Stochastic simulated annealing for directed feedback vertex set

Abstract

The minimum feedback vertex set problem consists in finding the minimum set of vertices that should be removed in order to make the graph acyclic. This is a well known NP-hard optimization problem with applications in various fields, such as VLSI chip design, bioinformatics and transaction processing. In this paper, we explore the complementary problem in directed graphs, i.e., how to construct the maximum directed acyclic graph (max-DAG). We show that the max-DAG problem is Poly-APX complete, which implies that even trying to obtain approximation algorithms for this problems is likely to be unfeasible. In light of these considerations, we introduce a new algorithmic solution, based on Simulated Annealing (SA), which combines techniques such as kernelization, efficient data-structures, novel heuristics to initialize the search process, a global re-structuring procedure, and a neighbor re-ordering technique to speed-up the local search step. We present an extensive experimental study that validates the key design and implementation choices undertaken in our proposal and compares it to state of the art SA-based solutions Galinier et al. (2013) and Tang et al. (2017). The proposed algorithm provides significant performance gains by obtaining feedback vertex sets up to 13.3× closer to the optimal solution in a wide variety of synthetic and real-world graphs.