CSE 326: Data Structures: Graphs

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CSE 326 Data Structures Graphs

• Lecture 24Friday, March 7th, 2003

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Today

• Finish NP complete problems
• Course evaluation forms

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NP-Complete Problems

• Recall
• A Hamiltonean path a paths that goes through
  each node exactly once
• How to find one ? Try out all paths.
• Exponential time, and nobody knows better
• Why dont we prove that there is no better
  algorithm ?
• Because we dont know how to prove it

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P and NP

• Recall
• A problem is in P (or PTIME) if we can solve it
  in time O(nk), for some k gt 0
• A problem is in NP if we can check a candidate
  solution in P
• Hamiltonean cycle (HC) is in NP
• In fact, P ? NP
• But what about NP ? P or NP ? P ? Nobody knows.

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Another NP Problem

• SAT Given a formula in Boolean logic, e.g.
• determine if there is an assignment of values to
  the variables that makes the formula true (1).
• Why is it in NP?

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SAT is NP-Complete

• Cook (1971) showed the following
• Including Hamiltonean Cycle (HC)
• In some sense, SAT is the hardest problem in NP
• We say that SAT is NP-Hard
• A problem that is NP-Hard and in NP is called
  NP-complete

Theorem Suppose that we can solve the SAT problem
in polynomial time. Then there is a way to solve
ANY NP problem in polynomial time !!!
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SAT is NP-Complete

• Proof of Cooks theorem
• Suppose we can solve SAT in time O(m7), where m
  is the size of the formula
• Let some other problem in NP we can check a
  candidate solution in, say, time O(n5), where n
  is the size of the problems input

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SAT is NP-Complete Proof

• To solve that other problem, do the following
• We have a program A that checks some candidate
  solution in time O(n5)
• Construct a HUGE boolean formula that represents
  the execution of A its variables are the
  candidate solution (which we dont know) plus all
  memory bits
• Then check if this formula is satisfiable (i.e.
  there exists some candidate solution)

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SAT is NP-Complete Proof
Programcounter
Memory(at most n5 memory words (why ?))
Candidatesolution (unknown)
Input
Time 0


Time 1


Time n5


Answer (0 or 1)
HUGE boolean formula of size O(n5 ? n5 ? n5) ?
check satisfiability in time O((n5 ? n5 ? n5)7)
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The Graph of NP-Completeness

• What is special about SAT ?
• Nothing ! There are hundreds of NP-complete
  problems
• Directed Hamiltonean Path (DHP)
• Vertex Cover
• Clique
• etc, etc, ...

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Directed Hamiltonean Path is NP-Complete

• Proof by reducing SAT to DHP
• Then use transitivity to argue than we can solve
  any NP problem in polynomial time
• Ill show you how to prove the lemma...

Theorem Directed Hamiltonean Path (DHP) is NP
Complete
Lemma If we can solve DHP in polynomial time,
thenwe can solve SAT in polynomial time
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Directed Hamiltonean Path is NP-Complete

• Suppose you are given a boolean formula in
  conjunctive normal form
• Construct a directed graph G s.t. it admits a
  Hamiltonean cycle iff the formula is satisfiable

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c1
c2
c3
Step 1 construct this subgraph
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Step 2 now replicate it once for each boolean
variable
a
b
c
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Step 3 now add a new node for each clause c1,
c2, ...
c1
a
c2
c3
b
c
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Step 4 now connect the variable graphs to the
clause nodes in clever wayE.g. for c2
Right-left for?a
c1
a
c2
c3
b
Left-right for c
c
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c1
c2
c3
Step 5 finally, the formula is satisfiable iff
there exists a Hamiltonean path ! E.g a1, b0,
c1
c1 (true becauseof a)
a?
c1
c2
c3
c2(true becauseof c)
b ?
c1
c2
c3
c3(true becauseof c)
c ?
c1
c2
c3
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c1
c2
c3
E.g a1, b0, c1
c1 (true becauseof a)
a?
c1
c2
c3
c2(true becauseof c)
b ?
c1
c2
c3
c3(true becauseof c)
c ?
c1
c2
c3
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A Great Book You Should Own!

• Computers and Intractability A Guide to the
  Theory of NP-Completeness, by Michael S. Garey
  and David S. Johnson

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Your Chance to Win a Turing Award P NP?

• Nobody knows whether NP ? P
• Proving or disproving this will bring you instant
  fame!
• It is generally believed that P ? NP, i.e. there
  are problems in NP that are not in P
• But no one has been able to show even one such
  problem!
• Practically all of modern complexity theory is
  premised on the assumption that P ? NP
• A very large number of useful problems are in NP

Alan Turing(1912-1954)
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P, NP, and Exponential Time Problems

• All currently known algorithms for NP-complete
  problems run in exponential worst case time
• Finding a polynomial time algorithm for any NPC
  problem would mean
• Diagram depicts relationship between P, NP, and
  EXPTIME (class of problems that provably require
  exponential time to solve)

EXPTIME
NPC
NP
P
It is believed that P ? NP ? EXPTIME
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Coping with NP-Completeness

• Settle for algorithms that are fast on average
  Worst case still takes exponential time, but
  doesnt occur very often.
• But some NP-Complete problems are also
  average-time NP-Complete!
• Settle for fast algorithms that give near-optimal
  solutions But finding even approximate solutions
  to some NP-Complete problems is NP-Complete!
• Just get the exponent as low as possible! Much
  work on exponential algorithms for Boolean
  satisfiability in practice can often solve
  problems with 1,000 variables
• But even 2n/100 will eventual hit the exponential
  curve!