A combinatorial approach to P vs NP Shachar Lovett Computation Input Memory Program Code Program code is constant Input has variable length n Run time memory grow with input length ID: 550222
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Slide1
Circuit Lower BoundsA combinatorial approach to P vs NP
Shachar
LovettSlide2
Computation
Input
Memory
Program
Code
Program code is
constant
Input has
variable length (n)
Run time, memory – grow with input length
Efficient algorithms = run time, memory
poly(n)Slide3
P vs NPP = problems we can solve
= efficient algorithm to
find solution
NP = problems we want to solve
= efficient algorithm to verify solutionExamples: graph 3-coloring, satisfiability ,…Slide4
ChallengeHow can you prove that some computational problems require >> polynomial time?In particular, one in NP
Combinatorial approach:
circuits
Replace “uniform computation” by a more combinatorial object
Slide5
CircuitsComplex computation = iteration of many small
simple
computations
x
Y
Z
AND
AND
AND
OR
Majority(X,Y,Z)Slide6
CircuitsComplex computation = iteration of many small
simple
computations
Simple
= any complete basis (e.g. AND,OR,NOT)
x1x2
x
3
x
4
x
5
x
6
x
7
x
8
…
x
n
f(X
1
,…,
X
n
)Slide7
Algorithms vs circuits
Circuits are as powerful
*
as algorithms:
Problems with efficient (poly-time) algorithms also have poly-size circuitsRevised challenge: show poly-size circuits cannot solve all interesting computational problems
Input
Memory
Code
x
1
x
2
x
3
x
4
x
5
x
6
x
7
x
8
…
x
n
f(X
1
,…,
X
n
)Slide8
Lower boundsGoal: show poly-size circuits cannot solve NP
Can prove lower bounds for restricted circuit models
Monotone circuits
Bounded depth circuits
General technique: Approximate circuit by a nice mathematical model Show the
mathematical model cannot solve the problem (not even approximately)Slide9
Monotone circuitsMonotone circuits
: circuits with just
AND-OR
gates (no NOT gates)
Compute monotone functions (e.g clique)Can clique have poly-size
monotone circuits?[Razborov’85, Alon-Boppana’87]: No. Clique requires exponential size monotone circuitsSlide10
Monotone circuits
x
1
x
2
x3x4
x
5
x
6
x
7
x
8
…
x
n
f(X
1
,…,
X
n
)
Input: n edges of
graph G
on m
n
1/2
vertices
Output: does G have large clique?
Circuit: poly-size with
AND-OR
gates
Step 1: approximate
AND-OR
circuit by
lattice
Step 2: show
lattice
cannot approximate cliqueSlide11
Bounded depth circuits
x
1
x
2
x3x4
x
5
x
6
x
7
x
8
…
x
n
f(X
1
,…,
X
n
)
Small
depth
= parallel computation
Efficient algorithms =
poly(n) depth
Can prove lower bounds for
depth << log(n
)
depthSlide12
Lower bounds for AND-OR-NOT circuitsParity(x
1
,…,
xn
) = sum of bits modulo 2Computed by small AND-OR-NOT circuits of depth log(n)Can the depth be reduced, while maintaining small size?
[Ajtai’83, Furst-Saxe-Sipser’84]: No. small (sub-exponential) AND-OR-NOT circuits of depth <<log(n) cannot compute parity[Yao’85, Hastad’86]: not even approximatelySlide13
Lower bounds for AND-OR-NOT circuitsMain idea:
random restrictions
of input
set
most inputs bits to random 0,1 values; leave remaining variables “alive”Simple computations: AND, OR, NOTGates with many inputs are
fixed by random restrictionIterate to make entire circuit simple (decision tree)Parity doesn’t simplify (becomes parity of fewer inputs)
X
1
AND
X
n
…Slide14
Lower bounds for AND-OR-NOT-PARITY circuitsWhat if we also allow parity gates as simple computations?
MOD
3
(x
1,…,xn) = sum of bits modulo 3Intuition: parities shouldn’t help compute MOD3[Razborov’87, Smolensky’87]: small (sub-exponential)
AND-OR-NOT-PARITY circuits of depth <<log(n) cannot compute MOD3Slide15
Lower bounds for AND-OR-NOT-PARITY circuitsLocal computation:
AND, OR, NOT, PARITY
Random restrictions fail: don’t simplify parity
Can approximate local computations by
low-degree polynomials modulo 2 (and by composition, approximate the entire circuit)Low degree polynomials modulo 2 cannot compute MOD3Slide16
Lower bounds for AND-OR-NOT-PARITY-MOD3 circuits
What if we allow both
PARITY
and
MOD3 gates as simple computation?Conjecture: cannot compute MOD5
in small size and depth <<log(n)[Williams’10]: cannot compute all NEXP - exponential analog of NP (problems whose solution can be verified in exponential time)