Abstract
Recent research has revealed new applications of network control science within bio-medicine, pharmacology, and medical therapeutics. These new insights and new applications generated in turn a rediscovery of some old, unresolved algorithmic questions, this time with a much stronger motivation for their tackling. One of these questions regards the so-called Structural Target Control optimization problem, known in previous literature also as Structural Output Controllability problem. Given a directed network (graph) and a target subset of nodes, the task is to select a small (or the smallest) set of nodes from which the target can be independently controlled, i.e., it can be driven from any given initial configuration to any desired final one, through a finite sequence of input values. In recent work, this problem has been shown to be NP-hard, and several heuristic algorithms were introduced and analyzed, both on randomly generated networks, and on bio-medical ones. In this paper, we show that the Structural Target Controllability problem is fixed parameter tractable when parameterized by the number of target nodes. We also prove that the problem is hard to approximate at a factor better than \(O(\log {n})\).
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Notes
- 1.
It is beyond the goal of this paper to define the topological notion of thin sets; we only give here the intuition that such sets consist of isolated cases that may be easily replaced with nearby favorable cases.
- 2.
We implicitly interchange the usage of \(x_i\) and i for matrix indices.
- 3.
We use the following upper bound for the binomial coefficient \({{n+k} \atopwithdelims ()k} \le {(\frac{e(n+k)}{k})}^k\).
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Acknowledgments
This work was supported by the Academy of Finland through grant 272451, by the Finnish Funding Agency for Innovation through grant 1758/31/2016, by the Romanian National Authority for Scientific Research and Innovation, through the POC grant P 37 257, and by the (Romanian) Ministry of Research and Innovation through the Institutional Research Programme PN 1819, project PN 1819-01-01.
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Czeizler, E., Popa, A., Popescu, V. (2018). Fixed Parameter Algorithms and Hardness of Approximation Results for the Structural Target Controllability Problem. In: Jansson, J., Martín-Vide, C., Vega-Rodríguez, M. (eds) Algorithms for Computational Biology. AlCoB 2018. Lecture Notes in Computer Science(), vol 10849. Springer, Cham. https://doi.org/10.1007/978-3-319-91938-6_9
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