Computational Method for k-Distance Limited Minimum Feedback Vertex Set and Its Application to Biological Networks

Abstract Background Feedback vertex set (FVS), a graph theory concept, is used to identify critical nodes in network and system control. Extended versions of FVS are continuously explored and have found applications in biological data analysis and system control. Then, the minimum feedback vertex set (MFVS), defined as the FVS with the smallest number of elements among all feedback vertex sets, is important when controlling a system with the least number of nodes. Results In this paper, we introduce the concept of the k-distance limited Minimum Feedback Vertex Set (kMFVS) as an extension to MFVS. We present a novel approach for obtaining the exact solutions for kMFVS using Integer Linear Programming (ILP). Simulations offer a means to determine the value of k for kMFVS in random graphs, depending on the number of nodes in the graph. Applying kMFVS to real biological data, we show that compared to MFVS, kMFVS improves recall by using only information about the network structure. Conclusion kMFVS can extract important nodes (genes) from pure network structures better than MFVS. Fur- thermore, we prove that in a discrete-time network, by controlling nodes chosen as kMFVS, the system can be transitioned to an arbitrary static steady state in at most k discrete times. Despite the computational complexity of the kMFVS problem, the proposed ILP-based method is capable of identifying exact optimal solutions even for scale-free graphs with approximately 3000 nodes.