Use inclusion-exclusion to count set covers in O(2^m poly(n))
Analyze the use inclusion-exclusion to count set covers in o(2^m poly(n)).
Examples
Input:"test_input_1"
Output:"output_1"
Input:"test_input_2"
Output:"output_2"
Hints
Start by recalling the inclusion-exclusion principle and how it can be applied to count the number of set covers by considering the union of all possible combinations of sets.
Consider the exponential-time dynamic programming approach for set cover, then optimize it using inclusion-exclusion to reduce the complexity to o(2^m) while maintaining polynomial time in n.
Use Möbius inversion over the subset lattice of the universe to derive a closed-form expression for the number of set covers, then analyze its computational feasibility to achieve o(2^m) poly(n) time.
Use inclusion-exclusion to count set covers in O(2^m poly(n))
Analyze the use inclusion-exclusion to count set covers in o(2^m poly(n)).