Constant ratio FPT approximation of hypertree width parameters for hypergraphs of bounded rank

TL;DR

This paper introduces a fixed-parameter tractable algorithm that approximates the hypertree width of hypergraphs with bounded rank, providing guarantees on the width or indicating when it exceeds a threshold.

Contribution

The paper presents the first FPT algorithm for approximating hypertree width parameters within a constant ratio for hypergraphs of bounded rank.

Findings

Algorithm runs in FPT time in parameters k and r.

Provides a 4k-approximation for generalized hypertree width.

Extends to fractional hypertree width with a 4k+1 approximation.

Abstract

We propose an algorithm whose input are parameters $k$ and $r$ and a hypergraph $H$ of rank at most $r$. The algorithm either returns a tree decomposition of $H$ of generalized hypertree width at most $4k$ or 'NO'. In the latter case, it is guaranteed that the hypertree width of $H$ is greater than $k$. Most importantly, the runtime of the algorithm is \emph{FPT} in $k$ and $r$. The approach extends to fractional hypertree width with a slightly worse approximation ($4k+1$ instead of $4k$). We hope that the results of this paper will give rise to a new research direction whose aim is design of FPT algorithms for computation and approximation of hypertree width parameters for restricted classes of hypergraphs.