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CSE 326: Data Structures: Graphs

Lecture 24:Friday, 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 don’t we prove that there is no better algorithm ?• Because we don’t know how to prove it

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P and NPRecall:• A problem is in P (or PTIME) if we can solve it in time

O(nk), for some k > 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?

a b a c b c

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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

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

TheoremSuppose 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 Cook’s 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 problem’s 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 don’t 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

Time = 0

Memory(at most n5 memory words (why ?))

Programcounter Input

Candidatesolution (unknown)

Time = 1

Time = n5

Boolean expressionsize = n5 n5

Boolean expressionsize = n5 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

• I’ll show you how to prove the lemma...

Theorem Directed Hamiltonean Path (DHP) is NP CompleteTheorem Directed Hamiltonean Path (DHP) is NP Complete

Lemma If we can solve DHP in polynomial time, thenwe can solve SAT in polynomial time

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

a b a c b c

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a b a c b c c 1 c 2 c 3

c 1 c 2 c 3

Step 1: construct this subgraph

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a b a c b c c 1 c 2 c 3

c 1 c 2 c 3

Step 2: now replicate it once for each boolean variable

c 1 c 2 c 3

c 1 c 2 c 3

a

b

c

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a b a c b c c 1 c 2 c 3

c 1 c 2 c 3

Step 3: now add a new node for each clause c1, c2, ...

c 1 c 2 c 3

c 1 c 2 c 3

a

b

c

c1

c2

c3

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a b a c b c c 1 c 2 c 3

c 1 c 2 c 3

Step 4: now connect the variable graphs to the clause nodes in clever wayE.g. for c2:

c 1 c 2 c 3

c 1 c 2 c 3

a

b

c

c1

c2

c3

Right-left fora

Left-right for c

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a b a c b c c 1 c 2 c 3

c 1 c 2 c 3

Step 5: finally, the formula is satisfiable iff there exists a Hamiltonean path !E.g a=1, b=0, c=1

c 1 c 2 c 3

c 1 c 2 c 3

a

b

c

c1

(true becauseof a)

c2

(true becauseof c)

c3

(true becauseof c)

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a b a c b c c 1 c 2 c 3

c 1 c 2 c 3

E.g a=1, b=0, c=1

c 1 c 2 c 3

c 1 c 2 c 3

a

b

c

c1

(true becauseof a)

c2

(true becauseof c)

c3

(true becauseof c)

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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)

It is believed that P NP EXPTIME

EXPTIME

NP

P

NPC

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Coping with NP-Completeness

1. Settle for algorithms that are fast on average: Worst case still takes exponential time, but doesn’t occur very often. But some NP-Complete problems are also average-time NP-Complete!

2. Settle for fast algorithms that give near-optimal solutions: But finding even approximate solutions to some NP-Complete problems is NP-Complete!

3. 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+ variablesBut even 2n/100 will eventual hit the exponential curve!