1. Hardness Of Approximation
An Introduction To PCP

2. Motivation
Say you want to put together a team.
You have a group of people,
So for every two you can tell if they get along.
Each team member must get along with every other team-member.
Now say there are 100 candidates, and you should find a team of maximum size.
How do you like the job?

3. Motivation
Since by now you already know a thing or two about complexity, you probably won’t even bother to try.
After all, you know CLIQUE is NP-hard.
But what if you are willing to compromise?
What if you can put up with a team whose size is not maximal, but close enough,
say, half the size of the optimal solution.
This seems easier, or doesn’t it?

4. Preview
In this lecture we’ll introduce a way of showing hardness of approximation problems.
We’ll see an NP-hard problem, which can be easily reduced to various approximation problems, and thus imply they are hard too.

5. Optimization Problems
Many of the problems we’ve encountered are actually optimization problems,
i.e. problems in which the aim is to find a maximal/minimal solution.
For instance, MAX-CLIQUE, MIN-VERTEX-COVER, MAX-CUT etc.. (Do you remember others?)

6. Approximation Problems
For such there are natural approximation versions,
usually taking the form of: “Approximate the solution within factor c”.
This means we wish to output an answer R, which satisfies:
1/c·A  R  c·A,
where A is the real solution.

7. From Approximation To Decision
A technical issue arises when dealing with approximations:
We don’t know how to cope with problems other than decision problems.
Let’s see what we have done in such cases in the past.

8. From Maximization To Decision
“What’s the maximal size of clique in this graph?”
“Is there a clique of size at least k?”
If we could have solved that...
We could have also solve this.
But this is NP-hard!

9. Can you come up with an analog for approximation problems?

10. Here’s The Idea
If we can show it’s NP-hard to distinguish between two far off cases,
then it’s also hard to even approximate the solution.
the size of the max-clique is tremendously big
the size of the max-clique is extremely small

11. Gap-Problems Such problems are called gap-problems.
There are two disjoint, but not exhausting, cases,
The aim is to say if the instance belongs to either one of them.
If the instance belongs to neither, any answer is acceptable.

12. A Gap Version For SAT As usual, our starting point is 3SAT.
Given a 3CNF formula, i.e a formula of the form
(123)...(m/3-2m/3-1m/3)
where each literal i{xj,xj}1jn, the task is to determine whether it’s satisfiable.
Can you think of an appropriate gap-version?

13. Gap-3SAT YES NO Definition (Gap-3SAT):
Instance: a 3CNF-formula with m clauses and n variables.
Problem: to distinguish between:
The formula is satisfiable
No more than an  fraction of the clauses can be satisfied simultaneously.
YES NO