CPS 590.4. Computational problems, algorithms, runtime, hardness (a ridiculously brief introduction to theoretical computer science) Vincent Conitzer. Set Cover (a computational problem ). We are given: A finite set S = {1, …, n} A collection of subsets of S: S 1 , S 2 , …, S m
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CPS 590.4
Computational problems, algorithms, runtime, hardness
(a ridiculously brief introduction to theoretical computer science)
Vincent Conitzer
2
3
4
1
6
5
2n
2n2
runtime
run of algorithm 1
run of algorithm 2
Algorithm 1 is O(n2)
(a polynomial-time algorithm)
Algorithm 2 is not O(nk) for any constant k
(not a polynomial-time algorithm)
The problem is in P
n = |x|
To show problem Q is easy:
reduce
Problem known to be easy (e.g., LP)
Q
To show problem Q is (NP-)hard:
reduce
Problem known to be (NP-)hard
(e.g., set cover, (M)IP)
Q
ABSOLUTELY NOT A PROOF OF NP-HARDNESS:
reduce
Q
MIP
2
1
3
, k=4
5
4
6
9
8
7
3
4
5
2
1
6
1
3
7