The Constant Inapproximability of the Parameterized Dominating Set Problem

TL;DR

This paper proves that the parameterized dominating set problem cannot be approximated within any constant ratio by fixed-parameter tractable algorithms unless FPT equals W[1], highlighting fundamental computational hardness.

Contribution

It establishes the inapproximability of the parameterized dominating set problem within any constant factor under standard complexity assumptions, using a novel hardness reduction.

Findings

No fixed-parameter tractable algorithm can approximate within any constant ratio

The result holds unless FPT=W[1]

Classical dominating set has no polynomial-time constant approximation under ETH

Abstract

We prove that there is no fpt-algorithm that can approximate the dominating set problem with any constant ratio, unless FPT= W[1]. Our hardness reduction is built on the second author's recent W[1]-hardness proof of the biclique problem. This yields, among other things, a proof without the PCP machinery that the classical dominating set problem has no polynomial time constant approximation under the exponential time hypothesis.