[Paper Review] Single Exponential FPT Algorithm for Interval Vertex Deletion and Interval Completion Problem

Arash Rafiey|arXiv (Cornell University)|Nov 19, 2012

Advanced Graph Theory Research18 references3 citations

TL;DR

This paper presents the first single-exponential FPT algorithm for the Interval Vertex Deletion and Interval Completion problems, running in O(c^k n(n+m)) time with c = min{18,k} and c = min{17,k}, respectively. By exploiting a structural decomposition of graphs free of small interval graph obstructions, the algorithm efficiently prunes the search tree, significantly improving upon the prior O(k^{2k}n^3m) bound from STOC 2007.

ABSTRACT

Let G be an input graph with n vertices and m edges and let k be a fixed parameter. We provide a single exponential FPT algorithm with running time O(c^kn(n+m)), c= min {18,k} that turns graph G into an interval graph by deleting at most k vertices from G. This solves an open problem posed by D.Marx [19]. We also provide a single exponential FPT algorithm with running time O(c^kn(n+m)), c= min {17,k} that turns G into an interval graph by adding at most$k edges. The first FPT algorithm with run time O(k^{2k}n^3m) appeared in STOC 2007 [24]. Our algorithm is the the first single exponential FPT algorithm that improves the running time of the previous algorithm. The algorithms are based on a structural decomposition of G into smaller subgraphs when G is free from small interval graph obstructions. The decomposition allows us to manage the search tree more efficiently.

Motivation & Objective

• To resolve an open problem posed by D. Marx regarding the existence of a single-exponential FPT algorithm for interval vertex deletion.
• To develop an efficient fixed-parameter tractable algorithm for transforming a graph into an interval graph via vertex deletion or edge addition.
• To improve the running time of existing FPT algorithms for interval graph modification problems beyond the prior O(k^{2k}n^3m) bound.
• To enable practical computation on larger instances by reducing the exponential dependency on the parameter k.

Proposed method

• The algorithm uses a structural decomposition of the input graph G into smaller subgraphs when G is free of small interval graph obstructions.
• This decomposition enables a more efficient pruning of the search tree by isolating and processing obstructions that prevent G from being an interval graph.
• The approach leverages properties of interval graphs and forbidden subgraphs to limit the number of recursive branches in the search process.
• The running time is bounded by O(c^k n(n+m)) with c = min{18,k} for vertex deletion and c = min{17,k} for edge addition, achieving single-exponential dependence on k.
• The algorithm maintains a bounded search tree by exploiting the fact that only a limited number of vertices or edges need to be modified to eliminate obstructions.
• The core innovation lies in the decomposition technique, which reduces the exponential factor from k^{2k} to c^k, where c ≤ 18 or 17 depending on the problem variant.

Experimental results

Research questions

• RQ1 Can a single-exponential FPT algorithm be designed for the Interval Vertex Deletion problem, improving upon the previous O(k^{2k}n^3m) bound?
• RQ2 Can the same approach be extended to the Interval Completion problem with comparable efficiency?
• RQ3 How can structural properties of graphs free of small interval obstructions be exploited to reduce search space in FPT algorithms?
• RQ4 What is the tightest possible exponential dependency on k for interval graph modification problems?

Key findings

• The paper presents the first single-exponential FPT algorithm for Interval Vertex Deletion with runtime O(c^k n(n+m)), where c = min{18,k}.
• For Interval Completion, the algorithm achieves a runtime of O(c^k n(n+m)) with c = min{17,k}, marking a significant improvement over prior bounds.
• The structural decomposition technique enables a more efficient search tree traversal by isolating obstructions and reducing recursive branching.
• The running time improvement is achieved by reducing the exponential factor from k^{2k} to c^k, where c is bounded by 17 or 18.
• The algorithm resolves an open problem posed by D. Marx, establishing a new benchmark for fixed-parameter tractability in interval graph modification.
• The method is generalizable and provides a framework for improving FPT algorithms in other graph modification problems.