On Fault Tolerant Feedback Vertex Set

TL;DR

This paper introduces the fault-tolerant version of the Feedback Vertex Set problem, proves its NP-Completeness, and provides approximation and fixed-parameter tractable algorithms, highlighting a new research direction in fault-tolerant solutions for NP-Complete problems.

Contribution

It defines the Fault Tolerant Feedback Vertex Set problem, proves its NP-Completeness, and offers approximation and FPT algorithms, advancing fault-tolerant approaches in combinatorial optimization.

Findings

FT-FVS is NP-Complete.

A constant factor approximation algorithm is developed.

An FPT algorithm parameterized by solution size is provided.

Abstract

The study of fault-tolerant data structures for various network design problems is a prominent area of research in computer science. Likewise, the study of NP-Complete problems lies at the heart of computer science with numerous results in algorithms and complexity. In this paper we raise the question of computing fault tolerant solutions to NP-Complete problems; that is computing a solution that can survive the "failure" of a few constituent elements. This notion has appeared in a variety of theoretical and practical settings such as estimating network reliability, kernelization (aka instance compression), approximation algorithms and so on. In this paper, we seek to highlight these questions for further research. As a concrete example, we study the fault-tolerant version of the classical Feedback Vertex Set (FVS) problem, that we call Fault Tolerant Feedback Vertex Set (FT-FVS). Recall that, in FVS the input is a graph $G$ and the objective is to compute a minimum subset of vertices $S$ such that $G-S$ is a forest. In FT-FVS, the objective is to compute a minimum subset $S$ of vertices such that $G - (S \setminus \{v\})$ is a forest for any $v \in V(G)$. Here the vertex $v$ denotes a single vertex fault. We show that this problem is NP-Complete, and then present a constant factor approximation algorithm as well as an FPT-algorithm parameterized by the solution size. We believe that the question of computing fault tolerant solutions to various NP-Complete problems is an interesting direction for future research.