Judging from the number of similar questions, I've found myself in a rather common situation: I've come up with a problem, encountered a dead end and am now searching for the name of the problem in order to learn more about it.
An example of the problem as follows:
Suppose we have a number of strings of ones and zeroes. For example:
$s_1: \fbox{0}\fbox{1}\fbox{0}\fbox{1}\fbox{0}$
$s_2: \fbox{0}\fbox{0}\fbox{1}\fbox{0}\fbox{1}\fbox{0}\fbox{0}$
$s_3: \fbox{0}\fbox{0}\fbox{0}\fbox{1}\fbox{0}\fbox{0}\fbox{0}\fbox{1}\fbox{0}\fbox{0}\fbox{0}$
Now we construct a new string $H$ by horizontally shifting each sequence $s_n$ by some amount $t_n$ and applying logical disjunction to the columns. What is the longest possible sequence of consecutive ones in the resulting string, if we're allowed to choose $t_n$? By exhaustive search, a solution to this particular problem would be 5, which is given by $t_1=2$; $t_2=2$; $t_3=0$ :
$s_1: \phantom{0000}\fbox{0}\fbox{1}\fbox{0}\fbox{1}\fbox{0}$
$s_2: \phantom{0000}\fbox{0}\fbox{0}\fbox{1}\fbox{0}\fbox{1}\fbox{0}\fbox{0}$
$s_3: \fbox{0}\fbox{0}\fbox{0}\fbox{1}\fbox{0}\fbox{0}\fbox{0}\fbox{1}\fbox{0}\fbox{0}\fbox{0}$
$H: \fbox{0}\fbox{0}\fbox{0}\fbox{1}\fbox{1}\fbox{1}\fbox{1}\fbox{1}\fbox{0}\fbox{0}\fbox{0}\fbox{0}$
Clarification: Shorter strings can extend past the longer ones, as long as the final string has no gaps. In principle, if all strings contained only $1$'s then the maximum stretch would be just $\sum_{i=1}^{n}{\lvert s_i \rvert}$, where $\lvert s_i \rvert$ is the length of $i$-th string.
The closest I could find is the Restricted Strip Covering problem, which is a special case of Geometric Set Covering, but it's not quite what I'm looking for.
Progress: The problem can be stated in terms of discrete job scheduling problem:
Suppose we have a number of employees who can work within a certain time span. Within that time span they may take certain gaps for leisure. Given a set of such employees, what is the maximum stretch of time in which at least one employee is working? The distinction between this problem and job-scheduling problem is important though - finding the optimal values for $t_n$ is NP-Hard, but maybe it's possible to calculate the maximum time without calculating $t_n$ in polynomial time.