Linear Recognition of Almost Interval Graphs

Give a graph class G and a nonnegative integer k, we use G+kv, G+ke, and G−ke to denote the classes of graphs that can be obtained from some graph in G by adding k vertices, adding k edges, and deleting k edges, respectively. They are called almost (unit) interval graphs if G is the class of (unit) interval graphs. Almost (unit) interval graphs are well motivated from computational biology, where the data ought to be represented by a (unit) interval graph while we can only expect an almost (unit) 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 (unit) interval graph as evidence. When k is part of the input, all the recognition problems are NP-complete. Our results imply that all of them are fixed-parameter tractable parameterized by k, thereby resolving the long-standing open problem on the parameterized complexity of recognizing (unit) interval+ke, first asked by Bodlaender et al. [Comput. Appl. Biosci., 11(1):49–57, 1995]. Moreover, our algorithms for recognizing (unit-)interval+kv and (unit-)interval−ke have single-exponential dependence on k and linear dependence on the graph size, which significantly improve all previous algorithms for recognizing the same classes. In particular, we show that: (n and m stand for the numbers of vertices and edges respectively in the input graph) • interval−ke can be recognized in time O(6 · (n +m)), improved from O(k2k · n3m) [Heggernes et al., STOC 2007]; • unit-interval−ke can be recognized in time O(4 · (n+m)), improved from O(16 · (m+n)) [Kaplan et al., FOCS 1994]; • interval+kv can be recognized in time O(8 · (n +m)), improved from O(10 · n9) [Cao and Marx, SODA 2014]; and • unit-interval+kv can be recognized in time O(6 · (n+m)), improved from O(6 ·n6) [Villanger, IPEC 2010]. These problems have natural optimization versions, which are known as graph modification problems. For those related to interval graphs, we show that under certain condition, there always exist optimum solutions that preserve all modules of the input graph. Another important ingredient of our algorithms is combinatorial and algorithmic characterizations of graphs free of small non-interval graphs. These studies might be of their own interest.