Fixed Budget vs. Covering Target: The Partial Set Cover Boundary for Bounded VC-Dimension

2026-08-04

AI Summary

This work investigates parameterized approximation bounds for Partial Set Cover and Maximum Coverage in set systems with bounded VC dimension. By introducing structural parameters such as the shatter function exponent and downward intersection complexity, it establishes the first hardness result showing that Partial Set Cover admits no $(2-\delta)$-approximation in FPT time when VC dimension is at least 7, unless $\text{FPT} = \text{W[1]}$. Conversely, under bounded shatter function exponent, a $(k+1)$-approximation guarantee is recovered and extended to weighted settings, multi-criteria objectives, and matroid constraints. Leveraging parameterized reductions and structural characterizations, the study designs a $2^{O(\Gamma k \log k)} N$-time algorithm for Weighted Partial Set Cover and an EPAS running in $2^{\tilde{O}(kd/\varepsilon)} N^{O(1)}$ time for Weighted CC-MaxSAT under bounded VC dimension.

Abstract

Maximum Coverage and Partial Set Cover are fundamental parameterized covering problems. The former fixes a budget $k$ and maximizes coverage; the latter meets a target with as few sets as possible. Badanidiyuru, Kleinberg, and Lee (SoCG 2012) give an EPAS for the former on bounded-VC set systems, while Jain et al. (SODA 2023) show that on $K_{d,d}$-free incidence graphs, $k+1$ sets suffice whenever $k$ sets meet the target. We ask whether this guarantee extends to all bounded-VC set systems. Our first result is negative. Unless FPT = W[1], Partial Set Cover admits no parameterized $(2-δ)$-approximation even at VC-dimension seven. Under ETH, it has no parameterized approximation scheme there and no $2^{o(d)}$-approximation at VC-dimension $d$. On the positive side, bounded semi-ladder index restores this guarantee. It is stronger than bounded VC-dimension but strictly generalizes the $K_{d,d}$-free setting. For Weighted Partial Set Cover, if $k$ sets cover weight $W$, we find $k+1$ sets covering weight $W$ in $2^{O(Γk\log k)}N$ time, where $Γ$ is the downward intersection complexity and $N$ is the input size. The framework supports per-class targets and matroid independence, with applications to partial dominating set and geometric and bounded-size covering. Finally, we give a deterministic FPT reduction from Weighted CC-MaxSAT to a bounded family of Weighted Maximum Coverage instances, preserving incidence structure and approximation schemes with constant-factor accuracy loss. This gives an EPAS at bounded semi-ladder index. We improve the deterministic BKL bounded-VC implementation; combined with our reduction, it yields a $2^{\widetilde{O}(kd/\varepsilon)}N^{O(1)}$-time EPAS for bounded-VC Weighted CC-MaxSAT.

Problem

• Partial Set Cover
• Bounded VC-Dimension
• Approximation
• Parameterized Complexity
• Set Cover

Innovation

• bounded semi-ladder index
• parameterized approximation
• Partial Set Cover
• VC-dimension
• EPAS