Interval Completion is Fixed Parameter Tractable ∗ Yngve Villanger † Pinar Heggernes † Christophe Paul ‡ Jan Arne Telle † Abstract We present an algorithm with runtime O(k 2k n 3 m) for the following NP-complete problem [9, problem GT35]: Given an arbitrary graph G on n vertices and m edges, can we obtain an interval graph by adding at most k new edges to G? This resolves the long-standing open question [17, 7, 25, 14], first posed by Kaplan, Shamir and Tarjan, of whether this problem was fixed parameter tractable. The problem has applications in Profile Minimization for Sparse Matrix Computations [10, 26], and our results show tractability for the case of a small number k of zero elements in the envelope. Our algorithm performs bounded search among possible ways of adding edges to a graph to obtain an interval graph, and combines this with a greedy algorithm when graphs of a certain structure are reached by the search. Keywords: Interval graphs, profile minimization, edge completion, FPT algorithm, branching 1 Introduction and motivation Interval graphs are the intersection graphs of intervals of the real line and have a wide range of applications [13]. Connected with interval graphs is the following problem: Given an arbitrary graph G, what is the minimum number of edges that must be added to G in order to obtain an interval graph? This problem is NP-hard [18, 9]. The problem arises in Sparse Matrix Computations, where one of the standard methods for reordering a matrix to get as few non- zero elements as possible during Gaussian elimination, is to permute the rows and columns of the matrix so that non-zero elements are gathered close to the main diagonal [10]. The profile of a matrix is the smallest number of entries that can be enveloped within off-diagonal non-zero elements of the matrix. Translated to graphs, the profile of a graph G is exactly the minimum number of edges in an interval supergraph of G [26]. Originally, Physical Mapping of DNA was another motivation for this problem [12]. In this paper, we present an algorithm with runtime O(k 2k n 3 m) for the k-Interval Completion problem of deciding whether a graph on n vertices and m edges can be made into an interval graph by adding at most k edges. A parameterized problem with parameter k and input size x that can be solved by an algorithm having runtime f (k) · x O(1) is called fixed parameter tractable (FPT) (see [7] for an introduction to fixed parameter tractability and bounded search ∗ This work is supported by the Research Council of Norway and the French ANR project “Graph decomposition and algorithm”. An extended abstract of this paper was presented at STOC 2007. † Department of informatics, University of Bergen, Norway. Emails: pinar@ii.uib.no, telle@ii.uib.no, yngvev@ii.uib.no ‡ LIRMM, Universit´ e Montpellier II, France and McGill University, Canada Email: paul@lirmm.fr 1