Vertex cover problem studied by cavity method: Analytics and population dynamics

European Physical Journal B

Volumn 32, Issue 2, 2003, Pages 265-270

  Zhou, Haijun ^a  

^a MAX PLANCK INSTITUTE OF COLLOIDS AND INTERFACES   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

FREE ENERGY; FUNCTIONS; GROUND STATE; MATHEMATICAL MODELS; STATISTICAL METHODS; THERMAL EFFECTS;

FINITE CONNECTIVITY; ISING SPIN MODEL; VERTEX COVER PROBLEM;

PROBLEM SOLVING;

EID: 7244256985     PISSN: 14346028     EISSN: None     Source Type: Journal    
DOI: 10.1140/epjb/e2003-00096-4     Document Type: Article

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3. The cavity method as outlined in reference [2] is exact for a tree. If loops exist in the graph, this method is an approximation and is equivalent to the first-step replica symmetry breaking (RSB) solution. For the vertex cover problem, the present work suggests that one needs to go beyond this first-step RSB solution. To Extend the cavity formalism is both challenging and important
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20. j = -1 and the total energy will be lowered. However, choosing λ ≃ 1 might facilitate a search algorithm in finding the true energy minimum
21. Some more discussion about the cavity fields. At zero temperature, each state of the system corresponds to an ensemble of spin microscopic configurations. All these configurations have the same energy (corresponding to a minimum value of the energy functional Eq. (4)), and only finite number of spin flips is needed to transit from one to another configuration of them. The spins on some of the vertices are fixed at σ = +1 or σ = -1 in all these configurations, while the spins on the remaining vertices are not fixed. We say that those spins whose value fluctuates from one configuration to another do not feel any magnetic field in this state (h = 0). A spin whose value is fixed feels a non-zero magnetic field. The magnitude of this non-zero cavity field is defined as half the total increase in minimum energy associated with flipping this spin. For our present problem, an energy local minimum has value 2N↓ -N, where N↓, is the total number of negatively valued spins. Therefore, we conclude that (1) all the cavity fields must be integer-valued, and (2) the cavity fields could not exceed unity (h ≤ 1)
22. 2dφ(y)/dy changes sign from positive to negative as y is increased beyond some value. Therefore the largest y value is obtained by the equation dφ(y)/dy = 0, which corresponds to the maximum of φ(y)
23. c. = 0.05) leads to much bigger value
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