Abstract
Substrate cycles in metabolic networks play a role in various forms of homeostatic regulation, ranging from thermogenesis to the buffering and redistribution of steady-state populations of metabolites. While the general problem of enumerating these cycles is \(\#P\)-hard, it is unclear if this result holds for realistic networks where e.g. pathological vertex degree distributions or minors may not exist. We attempt to address this gap by showing that the problem of counting directed substrate cycles (\(\#DirectedCycle\)) remains \(\#P\)-complete (implying \(\#P\)-hardness for enumeration) for any superclass of cubic weakly-3-connected bipartite planar digraphs, and at the limit where all reactions are reversible, that the problem of counting undirected substrate cycles (\(\#UndirectedCycle\)) is \(\#P\)-complete for any superclass of cubic 3-connected bipartite planar graphs where the problem of counting Hamiltonian cycles is \(\#P\)-complete. Lastly, we show that unless \(NP=RP\), no FPRAS can exist for either counting problem whenever the Hamiltonian cycle decision problem is NP-complete.
The original version of this chapter was revised: Incorrect capitalization has been corrected. The erratum to this chapter is available at 10.1007/978-3-319-58741-7_37
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Barish, R.D., Suyama, A. (2017). Counting Substrate Cycles in Topologically Restricted Metabolic Networks. In: Kari, J., Manea, F., Petre, I. (eds) Unveiling Dynamics and Complexity. CiE 2017. Lecture Notes in Computer Science(), vol 10307. Springer, Cham. https://doi.org/10.1007/978-3-319-58741-7_14
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