Nearly Optimal NP-Hardness of Vertex Cover on k-Uniform k-Partite Hypergraphs

We study the problem of computing the minimum vertex cover on k-uniform k-partite hypergraphs when the k-partition is given. On bipartite graphs (k = 2), the minimum vertex cover can be computed in polynomial time. For general k, the problem was studied by Lov'asz, who gave a k/2 -approximation based on the standard LP relaxation. Subsequent work by Aharoni, Holzman and Krivelevich showed a tight integrality gap of (k/2 - o(1)) for the LP relaxation. While this problem was known to be NP-hard for k >= 3, the first non-trivial NP-hardness of approximation factor of k/4- \eps was shown in a recent work by Guruswami and Saket. They also showed that assuming Khot’s Unique Games Conjecture yields a k/2 - \eps inapproximability for this problem, implying the optimality of Lov'asz’s result. In this work, we show that this problem is NP-hard to approximate within k/2- 1 + 1/2k -\eps. This hardness factor is off from the optimal by an additive constant of at most 1 for k >= 4. Our reduction relies on the Multi-Layered PCP of Dinur et al. and uses a gadget - based on biased Long Codes - adapted from the LP integrality gap of Aharoni et al. The nature of our reduction requires the analysis of several Long Codes with different biases, for which we prove structural properties of the so called cross-intersecting collections of set families - variants of which have been studied in extremal set theory.

💡 Research Summary

The paper investigates the approximability of the Minimum Vertex Cover problem on k‑uniform, k‑partite hypergraphs when the k‑partition of the vertex set is part of the input. For k = 2 (i.e., bipartite graphs) the problem is polynomial‑time solvable via König’s theorem, but for k ≥ 3 it becomes computationally hard. Lovász gave a simple k⁄2‑approximation based on the natural linear‑programming (LP) relaxation, and Aharoni, Holzman, and Krivelevich later proved that this LP has an integrality gap of (k⁄2 − o(1)), showing that the approximation factor cannot be improved by the LP alone. Prior hardness results were limited: Guruswami and Saket proved NP‑hardness of approximation within a factor of k⁄4 − ε, while under the Unique Games Conjecture (UGC) one can obtain a k⁄2 − ε bound, matching Lovász’s algorithm. The present work closes the gap on the NP‑hardness side, establishing that it is NP‑hard to approximate the problem within a factor of

 k⁄2 − 1 + 1⁄(2k) − ε.

For every k ≥ 4 this bound is at most one unit away from the optimal k⁄2 factor, and for larger k the additive term 1 − 1⁄(2k) becomes negligible.

Technical Overview

The reduction builds on two powerful ingredients: (1) the Multi‑Layered Probabilistically Checkable Proof (PCP) construction of Dinur, Harsha, Kindler, and Minzer, and (2) a family of biased Long Code gadgets derived from the LP integrality‑gap example of Aharoni et al.

Multi‑Layered PCP. The authors use an L‑layer PCP where each layer contains a set of variables and a set of constraints that project assignments from one layer to the next. The projection property ensures that a correct proof can be “propagated” through all layers, while any erroneous proof incurs a noticeable loss of acceptance probability in at least one layer. By carefully choosing the number of layers and the projection parameters, the authors obtain a gap between completeness (the verifier accepts with probability close to 1) and soundness (any proof is accepted with probability at most 1⁄k + δ for a small δ). This multi‑layer structure is crucial because it allows the reduction to embed many independent copies of the Long Code gadget, one per layer, while preserving the global consistency required for the hypergraph instance.

Biased Long Code Gadgets. In the classic Long Code, each coordinate corresponds to a subset of the underlying domain, and the codeword indicates membership of the assignment in that subset. The authors introduce a bias p_i for each part i of the hypergraph, setting p_i = ½ + 1⁄(2k) − δ_i, where δ_i is a tiny slack term. The bias determines the probability that a coordinate takes value 1, and it directly influences the size of a vertex cover that can be constructed from a satisfying assignment. By assigning different biases to different parts, the gadgets become “cross‑intersecting”: any collection of selected coordinates from distinct parts must intersect in a prescribed number of positions. This property is essential for translating the PCP’s acceptance condition into a lower bound on the vertex cover size in the “NO” case.

Cross‑Intersecting Set Families. To analyze the interaction of multiple biased Long Codes, the authors prove a new extremal result about cross‑intersecting families of subsets. Roughly, if families F₁,…,F_k ⊆ 2^{