FPT Inapproximability of Directed Cut and Connectivity Problems

TL;DR

This paper explores the fixed-parameter tractability and inapproximability of directed cut and connectivity problems, providing new algorithms and hardness results based on the ETH and related hypotheses.

Contribution

It introduces FPT approximation algorithms for directed cut and connectivity problems and strengthens hardness results through gap-instances under ETH assumptions.

Findings

FPT approximation algorithms for several directed problems

Hardness results under ETH and Gap-ETH hypotheses

Circumventing previous intractability results with new algorithms

Abstract

(see paper for full abstract) Cut problems and connectivity problems on digraphs are two well-studied classes of problems from the viewpoint of parameterized complexity. After a series of papers over the last decade, we now have (almost) tight bounds for the running time of several standard variants of these problems parameterized by two parameters: the number $k$ of terminals and the size $p$ of the solution. When there is evidence of FPT intractability, then the next natural alternative is to consider FPT approximations. In this paper, we show two types of results for several directed cut and connectivity problems, building on existing results from the literature: first is to circumvent the hardness results for these problems by designing FPT approximation algorithms, or alternatively strengthen the existing hardness results by creating "gap-instances" under stronger hypotheses such as the (Gap-)Exponential Time Hypothesis (ETH).