A PERSONAL VIEW OF P VERSUS NP

Lance FortnowGeorgia Institute of Technology

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, 2010From: Deolalikar, VinayTo: 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…

NEW YORK TIMES AUGUST 16, 2010

$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

CLAY MATH MILLENNIUM PRIZES

FRIENDS AND ENEMIES

FRIENDS AND ENEMIES OF FRENEMY

DATING SERVICE

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We can efficiently find a matching even among millions of men and women avoiding having to search all the possibilities.

EFFICIENT ALGORITHMS

P

CLIQUE

CLIQUE

CLIQUE

CLIQUE: HARD TO FIND

CLIQUE: EASY TO VERIFY

Given a solution to a clique problem we can check it quickly

EFFICIENTLY VERIFIABLE

NP

EASY TO SOLVE

EASY TO VERIFY

P AND NP

NPP

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?

P = NP?

WRITING ABOUT P AND NP

THE P VERSUS NP PROBLEM

Two views of the problem Mathematical

) = ) ? World View

Can 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

Transition Function

Tape AlphabetBlank Symbol

Input Alphabet

State SpaceStart State

Accept State

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)

P = ) NP = )

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

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

NP-complete 1971: Leonid Levin similar work in Russia 1979: Garey and Johnson publish list of 100’s of

NP-complete problems Now thousands of NP-complete problems over

many disciplines

VERY SHORT HISTORY

OUTSIDE WORLD OF P VERSUS NP

WHAT HAPPENS IF P = NP?

WECURE

CANCER

CURING CANCER

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”.

OCCAM’S RAZOR

Rosetta Stone 196 BC Decree in three

languages• Greek• Deomotic• Hieroglyphic In 1822, Jean-François

Champollion found a simple grammar.

TRANSLATION

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?

Brute Force Heuristics Small Parameters Approximation Solve a Different Problem Give Up

DEALING WITH HARDNESS

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 Out Out Out In

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

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