HackerRank: ACM ICPC Team
You're spending a lot of time in item_value converting numbers to a binary string back to an int to get the sum. It would be a lot more expedient to just use str.count to get the number of 1s in the string. That saves you a lot of type conversions as well as a call to sum with a generator.

def item_value(e1, e2):
    c = bin(int(e1, 2) | int(e2, 2))
    return c[2:].count('1')


My results from this caused a reduction from 58.9s to 0.91s.