I came up with the following problem, but do not know how to analyze it.

Let $S$ be an ordered set of integers with size $n$ (i.e., $S=\{1,2,...,n\}$). An interval $INV(a,b)$ covers the elements in $S$ from $a$ to $b$ $(1 \leq a < b \leq n)$. For example, $INV(3,5)$ covers the element of $\{3,4,5\}$. Now I am trying to analyze the following problem:

Given a set of intervals $I$, select $k$ intervals $R$ from $I$ (i.e., $R\subseteq I$ and $|R|=k$), such that $R$ covers the most of $S$.

I am also interesting in a more generalized version where each element has a weight. But I wish to analyze the simpler version first. Specifically, I wish to know whether there exists polynomial time (wrt. $n$) for this problem. If not, is this problem NP-hard, and how to prove it ?

Any hints or discussions would be very helpful.

Thank you.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.