2024/07/12 by S., Karthik C., Lee, Euiwoong, Manurangsi, Pasin
#Computational Complexity (cs.CC) #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2407.08917
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 τ>0, and a parameter k and the goal is to determine if there exist k sets in the input whose union is at least τ 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 τ fraction of the universe or if the union of every k input sets is not much larger than τ⋅ (1-(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.