Min (A)cyclic Feedback Vertex Sets and Min Ones Monotone 3-SAT

TL;DR

This paper explores the complexity of feedback vertex set problems in directed graphs, linking them to monotone 3-SAT variants, establishing NP-completeness for restricted cases, and identifying graph classes where solutions are polynomial.

Contribution

It establishes NP-completeness of 2-Choice versions of feedback vertex set and monotone 3-SAT problems, and identifies graph classes where acyclic feedback vertex set problems are polynomial.

Findings

NP-completeness of 2-Choice MFVS and variants

Polynomial algorithms for flow reducible graphs and C1P-digraphs

Connections between feedback vertex sets and monotone 3-SAT variants

Abstract

In directed graphs, we investigate the problems of finding: 1) a minimum feedback vertex set (also called the Feedback Vertex Set problem, or MFVS), 2) a feedback vertex set inducing an acyclic graph (also called the Vertex 2-Coloring without Monochromatic Cycles problem, or Acyclic FVS) and 3) a minimum feedback vertex set inducing an acyclic graph (Acyclic MFVS). We show that these problems are strongly related to (variants of) Monotone 3-SAT and Monotone NAE 3-SAT, where monotone means that all literals are in positive form. As a consequence, we deduce several NP-completeness results on restricted versions of these problems. In particular, we define the 2-Choice version of an optimization problem to be its restriction where the optimum value is known to be either D or D+1 for some integer D, and the problem is reduced to decide which of D or D+1 is the optimum value. We show that the 2-Choice versions of MFVS, Acyclic MFVS, Min Ones Monotone 3-SAT and Min Ones Monotone NAE 3-SAT are NP-complete. The two latter problems are the variants of Monotone 3-SAT and respectively Monotone NAE 3-SAT requiring that the truth assignment minimize the number of variables set to true. Finally, we propose two classes of directed graphs for which Acyclic FVS is polynomially solvable, namely flow reducible graphs (for which MFVS is already known to be polynomially solvable) and C1P-digraphs (defined by an adjacency matrix with the Consecutive Ones Property).