1. Reductions
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2. Motivation
Reductions are our main tool for comparison between problems.
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3. Introduction Objectives: To formalize the notion of “reductions”.
Overview:
Karp reductions
HAMPATH p HAMCYCLE
Closeness under reductions
Cook reductions
Completeness
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4. Reductions – Concept p
IF problem A can be efficiently translated to problem B… A B p
THEN B is at least as hard as A.
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5. Reductions – Formal Definition
SIP 250
Reductions – Formal Definition
I.e – there exists a polynomial time TM which halts with f(w) on its tape when given input w.
Definition: Language A is polynomial time reducible to language B if…
a polynomial time computable function f:** exists,
where for every w, wA  f(w)B
Denote Ap B
f is referred to as a polynomial-time reduction of A to B.
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6. Reductions and Algorithms* B
* - B A f
* - A *
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7. Reductions and Algorithms
polynomial time algorithm for A f
polynomial time algorithm for B
f(w)B?
f(w)
wA? w
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8. How to Reduce? Construct f. Show f is polynomial time computable.
Prove f is a reduction, i.e show:
If wA then f(w)B
If f(w)B then wA
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9. Terminology
The type of reductions we’ve just described is also called:
Polynomial-time mapping reduction
Polynomial-time many-one reduction
Polynomial-time Karp reduction
We’ll always refer to such reductions unless otherwise specified.
Complexity ©D.Moshkovitz

10. Example
To demonstrate a reduction, we’ll revisit the example we gave in the introduction.
We’ll rephrase and formalize the seating and tour problems
And demonstrate a slightly different reduction between them.
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11. Hamiltonian Path
Instance: a directed graph G=(V,E) and two vertices stV.
Problem: To decide if there exists a path from s to t, which goes through each node once.
In the version presented in the introduction we asked for some path, without specifying the start and end vertices.
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12. Hamiltonian Cycle Instance: a directed graph G=(V,E).
Problem: To decide if there exists a simple cycle in the graph which goes through each node exactly once.
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13. HAMPATH p HAMCYCLE
f( <G=(V,E),s,t> ) = <G’=(V{u},E{(u,s),(t,u)})>
f( <G=(V,E),s,t> ) = <G’=(V{u},E{(u,s),(t,u)})>
f( <G=(V,E),s,t> ) = <G’=(V{u},E{(u,s),(t,u)})>
f( <G=(V,E),s,t> ) = <G’=(V{u},E{(u,s),(t,u)})> s t n s t p
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14. Correctness
(Completeness) If there exists a Hamiltonian path (v0=s,v1,…,vn=t) in the original graph, then (u,v0=s,v1,…,vn=t,u) is a Hamiltonian cycle in the new graph.
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15. Correctness
(Soundness) (u,s) and (t,u) must be in any Hamiltonian cycle in the constructed graph, thus removing u yields a Hamiltonian path from s to t.
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16.  How to Reduce?     Construct f.
Show f is polynomial time computable.
Prove f is a reduction, i.e show:
If wHAMPATH then f(w)HAMCYCLE
If f(w)HAMCYCLE then wHAMPATH 
easy to verify   
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17. Closeness Under Reductions
Definition: A complexity class C is closed under reductions if, whenever L is reducible to L’ and L’C, then L is also in C. C L’  L
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18. Observation
Theorem: P, NP, PSPACE and EXPTIME are closed under polynomial-time Karp reductions.
Proof: Filling in the exact details is left to the reader. (See sketch)
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19. f is referred to as a log-space reduction of A to B.
Log-Space Reductions
I.e – there exists a log-space TM which halts with f(w) on its tape when given input w.
Definition: Language A is log-space reducible to language B if…
a log-space computable function f:** exists,
Denote AL B
where for every w,
wA  f(w)B
f is referred to as a log-space reduction of A to B.
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20. Observation
Theorem: L, NL, P, NP, PSPACE and EXPTIME are closed under log-space reductions.
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21. Other Types of Reductions
Cook Reduction: Use an efficient “black box” (procedure) that decides B, to construct a polynomial-time machine which decides A.
polynomial time algorithm for B
polynomial time algorithm for A
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22. Cook reduction of HAMCYCLE to HAMPATH.
For each edge (u,v)E,
if there’s a Hamiltonian path from v to u in the graph where (u,v) is removed, output ‘YES’, exit
Output ‘NO’
Make sure it’s efficient!
HAMPATH oracle
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23. How do Cook and Karp Reductions Differ?
Cook reductions allow us to call the procedure many times (instead of 1).
When Karp reducing, we must output the answer of the procedure.
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24. On the Relation Between Cook and Karp Reductions
Observation: Karp reductions are special cases of Cook reductions.
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25. Food for Thought…
Which complexity classes are closed under Cook reductions?
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26. Completeness
Definition: Let C be a complexity class and let L be a language. We say L is C-Complete if:
L is in C
For every language AC, A is reducible to L.
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27. C-Complete Languages
hardness
C-Complete C
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28. Observation
Theorem: If two classes C and C’ are both closed under reductions (of some specific type) and there is a language L which is complete for both C and C’, then C=C’.
Proof: W.l.o.g we’ll only show CC’. All languages in C are reducible to L. Since C’ is closed under reductions, it follows that CC’. 
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29. Corollary ? If any NP-Complete problem is in P, then P=NP.
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30.  Summary In this lecture we’ve introduced many types of reductions:
Cook/Karp reductions
Polynomial-time reductions
We’ve defined the notion of completeness
And seen how it can help us prove equality of complexity classes.
Nevertheless, we are still not convinced complete languages even exist.
Complexity ©D.Moshkovitz