Slides23 - NPblerner/cs312/Slides/Slides23.pdf · 2019-04-29 · 51, 17 51, 16 RELPRIME Are x and y...

End of Year Celebration!!

Thursday, May 2, 12:00

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Gabi1-2



Gabi

Emilyann

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Slides23 - NP.key - April 26, 2019

Practically solvable?

Yes Probably no

Shortest path Longest path

Min cut Max cut

2-SAT 3-SAT

Bipartite vertex cover Vertex cover

Primality testing Factoring

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Classify problems according to relative difficulty.

Design algorithms. If Y ≤ P X and X can be solved in polynomial-time, then Y can also be solved in polynomial time.

Establish intractability. If Y ≤ P X and Y cannot be solved in polynomial-time, then X cannot be solved in polynomial time.

Establish equivalence. If X ≤ P Y and Y ≤ P X, we use notation X ≡ P Y.

Polynomial-Time Reduction3-1

Today’s Focus

Classify problems according to relative difficulty.

Design algorithms. If Y ≤ P X and X can be solved in polynomial-time, then Y can also be solved in polynomial time.

Establish intractability. If Y ≤ P X and Y cannot be solved in polynomial-time, then X cannot be solved in polynomial time.

Establish equivalence. If X ≤ P Y and Y ≤ P X, we use notation X ≡ P Y.

Polynomial-Time Reduction3-2


Self-Reducibility

Decision problem. Does there exist a vertex cover of size ≤ k?

Search problem. Find vertex cover of minimum cardinality.

Self-reducibility. Search problem ≤ P decision version.Applies to all (NP-complete) problems in this chapter.Justifies our focus on decision problems.

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Definition of PP: Decision problems for which there is a polynomial algorithm.

Problem Description Algorithm Yes No

MULTIPLE Is x a multiple of y?Grade school

division 51, 17 51, 16

RELPRIME Are x and y relatively prime?

Euclid (300 BCE) 34, 39 34, 51

PRIMES Is x prime? AKS (2002) 53 51

EDIT-DISTANCE

Is the edit distance between x and y less than

5?

Dynamic programming

niether neither

acgggt ttttta

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P, NP

P. Decision problems for which there is a polynomial time algorithm.

NP. Decision problems for which there is a polynomial time certifier.

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Solving vs. CertifyingConsider the independent set problem:

Solving => Find an independent set of size k for a graph G indep-set = SolveIndependentSet (G, k)

Certifying => Given a graph G and a set, certify whether the set is an independent set of size k is-independent = CertifyIndependentSet (G, k, indep-set)

No known polynomial time solution to solve independent setA purported solution to independent set can be certified in polynomial time

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Certifiers and Certificates: Non-Prime

COMPOSITES. Given an integer s, is s non-prime?

Certificate. A nontrivial factor t of s. Note that such a certificate exists iff s is non-prime. Moreover t ≤ s.

Certifier:

boolean C(s, t) { if (t ≤ 1 or t ≥ s) return false else if (s is a multiple of t) return true else return false }

Instance: s = 437,669.

Certificate: t = 541 or 809.

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Certifiers and Certificates: 3-Satisfiability

SAT. Given a CNF formula, is there a satisfying assignment?

(x1 ⋁ x2 ⋁ x3) ⋀ (x1 ⋁ x2 ⋁ x3) ⋀ (x1 ⋁ x2 ⋁ x4) ⋀ (x1 ⋁ x3 ⋁ x4)

Certificate. An assignment of truth values to the n boolean variables.

x1 = true, x2 = true, x3 = false, x4 = true

Certifier. Check that each clause has at least one true literal.

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Certifiers and Certificates: Hamiltonian Cycle

HAM-CYCLE. Given an undirected graph G = (V, E), does there exist a simple cycle C that visits every node?

Certificate. A permutation of the n nodes.

instance s

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Certifiers and Certificates: Hamiltonian Cycle

HAM-CYCLE. Given an undirected graph G = (V, E), does there exist a simple cycle C that visits every node?

Certificate. A permutation of the n nodes.

instance s

certificate t

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P, NP

Claim. P ⊆ NP.

Proof. Consider any problem X in P.By definition, there exists a polynomial time algorithm A(s) that solves X.Certificate: t = ε, certifier C(s, t) = A(s). ▪

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The Main Question: P Versus NP

Does P = NP?

Is the decision problem as easy as the certification problem?

Clay $1 million prize.

If yes: Efficient algorithms for 3-SAT, vertex cover, …

If no: No efficient algorithms possible for 3-SAT, vertex cover, …

EXP NPP

EXPP = NP

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“ The classes of problems which are respectively known and not known to have good algorithms are of great theoretical interest…. I conjecture that there is no good algorithm for the traveling salesman problem. My reasons are the same as for any mathematical conjecture:

(i) It is a legitimate mathematical possibility and (ii) I do not know.”

-- Jack Edmonds, 1966

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The Simpson's: P = NP?

Copyright © 1990, Matt Groening

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NP-Complete

NP-complete. A problem Y in NP with the property that for every problem X in NP, X ≤ p Y.

Theorem. Suppose Y is an NP-complete problem. Then Y is solvable in polynomial time iff P = NP.

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NP-Complete

Theorem. Suppose Y is an NP-complete problem. Then Y is solvable in polynomial time iff P = NP.

Proof of only-if part: If P = NP then Y can be solved in polynomial time since Y is in NP. Proof of if-part: Suppose Y can be solved in polynomial time.

Let X be any problem in NP. Since X ≤ p Y, we can solve X in polynomial time. This implies NP ⊆ P.We already know P ⊆ NP. Thus P = NP. ▪

Fundamental question. Do there exist "natural" NP-complete problems?

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∧

¬

∧ ∨

∧

∨

1 0 ? ? ?

output

inputshard-coded inputs

Circuit SatisfiabilityCIRCUIT-SAT. Given a combinational circuit built out of AND, OR, and NOT gates, is there a way to set the circuit inputs so that the output is 1?

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∧

¬

∧ ∨

∧

∨

1 0 ? ? ?

output

inputshard-coded inputs

yes: 1 0 1

Circuit SatisfiabilityCIRCUIT-SAT. Given a combinational circuit built out of AND, OR, and NOT gates, is there a way to set the circuit inputs so that the output is 1?

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All problems below are NP-complete and polynomial reduce to one another!

CIRCUIT-SAT

3-SAT

DIR-HAM-CYCLE

INDEPENDENT SET

VERTEX COVER

3-SAT re

duces to

INDEPENDENT SE

T

GRAPH 3-COLOR

HAM-CYCLE

TSP

SUBSET-SUM

SCHEDULINGPLANAR 3-COLOR

NP-Completeness

by definition of NP-completeness

SET COVER

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Some NP-Complete ProblemsSix basic genres of NP-complete problems and paradigmatic examples.

Packing problems: SET-PACKING, INDEPENDENT SET.Covering problems: SET-COVER, VERTEX-COVER.Constraint satisfaction problems: SAT, 3-SAT.Sequencing problems: HAMILTONIAN-CYCLE, TSP.Partitioning problems: 3D-MATCHING 3-COLOR.Numerical problems: SUBSET-SUM, KNAPSACK.

Practice. Most NP problems are either known to be in P or NP-complete.

Notable exceptions. Factoring, graph isomorphism

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Extent and Impact of NP-Completeness

Extent of NP-completeness. Prime intellectual export of CS to other disciplines.6,000 citations per year (title, abstract, keywords).

more than "compiler", "operating system", "database"Broad applicability and classification power.

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General solutions get you a 50% tip.

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More Hard Computational Problems

Aerospace engineering: optimal mesh partitioning for finite elements.Biology: protein folding.Chemical engineering: heat exchanger network synthesis.Civil engineering: equilibrium of urban traffic flow.Economics: computation of arbitrage in financial markets with friction.Electrical engineering: VLSI layout. Environmental engineering: optimal placement of contaminant sensors.Financial engineering: find minimum risk portfolio of given return.Game theory: find Nash equilibrium that maximizes social welfare.Genomics: phylogeny reconstruction.Mechanical engineering: structure of turbulence in sheared flows.Medicine: reconstructing 3-D shape from biplane angiocardiogram.Operations research: optimal resource allocation. Physics: partition function of 3-D Ising model in statistical mechanics.Politics: Shapley-Shubik voting power.Statistics: optimal experimental design.

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Status of P = NP questionIn 1971, Stephen Cook wrote a paper that precisely formulated the problem

In 2000, it became one of the Millennium Prizes

August 8, 2010: Announcement of proof that P ≠ NP

August 10, 2010: Concluded not proven

Currently, computer scientists widely suspect that P ≠ NP, but there is no proof. If P = NP, public key cryptography, which all ecommerce

depends on, will be insecure.

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Will it be on the exam? 24-1


Will it be on the exam?The facts you should know for the exam:

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A problem is in P if there is a polynomial time algorithm to solve the problem

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A problem is in P if there is a polynomial time algorithm to solve the problemA problem is in NP if we can check a purported solution in polynomial time

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A problem is in P if there is a polynomial time algorithm to solve the problemA problem is in NP if we can check a purported solution in polynomial timeP ⊆ NP

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A problem is in P if there is a polynomial time algorithm to solve the problemA problem is in NP if we can check a purported solution in polynomial timeP ⊆ NPNobody knows if P = NP or not

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A problem is in P if there is a polynomial time algorithm to solve the problemA problem is in NP if we can check a purported solution in polynomial timeP ⊆ NPNobody knows if P = NP or notWhile it has not been proven, computer scientist believe that P ≠ NP.

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A problem is in P if there is a polynomial time algorithm to solve the problemA problem is in NP if we can check a purported solution in polynomial timeP ⊆ NPNobody knows if P = NP or notWhile it has not been proven, computer scientist believe that P ≠ NP.If P = NP, a large number of problems with no known polynomial time algorithms actually can be solved in polynomial time.

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A problem is in P if there is a polynomial time algorithm to solve the problemA problem is in NP if we can check a purported solution in polynomial timeP ⊆ NPNobody knows if P = NP or notWhile it has not been proven, computer scientist believe that P ≠ NP.If P = NP, a large number of problems with no known polynomial time algorithms actually can be solved in polynomial time.If P = NP, public key encryption, which is the basis for ecommerce, will no longer be secure

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P = NP in Popular CultureIn 1995, the Simpsons Treehouse of Horror VI shows the P=NP equation when Homer enters the 3D universe

In a 2011 episode of Suits, two lawyers solve an NP-complete problem.

Traveling Salesman - a movie released in 2012 in which 4 mathematicians are hired by the US government to solve P=NP (presumably so they could break encryption)

A 2013 episode of Elementary is about a mathematician who is murdered who was trying to solve P=NP

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Slides23 - NPblerner/cs312/Slides/Slides23.pdf · 2019-04-29 · 51, 17 51, 16 RELPRIME Are x and y...

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