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

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: 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 Easy to Verify P NP P AND 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 ) = k NTIME(n k )? 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 State Space Start State Accept State Tape Alphabet Transition Function Input Alphabet 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) P = NP = DTIME(n k k ) NTIME(n k k ) 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 (14 th Century) Entia non sunt multiplicanda praeter necessitatem

OCCAM S RAZOR William of Ockham English Franciscan Friar Occam s Razor (14 th 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 S 1 In Out In Out In In S 2 Out In Out Out In Out S 3 Out Out Out Out Out Out S 4 In Out In Out In Out S 5 In In In In In In S 6 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 n 1.8 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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