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Solution. exact. approximate. fast. Speed. slow. Algorithms. Tradeoff : Execution speed vs. solution quality. “Short & sweet”. “Quick & dirty”. “Slowly but surely”. “Too little, too late”. Computational Complexity. Problem : Avoid getting trapped in local minima. Global optimum.

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algorithms

Solution

exact

approximate

fast

Speed

slow

Algorithms

Tradeoff: Execution speed vs. solution quality

“Short & sweet”

“Quick & dirty”

“Slowly but surely”

“Too little, too late”

computational complexity
Computational Complexity

Problem: Avoid getting trapped in local minima

Global optimum

slide3

Approximation Algorithms

Idea: Some intractable problems can be efficiently approximated within close to optimal!

Fast:

  • Simple heuristics (e.g., greed)
  • Provably-good approximations

Slower:

  • Branch-and-bound approaches
  • Integer Linear Programming relaxation
slide4

Approximation Algorithms

Wishful:

  • Simulated annealing
  • Genetic algorithms
slide5

Minimum Vertex Cover

Minumum vertex cover problem: Given a graph, find a minimum set of vertices such that each edge is incident to at least one vertex of these vertices.

Example:

Input graph

Heuristic solution

Optimal solution

Applications: bioinformtics, communications,

civil engineering, electrical engineering, etc.

  • One of Karp’s original NP-complete problems
slide7

Approximate Vertex Cover

Theorem: The minimum vertex cover problem is NP-complete (even in planar graphs of max degree 3).

Theorem: The minimum vertex cover problem can be solved exactly within exponential time nO(1)2O(n).

Theorem: The minimum vertex cover problem can not be approximated within £ 1.36*OPT unless P=NP.

Theorem: The minimum vertex cover problem can be approximated (in linear time) within 2*OPT.

Idea: pick an edge, add its endpoints, and repeat.

slide8

x

y

Approximate Vertex Cover

Algorithm: Linear time 2*OPT approximation for the minimum vertex cover problem:

  • Pick random edge (x,y)
  • Add {x,y} to the heuristic solution
  • Eliminate x and y from graph
  • Repeat until graph is empty

Best approximation

bound known for VC!

Idea: one of {x,y} must be in any optimal solution.

Þ Heuristic solution is no worse than 2*OPT.

slide9

A

B

A

B

A

B

E

C

E

E

D

C

C

D

D

Maximum Cut

Maximum cut problem: Given a graph, find a partition

of the vertices maximizing the # of crossing edges.

Example:

cut size = 2

cut size = 4

cut size = 5

Input graph

Heuristic solution

Optimal solution

Applications: VLSI circuit design, statistical physics,

communication networks.

  • One of Karp’s original NP-complete problems.
slide10

Maximum Cut

Theorem [Karp, 1972]: The minimum vertex cover

problem is NP-complete.

Theorem: The maximum cut problem can be solved

in polynomial time for planar graphs.

Theorem: The maximum cut problem can not

be approximated within £ 17/16*OPT unless P=NP.

Theorem: The maximum cut problem can be

approximated in polynomial time within 2*OPT.

Theorem: The maximum cut problem can be

approximated in polynomial time within 1.14*OPT.

=1.0625*OPT

slide11

A

B

E

C

D

A

B

A

B

E

E

C

C

D

D

Maximum Cut

Algorithm: 2*OPT approximation for maximum cut:

Start with an arbitrary node partition

  • If moving an arbitrary node across the partition

will improve the cut, then do so

  • Repeat until no further improvement is possible

cut size = 2

cut size = 3

cut size = 5

Input graph

Heuristic solution

Optimal solution

Idea: final cut must contain at least half of all edges.

Þ Heuristic solution is no worse than 2*OPT.

slide12

Approximate Traveling Salesperson

Traveling salesperson problem: given a pointset, find

shortest tour that visits every point exactly once.

2*OPT metric TSP heuristic:

  • Compute MST
  • T = Traverse MST
  • S = shortcut tour
  • Output S

Analysis:

S < T

=

2*

MST < OPT TSP

2*

triangle inequality!

T covers minimum spanning tree twice

TSPminus an edge is a spanning tree

slide13

Non-Approximability

  • NP transformations typically do not preserve the

approximability of the problem!

  • Some NP-complete problems can be approximated

arbitrarily close to optimal in polynomial time.

Theorem: Geometric TSP can be approximated in

polynomial time within (1+e)*OPT for any e>0.

  • Other NP-complete problems can not be approximated

within any constant in polynomial time (unless P=NP).

Theorem: General TSP can not be approximated

efficiently within K*OPT for any K>0 (unless P=NP).

slide14

Graph Isomorphism

Definition: two graphs G1=(V1,E1) and G2=(V2,E2) are

isomorphic iff $ bijection ƒ:V1®V2such that

"vi,vjÎV1 (vi,vj)ÎE1Û (ƒ(vi),ƒ(vj))ÎE2

Isomorphism ºedge-preserving vertex permutation

Problem: are two given graphs isomorphic?

Note: Graph isomorphism ÎNP, but not known to be in P

slide15

Graph Isomorphism

slide16

G2

G1

G

Zero-Knowledge Proofs

Idea: proving graph isomorphism without disclosing it!

Premise: Everyone knows G1 and G2 but not ≈

≈must remain secret!

Create random G ≈ G1

Note: ≈ is≈(≈)

Broadcast G

Verifier asks for≈or≈

Broadcast≈or≈

Verifier checks G≈G1 or G≈G2

Repeat k times

ÞProbability of cheating: 2-k

Approximating a “proof”!

slide17

G2

G1

χ

Zero-Knowledge Proofs

Idea: prove graph 3-colorable without disclosing how!

Premise: Everyone knows G1 but not its 3-coloring χ

which must remain secret!

χ'

Create random G2≈ G1

Note: 3-coloring χ'(G2) is ≈(χ(G1))

Broadcast G2

Verifier asks for≈or χ'

Broadcast≈or χ'

Verifier checks G1≈G2 or χ'(G2)

Repeat k times

ÞProbability of cheating: 2-k

χ

Interactive proof!

slide18

χ

Zero-Knowledge Caveats

  • Requires a good random number generator
  • Should not use the same graph twice
  • Graphs must be large and complex enough

Applications:

  • Identification friend-or-foe (IFF)
  • Cryptography
  • Business transactions
slide19

Zero-Knowledge Proofs

Idea: prove that a Boolean formula P is satisfiable

without disclosing a satisfying assignment!

Premise: Everyone knows P but not its secret

satisfying assignment V !

P = (x+y+z)(x'+y'+z)(x'+y+z')

ƒ

Convert P into a graph 3-colorability

instance G =ƒ(P)

Publically broadcastƒ and G

Use zero-knowledge protocol

to show that G is 3-colorable

Þ P is satisfiable iff G is 3-colorable

Þ P is satisfiable with probability1-2-k

G =

Interactive proof!

slide20

Interactive Proof Systems

  • Prover has unbounded power and may be malicious
  • Verifier is honest and has limited power

Completeness: If a statement is true, an honest verifier

will be convinced (with high prob) by an honest prover.

Soundness: If a statement is false, even an omnipotent

malicious prover can not convince an honest verifier that

the statement is true (except with a very low probability).

  • The induced complexity class depends on the verifier’s

abilities and computational resources:

Theorem: For a deterministic P-time verifier, class is NP.

Def: For a probabilistic P-time verifier, induced class is IP.

Theorem [Shamir, 1992]: IP = PSPACE

slide21

Concepts, Techniques, Idea & Proofs

  • 2-SAT
  • 2-Way automata
  • 3-colorability
  • 3-SAT
  • Abstract complexity
  • Acceptance
  • Ada Lovelace
  • Algebraic numbers
  • Algorithms
  • Algorithms as strings
  • Alice in Wonderland
  • Alphabets
  • Alternation
  • Ambiguity
  • Ambiguous grammars
  • Analog computing
  • Anisohedral tilings
  • Aperiodic tilings
  • Approximate min cut
  • Approximate TSP
  • Approximate vertex cover
  • Approximations
  • Artificial intelligence
  • Asimov’s laws of robotics
  • Asymptotics
  • Automatic theorem proving
  • Autonomous vehicles
  • Axiom of choice
  • Axiomatic method
  • Axiomatic system
  • Babbage’s analytical engine
  • Babbage’s difference engine
  • Bin packing
  • Binary vs. unary
  • Bletchley Park
  • Bloom axioms
  • Boolean algebra
  • Boolean functions
  • Bridges of Konigsberg
  • Brute fore
  • Busy beaver problem
  • C programs
  • Canonical order
  • Cantor dust
  • Cantor set
  • Cantor’s paradox
  • CAPCHA
  • Cardinality arguments
  • Cartesian coordinates
  • Cellular automata
  • Chaos
  • Chatterbots
  • Chess-playing programs
  • Chinese room
  • Chomsky hierarchy
  • Chomsky normal form
  • Chomskyan linguistics
  • Christofides’ heuristic
  • Church-Turing thesis
  • Clay Mathematics Institute
slide22

Concepts, Techniques, Ideas & Proofs

  • Clique problem
  • Cloaking devices
  • Closure properties
  • Cogito ergo sum
  • Colorings
  • Commutativity
  • Complementation
  • Completeness
  • Complexity classes
  • Complexity gaps
  • Complexity Zoo
  • Compositions
  • Compound pendulums
  • Compressibility
  • Computable functions
  • Computable numbers
  • Computation and physics
  • Computation models
  • Computational complexity
  • Computational universality
  • Computer viruses
  • Concatenation
  • Co-NP
  • Consciousness and sentience
  • Consistency of axioms
  • Constructions
  • Context free grammars
  • Context free languages
  • Context sensitive grammars
  • Context sensitive languages
  • Continuity
  • Continuum hypothesis
  • Contradiction
  • Contrapositive
  • Cook’s theorem
  • Countability
  • Counter automata
  • Counter example
  • Cross- product
  • Crossing sequences
  • Cross-product construction
  • Cryptography
  • DARPA Grand Challenge
  • DARPA Math Challenges
  • De Morgan’s law
  • Decidability
  • Deciders vs. recognizers
  • Decimal number system
  • Decision vs. optimization
  • Dedekind cut
  • Denseness of hierarchies
  • Derivations
  • Descriptive complexity
  • Diagonalization
  • Digital circuits
  • Diophantine equations
  • Disorder
  • DNA computing
  • Domains and ranges
slide23

Concepts, Techniques, Ideas & Proofs

  • Dovetailing
  • DSPACE
  • DTIME
  • EDVAC
  • Elegance in proof
  • Encodings
  • Enigma cipher
  • Entropy
  • Enumeration
  • Epsilon transitions
  • Equivalence relation
  • Euclid’s “Elements”
  • Euclid’s axioms
  • Euclidean geometry
  • Euler’s formula
  • Euler’s identity
  • Eulerian tour
  • Existence proofs
  • Exoskeletons
  • Exponential growth
  • Exponentiation
  • EXPSPACE
  • EXPSPACE
  • EXPSPACE complete
  • EXPTIME
  • EXPTIME complete
  • Extended Chomsky hierarchy
  • Fermat’s last theorem
  • Fibonacci numbers
  • Final states
  • Finite automata
  • Finite automata minimization
  • Fixed-point theorem
  • Formal languages
  • Formalizations
  • Four color problem
  • Fractal art
  • Fractals
  • Functional programming
  • Fundamental thm of Algebra
  • Fundamental thm of Arith.
  • Gadget-based proofs
  • Game of life
  • Game theory
  • Game trees
  • Gap theorems
  • Garey & Johnson
  • General grammars
  • Generalized colorability
  • Generalized finite automata
  • Generalized numbers
  • Generalized venn diagrams
  • Generative grammars
  • Genetic algorithms
  • Geometric / picture proofs
  • Godel numbering
  • Godel’s theorem
  • Goldbach’s conjecture
  • Golden ratio
  • Grammar equivalence
slide24

Concepts, Techniques, Ideas & Proofs

  • Grammars
  • Grammars as computers
  • Graph cliques
  • Graph colorability
  • Graph isomorphism
  • Graph theory
  • Graphs
  • Graphs as relations
  • Gravitational systems
  • Greibach normal form
  • “Grey goo”
  • Guess-and-verify
  • Halting problem
  • Hamiltonian cycle
  • Hardness
  • Heuristics
  • Hierarchy theorems
  • Hilbert’s 23 problems
  • Hilbert’s program
  • Hilbert’s tenth problem
  • Historical perspectives
  • Household robots
  • Hung state
  • Hydraulic computers
  • Hyper computation
  • Hyperbolic geometry
  • Hypernumbers
  • Identities
  • Immerman’s Theorem
  • Incompleteness
  • Incompressibility
  • Independence of axioms
  • Independent set problem
  • Induction & its drawbacks
  • Infinite hotels & applications
  • Infinite loops
  • Infinity hierarchy
  • Information theory
  • Inherent ambiguity
  • Initial state
  • Intelligence and mind
  • Interactive proofs
  • Intractability
  • Irrational numbers
  • JFLAP
  • Karp’s paper
  • Kissing number
  • Kleene closure
  • Knapsack problem
  • Lambda calculus
  • Language equivalence
  • Law of accelerating returns
  • Law of the excluded middle
  • Lego computers
  • Lexicographic order
  • Linear-bounded automata
  • Local minima
  • LOGSPACE
  • Low-deg graph colorability
  • Machine enhancements
slide25

Concepts, Techniques, Ideas & Proofs

  • Machine equivalence
  • Mandelbrot set
  • Manhattan project
  • Many-one reduction
  • Matiyasevich’s theorem
  • Mechanical calculator
  • Mechanical computers
  • Memes
  • Mental poker
  • Meta-mathematics
  • Millennium Prize
  • Minimal grammars
  • Minimum cut
  • Modeling
  • Multiple heads
  • Multiple tapes
  • Mu-recursive functions
  • MAD policy
  • Nanotechnology
  • Natural languages
  • Navier-Stokes equations
  • Neural networks
  • Newtonian mechanics
  • NLOGSPACE
  • Non-approximability
  • Non-closures
  • Non-determinism
  • Non-Euclidean geometry
  • Non-existence proofs
  • NP
  • NP completeness
  • NP-hard
  • NSPACE
  • NTIME
  • Occam’s razor
  • Octonions
  • One-to-one correspondence
  • Open problems
  • Oracles
  • P completeness
  • P vs. NP
  • Parallel postulate
  • Parallel simulation
  • Dovetailing simulation
  • Parallelism
  • Parity
  • Parsing
  • Partition problem
  • Paths in graphs
  • Peano arithmetic
  • Penrose tilings
  • Physics analogies
  • Pi formulas
  • Pigeon-hole principle
  • Pilotless planes
  • Pinwheel tilings
  • Planar graph colorability
  • Planarity testing
  • Polya’s “How to Solve It”
  • Polyhedral dissections
slide26

Concepts, Techniques, Ideas & Proofs

  • Polynomial hierarchy
  • Polynomial-time
  • P-time reductions
  • Positional # system
  • Power sets
  • Powerset construction
  • Predicate calculus
  • Predicate logic
  • Prime numbers
  • Principia Mathematica
  • Probabilistic TMs
  • Proof theory
  • Propositional logic
  • PSPACE
  • PSPACE completeness
  • Public-key cryptography
  • Pumping theorems
  • Pushdown automata
  • Puzzle solvers
  • Pythagorean theorem
  • Quantifiers
  • Quantum computing
  • Quantum mechanics
  • Quaternions
  • Queue automata
  • Quine
  • Ramanujan identities
  • Ramsey theory
  • Randomness
  • Rational numbers
  • Real numbers
  • Reality surpassing Sci-Fi
  • Recognition and enumeration
  • Recursion theorem
  • Recursive function theory
  • Recursive functions
  • Reducibilities
  • Reductions
  • Regular expressions
  • Regular languages
  • Rejection
  • Relations
  • Relativity theory
  • Relativization
  • Resource-bounded comput.
  • Respect for the definitions
  • Reusability of space
  • Reversal
  • Reverse Turing test
  • Rice’s Theorem
  • Riemann hypothesis
  • Riemann’s zeta function
  • Robots in fiction
  • Robustness of P and NP
  • Russell’s paradox
  • Satisfiability
  • Savitch’s theorem
  • Schmitt-Conway biprism
  • Scientific method
  • Sedenions
slide27

Concepts, Techniques, Ideas & Proofs

  • Self compilation
  • Self reproduction
  • Set cover problem
  • Set difference
  • Set identities
  • Set theory
  • Shannon limit
  • Sieve of Eratosthenes
  • Simulated annealing
  • Simulation
  • Skepticism
  • Soundness
  • Space filling polyhedra
  • Space hierarchy
  • Spanning trees
  • Speedup theorems
  • Sphere packing
  • Spherical geometry
  • Standard model
  • State minimization
  • Steiner tree
  • Stirling’s formula
  • Stored progam
  • String theory
  • Strings
  • Strong AI hypothesis
  • Superposition
  • Super-states
  • Surcomplex numbers
  • Surreal numbers
  • Symbolic logic
  • Symmetric closure
  • Symmetric venn diagrams
  • Technological singularity
  • Theory-reality chasms
  • Thermodynamics
  • Time hierarchy
  • Time/space tradeoff
  • Tinker Toy computers
  • Tractability
  • Tradeoffs
  • Transcendental numbers
  • Transfinite arithmetic
  • Transformations
  • Transition function
  • Transitive closure
  • Transitivity
  • Traveling salesperson
  • Triangle inequality
  • Turbulance
  • Turing complete
  • Turing degrees
  • Turing jump
  • Turing machines
  • Turing recognizable
  • Turing reduction
  • Turing test
  • Two-way automata
  • Type errors
  • Uncomputability
slide28

Concepts, Techniques, Ideas & Proofs

  • Uncomputable functions
  • Uncomputable numbers
  • Uncountability
  • Undecidability
  • Universal Turing machine
  • Venn diagrams
  • Vertex cover
  • Von Neumann architecture
  • Von Neumann bottleneck
  • Wang tiles & cubes
  • Zero-knowledge protocols

.

.

.

.

“Make everything as simple

as possible, but not simpler.”

- Albert Einstein (1879-1955)