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Everything you always wanted to know about theoretical computer science (but were afraid to ask) Dimitris Achlioptas Microsoft Research

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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 Polynomial-time Robustness

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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

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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.

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Polynomial Time N vertices A s,t distance from s to t Find a spanning cycle of minimum length

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Polynomial Time Edge lengths are written in binary. Lists vs. Matrices Assume M could decide if OPT**
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P vs. NP P olynomial Time PTIME = [ k n k 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.

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P vs. NP P olynomial Time PTIME = [ k n k 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. N on-Deterministic P olynomial 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.

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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 NP coNP P

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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 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: (1) It’s a legitimate mathematical possibility, and (2) I do not know.” Hartmanis, Stearns 1965: “On the computational Complexity of Algorithms”, AMS Transactions

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Input for Problem B Output for Problem B Algorithm for Problem B Reduction from B to A Algorithm for B x R(x) Yes/No Reducibility

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Input for Problem B Output for Problem B Algorithm for Problem B Reduction from B to A Algorithm for B x R(x) Yes/No Shortest Spanning Path from s to t Shortest Spanning Cycle Reducibility

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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

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Cook’s Theorem Satisfiability is NP-Complete Idea: Let variable X p,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

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Integer Linear Programming : Ax · b Is there a satisfying vector x where all x i are non-negative INTEGERS? NP coNP P TSP PRIMALITY More hard problems

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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.

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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.

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Greed Problem: Find a spanning tree of minimum length

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Greed Problem: Find a spanning tree of minimum length Algorithm: Always add the cheapest allowable edge

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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.

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Duality Max-Flow = Min-Cut

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Linear Programming : Ax · b Is there a satisfying vector where all x i are non-negative rationals? Linear Programming

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Linear Programming : Ax · b Is there a satisfying vector where all x i are non-negative rationals? Linear Programming An integer matrix A is Totally Unimodular (TUM) if every square, nonsingular submatrix B has det(B)=±1

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Linear Programming : Ax · b Is there a satisfying vector where all x i 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!

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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?

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Semidefinite Programming Arrange n unit vectors so that their inner products satisfy a system of linear inequalities.

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Approximability MAX CUT Assign a unit vector v s to each vertex s. Maximize the sum of (1-v s v t )w st 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)).

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Non-worst-case Inputs Random graphs Random pointsets Random formulas

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Non-worst-case Inputs Random graphs Random pointsets Random formulas Theoretical CS Probabilistic Combinatorics

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Non-worst-case Inputs Random graphs Random pointsets Random formulas Theoretical CS Probabilistic Combinatorics Physics

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Part VI NP-Hardness. Lecture 23 Whats NP? Hard Problems.

Part VI NP-Hardness. Lecture 23 Whats NP? Hard Problems.

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