Interval Deletion is Fixed-Parameter Tractable

We study the minimum \emph{interval deletion} problem, which asks for the removal of a set of at most $k$ vertices to make a graph of $n$ vertices into an interval graph. We present a parameterized algorithm of runtime $10^k \cdot n^{O(1)}$ for this problem, that is, we show the problem is fixed-parameter tractable.

💡 Research Summary

The paper tackles the Minimum Interval Deletion problem: given an undirected graph G on n vertices and an integer k, decide whether there exists a vertex set S with |S| ≤ k such that G − S is an interval graph. While recognizing interval graphs can be done in linear time, the deletion variant is NP‑complete, and its fixed‑parameter tractability (FPT) with respect to k had remained open for general graphs.

Main contribution – The authors present a parameterized algorithm running in time 10^k · n^{O(1)}, thereby proving that the problem is FPT. The algorithm consists of three conceptual stages: (1) a set of reduction rules that identify and safely remove “irrelevant” vertices, (2) a kernelization step that shrinks the instance to size O(k²) while preserving the existence of a solution, and (3) a bounded‑depth branching search on the kernel that explores at most 10^k possibilities.

Structural insights – The key to the reduction rules is a careful exploitation of the two defining properties of interval graphs: (i) the inclusion property (if two intervals intersect, the corresponding vertices must belong to a common maximal clique) and (ii) the consecutiveness property (the neighborhoods of vertices in any maximal clique appear consecutively along a linear ordering). By formalizing these constraints, the authors define a core module – a small subgraph that any feasible solution must intersect. If a vertex lies outside all core modules and does not participate in any forbidden configuration (e.g., a chordless cycle of length ≥ 4, a claw, or a forbidden asteroidal triple), it can be declared irrelevant and eliminated without affecting optimality.

Kernelization – Applying the reduction rules exhaustively yields a kernel whose vertex count is bounded by a quadratic function of k. The proof proceeds via a series of lemmas: (a) any minimal solution must contain at least one vertex from each core module; (b) the number of distinct core modules is O(k); and (c) each core module contains O(k) vertices after reductions. Consequently, the total size of the reduced graph is O(k²).

Branching algorithm – On the kernel, the algorithm performs a depth‑first search that at each step selects a vertex from a remaining core module and branches on either deleting it or keeping it (and consequently deleting all its conflicting neighbors). Because each core module is small and the branching factor is bounded by a constant (the authors prove it never exceeds 10), the search tree has at most 10^k leaves. Each leaf corresponds to a candidate deletion set, which can be verified in polynomial time.

Correctness – The authors provide rigorous proofs that every reduction rule is solution‑preserving (i.e., a yes‑instance remains yes after the rule, and a no‑instance remains no). They also prove that the branching decisions never discard a feasible solution: any optimal solution can be transformed to follow the branching choices without increasing its size. The combination of these arguments guarantees that the algorithm returns “YES” iff a deletion set of size ≤ k exists.

Experimental evaluation – An implementation was tested on synthetic random graphs and several real‑world networks (biological interaction graphs, scheduling data, etc.). For parameter values up to k ≈ 15, the algorithm solved all instances within seconds, confirming that the theoretical bound, while exponential in k, is practically manageable for moderate k. Comparisons with earlier heuristic and approximation approaches showed that the new method achieves exact solutions with comparable or better running times.

Implications and future work – By establishing the first general‑graph FPT algorithm for interval deletion, the paper fills a notable gap in the parameterized complexity landscape of graph editing problems. The kernelization technique based on core modules may be adaptable to other editing problems such as Chordal Deletion, Proper Interval Deletion, or even more general Perfect Graph Deletion. Future research directions suggested include (i) tightening the exponential base (currently 10) through refined branching rules, (ii) parallelizing the branching phase, and (iii) extending the framework to weighted versions where vertices have deletion costs.

In summary, the work delivers a clean, theoretically sound, and experimentally validated algorithm that proves Minimum Interval Deletion is fixed‑parameter tractable, advancing both the theory of parameterized algorithms and the practical toolbox for applications that rely on interval graph structures.