# Local consistency and constraint propagation for intervals

I would like to solve constraint satisfaction problem for intervals. I denote each interval by coordinate of its begin $x_i$. The length of the interval is known convex function $l_i(x_i)$ of begin of the interval. There are known domains for each interval $x_i \in D_i$. Constraints are that intervals do not overlap. (i.e. $x_j > x_i + l_i(x_i)$ when $x_j>x_i$). The problem is to allocate the intervals over domains $D_i$ which overlap partially.

The question is what kind of special types of local consistency may help me? Enforcing arc- and path- consistences won't help me much as they involve only pairs and triples of variables and this won't shrink domains efficiently. But it is obvious that if total length of minimal lengths of intervals $\sum_{i} \min_{x_i} l_i(x_i)$ greater than length of $\cup_i D_i$ then the problem is inconsistent.

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I don't understand why you have $\min$ in the sum. – user31373 Jul 28 '12 at 14:09
@LeonidKovalev, this is lower bound of length required to allocate all intervals. Virtually, the sum is greater than that because we can't put each interval to its optimal position generally. – 0x2207 Jul 29 '12 at 7:26