Chapter 1 The Self-Reducibility Technique Matt Boutell and Bill Scherer CSC 486 April 4, 2001

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Chapter 1 The Self-Reducibility Technique Matt Boutell and Bill Scherer CSC 486 April 4, 2001. Historical Perspective. [Berman 1978]: P=NP   a tally set that is  m -hard for NP [Mahaney 1982]: P=NP   a sparse set that is  m -complete for NP

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Presentation Transcript
Historical Perspective

[Berman 1978]: P=NP 

 a tally set that is m-hard for NP

[Mahaney 1982]: P=NP 

 a sparse set that is m-complete for NP

[Ogiwara, Watanabe 1991]: today’s lecture

p

p

Proof Overview

p

Theorem: If  an NP btt-hard sparse set S,

then P = NP.

Technique: let L be an arbitrary language in NP. Then, using S and the reduction, we give a deterministic polynomial algorithm to decide L.

x

Accept

An Alternate Characterization of the Class NP

A language LNP  [AP, polynomial p | x*, xL

IFF (w)[wp(|x|)  x,wA].

x = input

w = witness = certificate = accepting path:

A = checking algorithm

x

Accept

Left Sets

The left set, denoted Left[A,p], is {x,y | x* 

yp(|x|)  (wp(|x|))[w lex y  x,wA]}.

Note that having the left set non-empty  the

existence of an accepting path.

Maximum Witnesses

The Maximum Witness for some input x, denoted

wmax(x), is max{y | yp(|x|)  x,yA]}.

Deciding xL  Determining if wmax(x) is defined.

(x*)(yp(|x|))[x,yLeft[A,p]  ylex wmax(x)].

(1.4)

Left[A,p]  NP

Left[A,p]  NP (by guessing wmax(x)), so since S

is NP-hard, Left[A,p] btt S via some function f.

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What does btt mean?

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Bounded truth table reductions, btt, are a type of

reduction that uses a very weak form of oracle.

Ak-ttB via some function f means that A = L(MB),

where M is a deterministic polynomial machine that on

input x precomputes up to k queries, asks them all in

parallel, and uses a k-ary Boolean function to compute

the output.

v1v2 v3 … vk

0 0 0 … 0 Yes

M: 0 0 0 … 1 No

1 1 1 … 1 No

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Back to the Proof…

Let u be a pair x,y. Then with our reduction,

uLeft[A,p]  S satisfies f(u).

Now, with query strings v1, v2, v3, … vk, let

(S(v1), S(v2), S(v3), … S(vk))  {Yes, No};

this line is a row in the truth table in our

reduction to S.

So f(u) is of the form , v1, v2, v3, … vk.

Intervals

The trick we will use is to generate a polynomially

bounded list of candidates for wmax(x). Once this list

is generated, we can use brute force computation to

see if any of these candidates are in fact witnesses.

We do this by keeping track of a set of pair-wise

disjoint intervals  in the range 0p(|X|)..1p(|X|), starting

initially with the entire range.

wmax(x)

The Interval Invariant

xL  wmax(x)UII (1.5)

wmax(x)

Last Definition: Covering

Let 1, 2 be two collections of pair-wise disjoint

intervals over p(|x|). Then 1 covers 2 with respect

to x if:

1) (I2) (J1)[IJ]

2) wmax(x)  UI1 wmax(x)  UI2

0

1

2

wmax(x)

Facts With Covering

Let 1, 2, 3, 4 be sets of pair-wise disjoint

intervals over p(|x|). Then (all with respect to x):

1) If the interval invariant holds for 1 and

2 is a cover of 1, it also holds for 2.

2) If 2 covers 1 and 3 covers 2, then 3

covers 1.

3) If 2 covers 1 and 4 covers 3, then 2U4

covers 1U3.

0

1

2

The Theorem, Restated

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If  an NP btt-hard sparse set S,

then P = NP.