On Equivalence of Parameterized Inapproximability of k-Median, k-Max-Coverage, and 2-CSP

TL;DR

The paper investigates parameterized inapproximability under $PIH$ and establishes bidirectional gap-preserving $\mathsf{FPT}$ reductions between $k$-MaxCoverage and $2$-$\mathsf{CSP}$, while also connecting $k$-median to multicolored $k$-MaxCoverage. Its core method is a three-step gap-preserving reduction from $k$-MaxCoverage to $2$-$\mathsf{CSP}$ (universe reduction, small-universe to Valued $2$-$\mathsf{CSP}$, then to standard $2$-$\mathsf{CSP}$), enabling transfer of hardness from coverage to CSP. Additionally, it provides an $\mathsf{FPT}$ gap-preserving reduction from $k$-median to multicolored $k$-MaxCoverage and uses this, together with existing work, to strengthen conditional hardness results for parameterized clustering objectives. The findings illustrate the power of $\mathsf{FPT}$ gap-preserving reductions to propagate approximation barriers across problems and reinforce the broader PIH-driven hardness landscape. Overall, the work deepens our understanding of how parameterized inapproximability interrelates across fundamental optimization problems.

Abstract

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 $τ\cdot (1-\frac{1}{e})$ 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.

Figures (1)

• Figure 1: In the above figure, we provide bidirectional $\mathsf{FPT}$ gap preserving reductions between $k$-clique, 2-$\mathsf{CSP}$, $k$-$\mathsf{maxcoverage}$, and $k$-median and $k$-means problems, whenever possible, with appropriate references. The reduction from $k$-median and $k$-means problems is to the multicolored version of the $k$-$\mathsf{maxcoverage}$ problem (see Remark \ref{['rem:mc']} for a discussion) and that is why the arrow is dashed.

Theorems & Definitions (20)

• Theorem 1: Informal statement; See Theorem \ref{['thm:mctocspformal']} for a formal statement
• Theorem 2
• Theorem 3
• Theorem 4: Chernoff Inequality
• Theorem 5
• Lemma 6
• Lemma 7
• Lemma 8
• proof : Proof of Theorem \ref{['thm:mctocspformal']}
• proof : Proof of Lemma \ref{['lem:uni-red']}