It is well known that constraint satisfaction problems (CSPs) in the phase transition region are most difficult for complete search algorithms. On the other hand, for incomplete hill-climbing algorithms, problems in the phase transition region are more difficult than problems beyond the phase transition region, i.e., more constrained problems. This result seems somewhat unnatural since these more constrained problems have fewer solutions than the phase transition problems. In this paper, we clarify the cause of this paradoxical phenomenon by exhaustively analyzing the state-space landscape of CSPs, which is formed by the evaluation values of states. The analyses show that by adding more constraints, while the number of solutions decreases, the number of local-minima also decreases, thus the number of states that are reachable to solutions increases. Furthermore, the analyses clarify that the decrease in local-minima is caused by a set of interconnected local-minima (basin) being divided into smaller regions by adding more constraints.
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