# Squeeze with difference

Given is a positive integer integer $$n$$, and integers $$a_1,b_1,\dots,a_n,b_n$$ with $$a_i\leq b_i$$ for each $$i$$. What is the complexity of deciding whether there exist integers $$c_1,\dots,c_n$$ such that $$a_i\leq c_i\leq b_i$$ for all $$i$$ and $$|c_i-c_j|\geq 2$$ for all $$i,j$$?

What can be observed is that a greedy algorithm, wherein we assume that $$a_1\leq\dots\leq a_n$$ and choose $$c_i$$ according to this order, does not necessarily work. For example, we may have $$a_1=1, b_1=4, a_2=b_2=2$$. Putting $$c_1=1$$ does not work (it leaves no room for $$c_2$$), but what works is putting $$c_1=4$$ and $$c_2=2$$. Is the problem perhaps NP-hard?