Unit Interval Editing is Fixed-Parameter Tractable

TL;DR

This paper presents fixed-parameter tractable algorithms for transforming a graph into a unit interval graph through limited vertex and edge modifications, improving efficiency over previous methods.

Contribution

It introduces new FPT algorithms for unit interval editing, including vertex and edge deletion variants, with significantly improved running times.

Findings

Algorithm runs in $2^{O(k \log k)} imes (n+m)$ time.

Edge deletion problem solved in $O(4^k imes (n+m))$ time.

Vertex deletion problem improved to $O(6^k imes (n+m))$ time.

Abstract

Given a graph~$G$ and integers $k_1$, $k_2$, and~$k_3$, the unit interval editing problem asks whether $G$ can be transformed into a unit interval graph by at most $k_1$ vertex deletions, $k_2$ edge deletions, and $k_3$ edge additions. We give an algorithm solving this problem in time $2^{O(k\log k)}\cdot (n+m)$, where $k := k_1 + k_2 + k_3$, and $n, m$ denote respectively the numbers of vertices and edges of $G$. Therefore, it is fixed-parameter tractable parameterized by the total number of allowed operations. Our algorithm implies the fixed-parameter tractability of the unit interval edge deletion problem, for which we also present a more efficient algorithm running in time $O(4^k \cdot (n + m))$. Another result is an $O(6^k \cdot (n + m))$-time algorithm for the unit interval vertex deletion problem, significantly improving the algorithm of van 't Hof and Villanger, which runs in time $O(6^k \cdot n^6)$.