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Everything you always wanted to know about theoretical computer science (but were afraid to ask). Dimitris Achlioptas Microsoft Research. Turing’s Machine. Alphabet Σ , state space K f : K × Σ → K × Σ × {←,→, ? } × { Halt,Yes,No} Language: L µ Σ * is decided by M L - PowerPoint PPT Presentation
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Everything you always wanted to know about theoretical computer science
(but were afraid to ask)
Dimitris AchlioptasMicrosoft Research
Turing’s Machine
• Alphabet Σ, state space K
• f : K×Σ → K×Σ×{←,→,?} ×{Halt,Yes,No}
• Language: L µ Σ* is decided by ML
• Polynomial-time Robustness
Any reasonable attempt to model mathematically computer algorithms and their performance is bound to end up with a model of computation and associated time cost that is equivalent to Turing machines within a polynomial.
Church’s Thesis
Any reasonable attempt to model mathematically computer algorithms and their performance is bound to end up with a model of computation and associated time cost that is equivalent to Turing machines within a polynomial.
Church’s Thesis
Turing’s Idea
Programmable Hardware:
Describe f on the leftmost part of the tape.
Polynomial Time• N vertices• As,t distance from s to t
Find a spanning cycle of minimum length
Polynomial Time
• Edge lengths are written in binary. Lists vs. Matrices
• Assume M could decide if OPT<B in polynomial time– Do binary search to determine OPT – Probe edges one-by-one, replacing their cost with OPT+1
• N vertices• As,t distance from s to t
Find a spanning cycle of minimum length
P vs. NP
Polynomial Time PTIME = [k nk
L is in P if there exists a Turing Machine M which for every x, decides if x is in L in a polynomial number of steps.
P vs. NP
Polynomial Time PTIME = [k nk
L is in P if there exists a Turing Machine M which for every x, decides if x is in L in polynomially many steps.
Non-Deterministic Polynomial Time
L is in NP if there exists a Turing Machine M s.t. for every x • If x is in L then there exists w s.t. M(x,w)→“Yes” in PTIME.• If x is not in L then there is no such w.
Complexity
• Is there a spanning cycle of cost < C ? NP• Is there a spanning tree of cost < C ? P• Is there no spanning tree of cost < C ? P • Is there no spanning cycle of cost < C ? coNP
NPcoNP
P
Perebor
• 50-60s: Many optimization problems appear amenable only to brute force, i.e. (near-)exhaustive enumeration.
• Edmonds 1966: “The classes of problems which are respectively known and not known tohave good algorithms are of great theoretical interest […] I conjecture
thatthere is no good algorithm for the traveling salesman problem. My
reasonsare the same as for any mathematical conjecture: (1) It’s a legitimate mathematical possibility, and (2) I do not know.”
• Hartmanis, Stearns 1965:
“On the computational Complexity of Algorithms”, AMS Transactions
Input for Problem B
Output for Problem B
Algorithm for Problem B
Reductionfrom B to A
Algorithmfor B
x R(x) Yes/No
Reducibility
Input for Problem B
Output for Problem B
Algorithm for Problem B
Reductionfrom B to A
Algorithmfor B
x R(x) Yes/No
Shortest Spanning Path from s to t
Shortest Spanning Cycle
Reducibility
NP-completeness
A problem Π is NP-hard if every
problem in NP has a polynomial-time reduction to Π.
If Π is in ΝP and NP-hard then Π is NP-complete.
Moral: At least as hard as any other problem in NP
Cook’s Theorem
Satisfiability is NP-Complete
Idea:
Let variable Xp,t denote the content of tape-cell p after t steps
• By now thousands of problems have been shown NP-complete
• NP-completeness has migrated to other sciences, e.g. economics
Integer Linear Programming:
Ax·bIs there a satisfying vector x where all xi are non-negative INTEGERS?
NPcoNP
P TSP
PRIMALITY
More hard problems
Approximation
• Vertex Cover: Find a smallest set of vertices that touches every edge.– Algorithm: If an edge is still untouched, add
both its endpoints to the set.
Approximation
• Vertex Cover: Find a smallest set of vertices that touches every edge.– Algorithm: If an edge is still untouched, add
both its endpoints to the set.
• Max Cut: Partition the vertices of the graph to maximize number of cut edges.– Algorithm: Partition randomly.
Greed
Problem: Find a spanning tree of minimum length
Greed
Problem: Find a spanning tree of minimum length
Algorithm: Always add the cheapest allowable edge
Greed
Problem: Find a spanning tree of minimum length
Algorithm: Always add the cheapest allowable edge
Proof:
The cheapest edge emanating from each tree is always good.
Duality
Max-Flow = Min-Cut
Linear Programming:
Ax·bIs there a satisfying vector where all xi are non-negative rationals?
Linear Programming
Linear Programming:
Ax·bIs there a satisfying vector where all xi are non-negative rationals?
Linear Programming
An integer matrix A is Totally Unimodular (TUM) if every square, nonsingular submatrix B has det(B)=±1
Linear Programming:
Ax·bIs there a satisfying vector where all xi are non-negative rationals?
Linear Programming
An integer matrix A is Totally Unimodular (TUM) if every square, nonsingular submatrix B has det(B)=±1
If A is TUM then the LP solution is integral!
Linear Programming II
• Simplex method:– Most known rules can be made to take an
exponential number of steps.
• Ellipsoid algorithm:– Number of steps depends on size of integers
Is there an algorithm that solves LPs with m equations in n variables in Poly(m,n) steps?
Semidefinite Programming
Arrange n unit vectors so that their inner products satisfy a system of
linear inequalities.
Approximability
MAX CUT
• Assign a unit vector vs to each vertex s.• Maximize the sum of (1-vsvt)wst over all vertices• Take a random hyperplane through the origin.• Get an 0.87… approximation
Vertex Cover
• Conjecture: Cannot beat stupid algorithm!• Theorem: Need at least 1.36..*OPT
• Tool: NP = PCP(O(logn), O(1)).
Non-worst-case Inputs
• Random graphs
• Random pointsets
• Random formulas
Non-worst-case Inputs
• Random graphs
• Random pointsets
• Random formulas
Theoretical CS ProbabilisticCombinatorics
Non-worst-case Inputs
• Random graphs
• Random pointsets
• Random formulas
Theoretical CS ProbabilisticCombinatorics
Physics