Lance Fortnow
Georgia Institute of Technology
A PERSONAL VIEW OF P VERSUS NP
NEW YORK TIMES AUGUST 16, 2010
Step 1: Post Elusive Proof. Step 2: Watch Fireworks.
By John Markoff
… Vinay Deolalikar, a mathematician and electrical
engineer at Hewlett-Packard, posted a proposed proof of
what is known as the “P versus NP” problem on a Web site,
and quietly notified a number of the key researchers.
Email: August 6, 2010
From: Deolalikar, Vinay
To: 22 people
Dear Fellow Researchers,
I am pleased to announce a proof that P is not equal to NP,
which is attached in 10pt and 12pt fonts…
CLAY MATH MILLENNIUM PRIZES
$1 Million Award for solving any of these
problems.
Birch
and Swinnerton-Dyer Conjecture
Hodge Conjecture
Navier-Stokes Equations
P vs NP
Poincaré Conjecture
Riemann Hypothesis
Yang-Mills Theory
FRIENDS AND ENEMIES
FRIENDS AND ENEMIES OF FRENEMY
DATING SERVICE
DATING SERVICE
DATING SERVICE
DATING SERVICE
DATING SERVICE
DATING SERVICE
DATING SERVICE
DATING SERVICE
DATING SERVICE
EFFICIENT ALGORITHMS
We can efficiently
find a matching
even among
millions of men
and women
avoiding having to
search all the
possibilities.
P
CLIQUE
CLIQUE
CLIQUE
CLIQUE: HARD TO FIND
CLIQUE: EASY TO VERIFY
EFFICIENTLY VERIFIABLE
Given a solution to a clique problem we can
check it quickly
NP
Easy to Solve
P
P AND NP
Easy to Verify
NP
P = NP
EVERY PROBLEM WE CAN VERIFY EFFICIENTLY WE CAN SOLVE EFFICIENTLY
P ≠ NP
THERE ARE PROBLEMS WE CAN VERIFY QUICKLY THAT WE CAN’T SOLVE QUICKLY
?
P = NP
CAN WE SOLVE EVERY PROBLEM QUICKLY IF THE SOLUTIONS ARE EASILY
VERIFIABLE?
WRITING ABOUT P AND NP
THE P VERSUS NP PROBLEM
Two views of the problem
Mathematical
𝑘) =
𝐷𝑇𝐼𝑀𝐸(𝑛
𝑘
World
Can
𝑘) ?
𝑁𝑇𝐼𝑀𝐸(𝑛
𝑘
View
we “efficiently” solve all problems where we can
“efficiently” check the solutions?
How does the world change if P = NP?
How do we deal with hard problems if P ≠ NP?
MATHEMATICAL VIEW OF P VS NP
TURING MACHINE
FORMALIZING THE TURING MACHINE
State Space
Start State
Accept State
Input Alphabet
Tape Alphabet
Transition Function
Blank Symbol
TRANSITIONS
Transition function
(state,
symbol) →(state, symbol, direction)
Nondeterministic
Can
map to multiple possibilities
DEFINING P AND NP
DTIME(t(n)) is the set of languages accepted by
deterministic Turing machines in time t(n)
NTIME(t(n)) is the set of languages accepted by
nondeterministic Turing machines in time t(n)
𝑘
= 𝑘 𝐷𝑇𝐼𝑀𝐸(𝑛 )
𝑘
NP = 𝑘 𝑁𝑇𝐼𝑀𝐸(𝑛 )
P
Does P = NP?
MATHEMATICALLY ROBUST
Instead of Turing machine
Multiple
tapes
Random access
λ – calculus
C++
LaTeX
Probabilistic and Quantum computers might
not define the same class
REDUCTIONS
A
B
NP-COMPLETE
Hardest problems in NP
Cook-Levin 1971
Boolean
Formula Satisfiability
u v w u w x v w x
NP-COMPLETE
VERY SHORT HISTORY
1935: Turing’s Machine
1962: Hartmanis-Stearns: Computation time depends on
size of problem
1966: Edmonds, Cobham: Models of efficient computation
1971: Steve Cook defines first NP-complete problem
1972: Richard Karp shows 22 common problems NPcomplete
1971: Leonid Levin similar work in Russia
1979: Garey and Johnson publish list of 100’s of NPcomplete problems
Now thousands of NP-complete problems over many
disciplines
OUTSIDE WORLD OF P VERSUS NP
WHAT HAPPENS IF P = NP?
WE
CURE
CANCER
CURING CANCER
OCCAM’S RAZOR
William of Ockham,
English Franciscan Friar
Occam’s Razor (14th Century)
Entia non sunt multiplicanda
praeter necessitatem
OCCAM’S RAZOR
William of Ockham
English Franciscan Friar
Occam’s Razor (14th Century)
Entities must not be multiplied
beyond necessity
The simplest explanation is
usually the best.
If P = NP we can find that
“simplest explanation”.
TRANSLATION
Rosetta Stone
196 BC Decree in three
languages
• Greek
• Deomotic
• Hieroglyphic
In 1822, Jean-François
Champollion found a
simple grammar.
MACHINE LEARNING
IF P = NP
IF P NP: CRYPTOGRAPHY
IF P NP: ZERO-KNOWLEDGE PROOFS
DEALING WITH HARDNESS
How do you deal with NP-completeness?
DEALING WITH HARDNESS
Brute Force
Heuristics
Small Parameters
Approximation
Solve a Different Problem
Give Up
HOW DO WE PROVE P ≠ NP?
WHAT DOESN’T WORK?
DIAGONALIZATION
1
2
3
4
5
6
S1
In
Out
In
Out
In
In
S2
Out
In
Out Out
In
Out
S3
Out Out Out Out Out Out
S4
In
Out
In
Out
In
Out
S5
In
In
In
In
In
In
S6
Out
In
In
Out Out Out
DIAGONALIZATION
NP doesn’t have enough power to simulate P
Relativized world where P = NP.
Can get weaker time/space results:
No
algorithm for satisfiability that uses logarithmic
space and n1.8 time.
CIRCUIT COMPLEXITY
Measure complexity by size of circuit.
Different circuits for each input length.
Efficient computation essentially equivalent to small
circuits.
CIRCUITS
Idea: Show no single gate changes things much
so needs lots of gates for NP-complete
problems
Works for circuits of limited depth or negations.
“Natural Proofs” give some limitations on this
technique.
PROOF COMPLEXITY
( x AND y ) OR (NOT x) OR (Not y)
If P = NP (or even NP = co-NP) then every
tautology has a short proof.
Try to show tautologies only have long proofs.
Works only for limited proof systems like
resolution.
THE FUTURE OF P V NP
THE GOLDEN TICKET
goldenticket.fortnow.com
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