[Paper Review] Polynomial-time Algorithms for the Subset Feedback Vertex Set Problem on Interval Graphs and Permutation Graphs

Charis Papadopoulos, Spyridon Tzimas|arXiv (Cornell University)|Jan 17, 2017

Advanced Graph Theory Research24 references4 citations

TL;DR

This paper presents the first polynomial-time algorithms for the weighted Subset Feedback Vertex Set problem on interval graphs and permutation graphs—two unrelated subclasses of AT-free graphs. By exploiting structural properties of these graphs and using dynamic programming on vertex partitions, the authors achieve efficient solutions, with implications for related problems like Multiway Cut.

ABSTRACT

Given a vertex-weighted graph $G=(V,E)$ and a set $S \subseteq V$, a subset feedback vertex set $X$ is a set of the vertices of $G$ such that the graph induced by $V \setminus X$ has no cycle containing a vertex of $S$. The extsc{Subset Feedback Vertex Set} problem takes as input $G$ and $S$ and asks for the subset feedback vertex set of minimum total weight. In contrast to the classical extsc{Feedback Vertex Set} problem which is obtained from the extsc{Subset Feedback Vertex Set} problem for $S=V$, restricted to graph classes the extsc{Subset Feedback Vertex Set} problem is known to be NP-complete on split graphs and, consequently, on chordal graphs. However as extsc{Feedback Vertex Set} is polynomially solvable for AT-free graphs, no such result is known for the extsc{Subset Feedback Vertex Set} problem on any subclass of AT-free graphs. Here we give the first polynomial-time algorithms for the problem on two unrelated subclasses of AT-free graphs: interval graphs and permutation graphs. As a byproduct we show that there exists a polynomial-time algorithm for circular-arc graphs by suitably applying our algorithm for interval graphs. Moreover towards the unknown complexity of the problem for AT-free graphs, we give a polynomial-time algorithm for co-bipartite graphs. Thus we contribute to the first positive results of the extsc{Subset Feedback Vertex Set} problem when restricted to graph classes for which extsc{Feedback Vertex Set} is solved in polynomial time.

Motivation & Objective

• Address the computational complexity of the Subset Feedback Vertex Set (SFVS) problem on graph classes where the classical Feedback Vertex Set is polynomial-time solvable.
• Overcome the NP-completeness of SFVS on split graphs and chordal graphs by identifying positive results on specific subclasses.
• Develop efficient algorithms for the weighted SFVS problem on interval graphs and permutation graphs, which are proper subclasses of AT-free graphs.
• Provide a foundation for solving related problems such as Multiway Cut on these graph classes through reduction to SFVS.
• Contribute to the understanding of SFVS complexity on AT-free graphs by establishing polynomial-time solvability on co-bipartite graphs as a byproduct.

Proposed method

• Design a dynamic programming approach based on vertex partitioning into four sets: X (S-vertices in solution), Y (non-S vertices adjacent to X), Z (S-vertices not in solution but adjacent to Y), and W (non-S vertices not adjacent to X or Z).
• Enumerate all possible configurations of these four sets that can form a maximal S-forest, ensuring acyclicity in the induced subgraph on S-vertices.
• Use the structural properties of interval and permutation graphs to bound the number of such configurations to O(n^4) per case, enabling polynomial-time enumeration.
• Apply a recursive decomposition strategy that leverages interval and permutation graph representations (e.g., interval ordering and permutation matrix structure) to guide the state space.
• Introduce a safe pruning rule: vertices not adjacent to any S-vertex in X or Z can be safely included in the solution if they do not create S-cycles.
• Prove correctness by showing that every maximal S-forest corresponds to one of the 22 canonical configurations, each of which can be checked and processed in polynomial time.

Experimental results

Research questions

• RQ1 Can the weighted Subset Feedback Vertex Set problem be solved in polynomial time on interval graphs?
• RQ2 Is the weighted Subset Feedback Vertex Set problem polynomial-time solvable on permutation graphs?
• RQ3 Does the existence of a polynomial-time algorithm for SFVS on interval and permutation graphs extend to other subclasses of AT-free graphs?
• RQ4 Can the SFVS problem on co-bipartite graphs be solved in polynomial time, given its structural similarity to interval and permutation graphs?
• RQ5 To what extent can the SFVS algorithm be used to solve related problems like Multiway Cut on these graph classes?

Key findings

• The paper presents the first polynomial-time algorithm for the weighted Subset Feedback Vertex Set problem on interval graphs.
• A similar polynomial-time algorithm is developed for permutation graphs, marking the first positive result for SFVS on any subclass of AT-free graphs where Feedback Vertex Set is polynomial-time solvable.
• The algorithm runs in O(n^4) time due to the enumeration of at most 22n^4 maximal S-forests, each corresponding to a valid configuration of vertex sets.
• The approach enables a polynomial-time solution for the Multiway Cut problem on interval and permutation graphs via reduction to SFVS.
• A polynomial-time algorithm is also established for co-bipartite graphs, a subclass of AT-free graphs, further expanding the class of graphs for which SFVS is tractable.
• The results demonstrate that SFVS is polynomial-time solvable on interval graphs and permutation graphs, despite being NP-complete on split graphs and chordal graphs.