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