On Equivalence of Parameterized Inapproximability of k-Median, k-Max-Coverage, and 2-CSP
<|reference_start|>On Equivalence of Parameterized Inapproximability of k-Median, k-Max-Coverage, and 2-CSP: Parameterized Inapproximability Hypothesis (PIH) is a central question in the field of parameterized complexity. PIH asserts that given as input a 2-CSP on $k$ variables and alphabet size $n$, it is W[1]-hard parameterized by $k$ to distinguish if the input is perfectly satisfiable or if every assignment to the input violates 1% of the constraints.   An important implication of PIH is that it yields the tight parameterized inapproximability of the $k$-maxcoverage problem. In the $k$-maxcoverage problem, we are given as input a set system, a threshold $\tau>0$, and a parameter $k$ and the goal is to determine if there exist $k$ sets in the input whose union is at least $\tau$ fraction of the entire universe. PIH is known to imply that it is W[1]-hard parameterized by $k$ to distinguish if there are $k$ input sets whose union is at least $\tau$ fraction of the universe or if the union of every $k$ input sets is not much larger than $\tau\cdot (1-\frac{1}{e})$ fraction of the universe.   In this work we present a gap preserving FPT reduction (in the reverse direction) from the $k$-maxcoverage problem to the aforementioned 2-CSP problem, thus showing that the assertion that approximating the $k$-maxcoverage problem to some constant factor is W[1]-hard implies PIH. In addition, we present a gap preserving FPT reduction from the $k$-median problem (in general metrics) to the $k$-maxcoverage problem, further highlighting the power of gap preserving FPT reductions over classical gap preserving polynomial time reductions.<|reference_end|>
@article{s.2024on,
    title={On Equivalence of Parameterized Inapproximability of k-Median,
  k-Max-Coverage, and 2-CSP},
    author={Karthik C. S., Euiwoong Lee, and Pasin Manurangsi},
    journal={arXiv preprint arXiv:2407.08917},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.08917},
    primaryClass={cs.CC cs.DS}
}