固定预算与覆盖目标：有界VC维下的部分集合覆盖边界
Fixed Budget vs. Covering Target: The Partial Set Cover Boundary for Bounded VC-Dimension
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中文总结 AI 辅助
本文探究最大覆盖与部分集合覆盖的边界，证明有界VC维下部分集合覆盖无相关近似保证，提出有界半梯索引条件，给出加权部分集合覆盖等问题的近似算法及应用。
中文摘要 AI 辅助
最大覆盖与部分集合覆盖是基础的参数化覆盖问题：前者固定预算k并最大化覆盖量，后者用尽可能少的集合达到覆盖目标。Badanidiyuru、Kleinberg和Lee（SoCG 2012）针对有界VC维集合系统给出了前者的EPAS（有效多项式时间近似方案）；而Jain等人（SODA 2023）证明，在无K_{d,d}关联图中，若k个集合可达到覆盖目标，则k+1个集合足够。本文探究该保证是否适用于所有有界VC维集合系统，首个结论为否定：除非FPT=W[1]，即使在VC维为7时，部分集合覆盖也不存在参数化(2-δ)-近似算法；在ETH（指数时间假设）下，其在该维度不存在参数化近似方案，且在VC维为d时不存在2^{o(d)}-近似算法。正面结果为：有界半梯索引可恢复上述保证，该条件强于有界VC维且严格泛化无K_{d,d}的场景。对于加权部分集合覆盖，若k个集合覆盖权重W，本文可在2^{O(Γk log k)}N时间内找到k+1个集合覆盖权重W，其中Γ为下交复杂度，N为输入规模；该框架支持按类目标和拟阵独立性，可应用于部分支配集、几何覆盖及有界大小覆盖。最后，本文给出从加权CC-MaxSAT到有界族加权最大覆盖实例的确定性FPT归约，保留关联结构与近似方案，且常数因子精度损失，由此在有界半梯索引下得到EPAS；本文还改进了确定性BKL有界VC实现，结合该归约，得到有界VC维加权CC-MaxSAT的2^{Õ(kd/ε)}N^{O(1)}-时间EPAS。
英文摘要
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.
