Qin, SM (reprint author), Chinese Acad Sci, Inst Theoret Phys, Zhong Guan Cun East Rd 55, Beijing 100190, Peoples R China.
A feedback vertex set (FVS) of an undirected graph contains vertices from every cycle of this graph. Constructing a FVS of sufficiently small cardinality is very difficult in the worst cases, but for random graphs this problem can be efficiently solved by converting it into an appropriate spin-glass model [H.-J. Zhou, Eur. Phys. J. B 86, 455 (2013)]. In the present work we study the spin-glass phase transitions and the minimum energy density of the random FVS problem by the first-step replica-symmetry-breaking (1RSB) mean-field theory. For both regular random graphs and Erdos-Renyi graphs, we determine the inverse temperature beta(1) at which the replica-symmetric mean-field theory loses its local stability, the inverse temperature beta(d) of the dynamical (clustering) phase transition, and the inverse temperature beta(s) of the static (condensation) phase transition. These critical inverse temperatures all change with the mean vertex degree in a nonmonotonic way, and beta(d) is distinct from beta(s) for regular random graphs of vertex degrees K > 60, while beta(d) are identical to beta(s) for Erdos-Renyi graphs at least up to mean vertex degree c = 512. We then derive the zero-temperature limit of the 1RSB theory and use it to compute the minimum FVS cardinality.
Physics, Fluids & Plasmas
; Physics, Mathematical
RANDOM REGULAR GRAPHS
National Basic Research Program of China [2013CB932804]
; National Natural Science Foundation of China [11121403, 11225526]
; Scientific Research Starting Foundation of Civil Aviation University of China [2015QD09S]