Small Even Covers, Locally Decodable Codes and Restricted Subgraphs of Edge-Colored Kikuchi Graphs

TL;DR

These works introduce the new technique that relates hypergraph even covers to cycles in the associated Kikuchi graphs, and gives a simple and purely combinatorial argument that recovers the best-known bound for Feige’s conjecture for even $k.

Abstract

Given a $k$-uniform hypergraph $H$ on $n$ vertices, an even cover in $H$ is a collection of hyperedges that touch each vertex an even number of times. Even covers are a generalization of cycles in graphs and are equivalent to linearly dependent subsets of a system of linear equations modulo $2$. As a result, they arise naturally in the context of well-studied questions in coding theory and refuting unsatisfiable $k$-SAT formulas. Analogous to the irregular Moore bound of Alon, Hoory, and Linial (2002), in 2008, Feige conjectured an extremal trade-off between the number of hyperedges and the length of the smallest even cover in a $k$-uniform hypergraph. This conjecture was recently settled up to a multiplicative logarithmic factor in the number of hyperedges (Guruswami, Kothari, and Manohar 2022 and Hsieh, Kothari, and Mohanty 2023). These works introduce the new technique that relates hypergraph even covers to cycles in the associated Kikuchi graphs. Their analysis of these Kikuchi graphs, especially for odd $k$, is rather involved and relies on matrix concentration inequalities. In this work, we give a simple and purely combinatorial argument that recovers the best-known bound for Feige's conjecture for even $k$. We also introduce a novel variant of a Kikuchi graph which together with this argument improves the logarithmic factor in the best-known bounds for odd $k$. As an application of our ideas, we also give a purely combinatorial proof of the improved lower bounds (Alrabiah, Guruswami, Kothari and Manohar, 2023) on 3-query binary linear locally decodable codes.

Figures (2)

• Figure 2: An illustration of Definitions \ref{['def:kikuchi']} and \ref{['def:HCdef']}. On the left, we have a pair of hyperedges $C,C'$ which both belong to a bucket. Then, on its right is depicted an edge $S \xleftrightarrow{(C,C')} T$ created by this pair. Then, on the further right are illustrated all hyperedges of $\mathcal{H}_c$ (where $c$ is the color of the hyperedge $C$) which are created in Definition \ref{['def:HCdef']} by the pair $(C,C')$. Here, red and blue correspond to labels $1$ and $2$ respectively (recall that $S,T \subseteq [n] \times [2]$ and $\mathcal{H}_c$ has vertex set $[n]\times [2]$).
• Figure 3: An illustration of a good flower gadget with $k=3$.

Theorems & Definitions (23)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Proposition 2.6: LDCs in normal form, see Theorem 8.1 in DvirNotes
• Lemma 2.7