A Parameterized Algorithmics Framework for Degree Sequence Completion Problems in Directed Graphs

TL;DR

This paper introduces a parameterized framework for solving degree sequence completion problems in directed graphs, focusing on arc insertions, and establishes fixed-parameter tractability and kernelization results based on maximum degree parameters.

Contribution

It extends parameterized complexity analysis from undirected to directed graphs by developing a general two-stage framework utilizing flow computations.

Findings

Achieves fixed-parameter tractability for directed degree sequence problems.

Establishes polynomial kernelization results based on maximum degree.

Demonstrates the applicability of flow-based methods in directed graph problems.

Abstract

There has been intensive work on the parameterized complexity of the typically NP-hard task to edit undirected graphs into graphs fulfilling certain given vertex degree constraints. In this work, we lift the investigations to the case of directed graphs; herein, we focus on arc insertions. To this end, we develop a general two-stage framework which consists of efficiently solving a problem-specific number problem and transferring its solution to a solution for the graph problem by applying flow computations. In this way, we obtain fixed-parameter tractability and polynomial kernelizability results, with the central parameter being the maximum vertex in- or outdegree of the output digraph. Although there are certain similarities with the much better studied undirected case, the flow computation used in the directed case seems not to work for the undirected case while $f$-factor computations as used in the undirected case seem not to work for the directed case.