1. Intro to Theory of Computation
LECTURE 27
Last time
Polynomial-time reductions
NP-completeness
Cook-Levin theorem
Today
Examples of NP-complete problems CS
464
Sofya Raskhodnikova
11/22/2018
S. Raskhodnikova; based on slides by N. Hopper and K. Wayne.

2. The class NP
NP is the class of languages that have polynomial-time verifiers.
11/22/2018

3. I-clicker question (frequency: AC)
Recall: The running time of a verifier V( 〈𝑤,𝑐〉 ) is measured only in terms of length of w.
If we allowed a verifier to run in time polynomial in the length of 𝑤,𝑐 , 𝐭𝐡𝐞 𝐜𝐥𝐚𝐬𝐬 𝐍𝐏 would
be smaller
be the same
contain more (decidable only) languages
contain more (even undecidable) languages
none of the above
t(n)= n log n
3/17/2016

4. I-clicker question (frequency: AC)
3SAT is an example of a problem that cannot be solved by an algorithm.
True
False
It is an open question
None of the above
t(n)= n log n
3/17/2016

5. I-clicker question (frequency: AC)
3SAT is an example of a problem that cannot be solved by a polynomial time algorithm.
True
False
It is an open question
None of the above
t(n)= n log n
3/17/2016

6. A language B is NP-complete if
Hardest problems in NP
A language B is NP-complete if
B∈𝐍𝐏
B is NP-hard,i.e.,
every language in NP is poly-time reducible to B. P NP B
11/22/2018

7. Establishing NP-completeness
Once we establish first "natural" NP-complete problems, others fall like dominoes.
Recipe to establish NP-completeness of problem Y.
Step 1. Show that Y is in NP.
Step 2. Choose an NP-complete problem X (e.g., 3SAT).
Step 3. Prove that X  p Y.
11/22/2018

8. SUBSET-SUM SSUM = { 𝑆,𝑡 ∣𝑆={ 𝑥 1 ,…, 𝑥 𝑟 },
and for some 𝑦 1 ,…, 𝑦 𝑟′ ⊆ 𝑥 1 ,…, 𝑥 𝑟 ,
we have 𝑦 1 +…+ 𝑦 𝑟′ =𝑡}
Repetitions allowed.
Examples: 5, 7, 23 ,46 〉
SSUM ∈ NP
∈𝑆𝑆𝑈𝑀
5, 15, 25 ,46 〉
∉𝑆𝑆𝑈𝑀
Warning: should remove ``repetitions allowed’’. Otherwise, the reduction in the book does not work.
Certificate: 𝑦 1 ,…, 𝑦 𝑟′
11/22/2018

9. 3SAT ≤ 𝒑 SUBSET-SUM SSUM = { 𝑆,𝑡 ∣𝑆={ 𝑥 1 ,…, 𝑥 𝑟 },
and for some 𝑦 1 ,…, 𝑦 𝑟′ ⊆ 𝑥 1 ,…, 𝑥 𝑟 ,
we have 𝑦 1 +…+ 𝑦 𝑟′ =𝑡}
``On input 〈𝜙〉, where 𝜙 is a 3CFN formula with variables 𝑥 1 ,…, 𝑥 ℓ and clauses 𝑐 1 ,…, 𝑐 𝑘 ,
Output a set S of numbers and target number t
(constructed on the board) ’’
11/22/2018

10. TSP (Traveling Salesman)
NP-Completeness
All problems below are NP-complete and hence poly-time reduce to one another!
by definition of NP-completeness
CIRCUIT-SAT
3SAT
3SAT reduces to INDEPENDENT SET
INDEPENDENT SET
HAMPATH
GRAPH 3-COLOR
SUBSET-SUM
Karp analyzed most juicy open problem in discrete math – showed most were NP-complete
x -> y means x reduces to y (if you can solve y, then you can solve x)
Then to join the NP complete club, you need a reduction from SAT (or any other current member) to you
VERTEX COVER
UHAMPATH
PLANAR 3-COLOR
SCHEDULING
SET COVER
TSP (Traveling Salesman)
11/22/2018

11. Some NP-Complete Problems
Six basic genres of NP-complete problems and paradigmatic examples.
Packing problems: SET-PACKING, INDEPENDENT SET.
Covering problems: SET-COVER, VERTEX-COVER.
Constraint satisfaction problems: SAT, 3-SAT.
Sequencing problems: HAMPATH, TSP.
Partitioning problems: 3D-MATCHIN,G 3-COLOR.
Numerical problems: SUBSET-SUM, KNAPSACK.
Most NP problems are either known to be in P or NP-complete.
Notable exceptions. Factoring, graph isomorphism, Nash equilibrium.
Nash equilibria in two-person non-zero-sum game always exist, but major open question to *find* such a solution in poly-time. [Is there a corresponding decision problem or is only the search problem interesting from a computational complexity viewpoint?]
11/22/2018

12. An encyclopedia of NP-complete problems
11/22/2018