When does FTP become FPT?

TL;DR

This paper investigates the computational complexity of the Fault-Tolerant Path problem, analyzing its fixed-parameter tractability and kernelization under various parameters related to graph edges and redundancy.

Contribution

It provides a comprehensive complexity landscape of FTP, identifying conditions under which the problem is fixed-parameter tractable or admits polynomial kernels.

Findings

Complexity varies with different parameterizations.

Identifies parameters leading to FPT algorithms.

Provides almost complete complexity classification.

Abstract

In the problem Fault-Tolerant Path (FTP), we are given an edge-weighted directed graph G = (V, E), a subset U \subseteq E of vulnerable edges, two vertices s, t \in V, and integers k and \ell. The task is to decide whether there exists a subgraph H of G with total cost at most \ell such that, after the removal of any k vulnerable edges, H still contains an s-t-path. We study whether Fault-Tolerant Path is fixed-parameter tractable (FPT) and whether it admits a polynomial kernel under various parameterizations. Our choices of parameters include: the number of vulnerable edges in the input graph, the number of safe (i.e, invulnerable) edges in the input graph, the budget \ell, the minimum number of safe edges in any optimal solution, the minimum number of vulnerable edges in any optimal solution, the required redundancy k, and natural above- and below-guarantee parameterizations. We provide an almost complete description of the complexity landscape of FTP for these parameters.