Existing random models for the constraint satisfaction problem (CSP) all require an extremely low constraint tightness in order to have non-trivial threshold behaviors and guaranteed hard instances at the threshold. We study the possibility of designing random CSP models that have interesting threshold and typical-case complexity behaviors while at the same time, allow a much higher constraint tightness. We show that random CSP models that enforce the constraint consistency have guaranteed exponential resolution complexity without putting much restriction on the constraint tightness. A new random CSP model is proposed to generate random CSPs with a high tightness whose instances are always consistent. Initial experimental results are also reported to illustrate the sensitivity of instance hardness to the constraint tightness in classical CSP models and to evaluate the proposed new random CSP model.
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