Statistical Mechanics of the L-Distance Minimal Dominating Set problem

Statistical mechanics is widely applied to solve hard optimization problem, the optimal strategy related to ground state energy that depends on low temperature. Common thermodynamic process is expected to approach the ground state energy if the temperature is lowered appropriately, but this belief is not always justified when the network contains more long loops in low temperature. Previously we always implement the canonical equilibrium process to predict the low-energy, but it doesn't work in L-distance (L>1) minimal dominating set problem, because the thermodynamical process can not guarantee to find the stable state of the system at the low temperature. Here, we employ energy-clamping strategy of cavity method ( micro canonical equilibrium process ) to predict low-energy and discover that the microcanonical process still find the stable state of given system at low temperature where canonical process work out. We develop Belief Propagation Decimation (BPD) and Greedy algorithm to calculate the L-distance ($2<L<7$) minimal dominating set, we find that the BPD algorithm results outperform the Greedy algorithm. We have witnessed the emergence of negative $\beta$ with different mean energy on different L-distance. The free energy has a discontinuous phase transition at $\beta = 0$. We predict the ground state energy by microcanonical cavity method, overcoming the limitation of canonical cavity method.