\gamma(\cdot)$ is capturing.

Theorems & Definitions (86)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Informal
• Theorem 2.1: Theorem 3.1 from LVW18
• Theorem 3.1: Theorem 3.1 from LVW18
• Theorem 3.1
• Lemma 3.2
• proof
• Corollary 3.3