Parameterized Enumeration with Ordering

TL;DR

This paper introduces a generic approach for ordered enumeration in parameterized complexity, leveraging neighborhood functions and Total-FPT algorithms to efficiently list solutions in problems like graph modifications.

Contribution

It presents a unified strategy to achieve Delay-FPT and Total-FPT enumeration algorithms for various problems using neighborhood functions.

Findings

Developed a generic algorithmic framework for ordered enumeration.

Applied the framework to graph modification problems.

Established conditions under which Delay-FPT algorithms exist.

Abstract

The classes Delay-FPT and Total-FPT recently have been introduced into parameterized complexity in order to capture the notion of efficiently solvable parameterized enumeration problems. In this paper we focus on ordered enumeration and will show how to obtain Delay-FPT and Total-FPT enumeration algorithms for several important problems. We propose a generic algorithmic strategy, combining well-known principles stemming from both parameterized algorithmics and enumeration, which shows that, under certain preconditions, the existence of a so-called neighbourhood function among the solutions implies the existence of a Delay-FPT algorithm which outputs all ordered solutions. In many cases, the cornerstone to obtain such a neighbourhood function is a Total-FPT algorithm that outputs all minimal solutions. This strategy is formalized in the context of graph modification problems, and shown to be applicable to numerous other kinds of problems.