Polynomial Kernels for Interval Completion

Abstract

An interval graph is the intersection graph of a family of intervals on the real line. \textsc{Interval Completion} asks whether a given graph can be transformed into an interval graph by adding at most $k$ edges. Although the problem is fixed-parameter tractable when parameterized by $k$, whether it admits a polynomial kernel has long been an open question. We resolve this question by giving the first polynomial kernel for \textsc{Interval Completion}. Our main contribution is a parameter-preserving polynomial-time reduction from \textsc{Interval Completion} to \textsc{Odd Cycle Transversal} (OCT). Combining this reduction with the known randomized and deterministic polynomial kernels for OCT and a polynomial-time reduction back to \textsc{Interval Completion}, we obtain a randomized kernel with $\tilde{O}(k^{18})$ vertices and $\tilde{O}(k^{36})$ edges, and a deterministic kernel with $O(k^{36})$ vertices and $O(k^{72})$ edges. Here, $\tilde{O}$ suppresses polylogarithmic factors in $k$.