Linear Recognition of Almost Interval Graphs

TL;DR

This paper presents linear-time algorithms for recognizing almost interval graphs, which are graphs close to interval graphs by a small number of modifications, and proves their fixed-parameter tractability.

Contribution

It introduces efficient algorithms for recognizing almost interval graphs with fixed parameters, resolving a long-standing open problem in parameterized complexity.

Findings

Linear-time algorithms for fixed k recognition

Fixed-parameter tractability of recognition problems

Improved algorithms with exponential dependence on k

Abstract

Let $\mbox{interval} + k v$, $\mbox{interval} + k e$, and $\mbox{interval} - k e$ denote the classes of graphs that can be obtained from some interval graph by adding $k$ vertices, adding $k$ edges, and deleting $k$ edges, respectively. When $k$ is small, these graph classes are called almost interval graphs. They are well motivated from computational biology, where the data ought to be represented by an interval graph while we can only expect an almost interval graph for the best. For any fixed $k$, we give linear-time algorithms for recognizing all these classes, and in the case of membership, our algorithms provide also a specific interval graph as evidence. When $k$ is part of the input, these problems are also known as graph modification problems, all NP-complete. Our results imply that they are fixed-parameter tractable parameterized by $k$, thereby resolving the long-standing open problem on the parameterized complexity of recognizing $\mbox{interval}+ k e$, first asked by Bodlaender et al. [Bioinformatics, 11:49--57, 1995]. Moreover, our algorithms for recognizing $\mbox{interval}+ k v$ and $\mbox{interval}- k e$ run in times $O(6^k \cdot (n + m))$ and $O(8^k \cdot (n + m))$, (where $n$ and $m$ stand for the numbers of vertices and edges respectively in the input graph,) significantly improving the $O(k^{2k}\cdot n^3m)$-time algorithm of Heggernes et al. [STOC 2007] and the $O(10^k \cdot n^9)$-time algorithm of Cao and Marx [SODA 2014] respectively.