Hardness Of Approximation

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Hardness Of Approximation

• An Introduction To PCP

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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?
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Motivation

• Since by now you already know a thing or two
  about complexity, you probably wont 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 doesnt it?

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Preview

• In this lecture well introduce a way of showing
  hardness of approximation problems.
• Well see an NP-hard problem, which can be easily
  reduced to various approximation problems, and
  thus imply they are hard too.

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Optimization Problems

• Many of the problems weve 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?)

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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/cA ? R ? cA,
• where A is the real solution.

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From Approximation To Decision

• A technical issue arises when dealing with
  approximations
• We dont know how to cope with problems other
  than decision problems.
• Lets see what we have done in such cases in the
  past.

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From Maximization To Decision
Is there a clique of size at least k?
Whats the maximal size of clique in this graph?
We could have also solve this.
If we could have solved that...
But this is NP-hard!
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Can you come up with an analog for approximation
problems?
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Heres The Idea

• If we can show its NP-hard to distinguish
  between two far off cases,
• then its 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
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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.

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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,?xj1?j?n,
  the task is to determine whether its
  satisfiable.
• Can you think of an appropriate gap-version?

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Gap-3SAT

• 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