Markov-Chain-Based Heuristics for the Feedback Vertex Set Problem for Digraphs

A feedback vertex set (FVS) of an undirected or directed graph G=(V, A) is a set F such that G-F is acyclic. The minimum feedback vertex set problem asks for a FVS of G of minimum cardinality whereas the weighted minimum feedback vertex set problem consists of determining a FVS F of minimum weight w(F) given a real-valued weight function w. Both problems are NP-hard [Karp72]. Nethertheless, they have been found to have applications in many fields. So one is naturally interested in approximation algorithms. While most of the existing approximation algorithms for feedback vertex set problems rely on local properties of G only, this thesis explores strategies that use global information about G in order to determine good solutions. The pioneering work in this direction has been initiated by Speckenmeyer [Speckenmeyer89]. He demonstrated the use of Markov chains for determining low cardinality FVSs. Based on his ideas, new approximation algorithms are developed for both the unweighted and the weighted minimum feedback vertex set problem for digraphs. According to the experimental results presented in this thesis, these new algorithms outperform all other existing approximation algorithms. An additional contribution, not related to Markov chains, is the identification of a new class of digraphs G=(V, A) which permit the determination of an optimum FVS in time O(|V|^4). This class strictly encompasses the completely contractible graphs [Levy/Low88]

Markov-Chain-Based Heuristics for the Minimum Feedback Vertex Set Problem

Let G be a directed graph. A vertex set F is called feedback vertex set (FVS) if its removal from G results in an acyclic graph. Because determining a minimum cardinality FVS is known to be NP hard, [Karp72], one is interested in designing fast approximation algorithms determining near-optimum FVSs. The paper presents deterministic and randomised heuristics based on Markov chains. In this regard, an earlier approximation algorithm developed in [Speckenmeyer89] is revisited and refined. Experimental results demonstrate the overall performance superiority of our algorithms compared to other algorithms known from literature with respect to both criteria, the sizes of solutions determined, as well as the consumed runtimes

Markov-Chain-Based Heuristics for the Minimum Feedback Vertex Set Problem

Let G be a directed graph. A vertex set F is called feedback vertex set (FVS) if its removal from G results in an acyclic graph. Because determining a minimum cardinality FVS is known to be NP hard, Karp72, one is interested in designing fast approximation algorithms determining near-optimum FVSs. The paper presents deterministic and randomised heuristics based on Markov chains. In this regard, an earlier approximation algorithm developed in Speckenmeyer89 is revisited and refined. Experimental results demonstrate the overall performance superiority of our algorithms compared to other algorithms known from literature with respect to both criteria, the sizes of solutions determined, as well as the consumed runtimes