Mim-Width is paraNP-complete

TL;DR

This paper proves that computing mim-width and related parameters is paraNP-complete, meaning it remains NP-hard even when restricted to graphs with bounded mim-width or sim-width.

Contribution

It establishes the paraNP-completeness of mim-width, sim-width, and their linear variants, highlighting their computational intractability even with bounded parameters.

Findings

NP-hard to distinguish graphs with small mim-width from those with large sim-width

Mim-Width and related parameters are paraNP-complete to compute

Computational intractability persists even with upper bounds on these parameters

Abstract

We show that it is NP-hard to distinguish graphs of linear mim-width at most 1211 from graphs of sim-width at least 1216. This implies that Mim-Width, Sim-Width, One-Sided Mim-Width, and their linear counterparts are all paraNP-complete, i.e., NP-complete to compute even when upper bounded by a constant.