CMPUT 671 Hard Problems

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CMPUT 671Hard Problems

• Winter 2002
• Joseph Culberson Home Page

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What are you doing?
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What are you doing?
What am I doing?
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What is Hard?

• Problems and Complexity
• P, NP, PSPACE between and beyond
• Worst case, Randomized, Average case

What are you doing?
What am I doing?
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What is Hard?

• Problems and Complexity
• P, NP, PSPACE between and beyond
• Worst case, Randomized, Average case
• Instances and Algorithms
• Some instances may be easy for some algorithms
• Hard instances for classes of algorithms

What are you doing?
What am I doing?
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What is Hard?

• Problems and Complexity
• P, NP, PSPACE between and beyond
• Worst case, Randomized, Average case
• Instances and Algorithms
• Some instances may be easy for some algorithms
• Hard instances for classes of algorithms
• Ensembles and Phase Transitions
• - Most instances may be easy
• - Hard instances may be concentrated
• - Hard random instances for classes of
  algorithms
• - Generation

What are you doing?
What am I doing?
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Background

• Asymptotic notation and algorithm analysis
• Problems, Complexity Classes, Reductions and
  Completeness (P to NPc)
• Backtrack Search, Stochastic Search
• Graphs and some graph theory
• Probability (as background to random graphs and
  structures)

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Problems

• A Problem is a set of instances and queries.
• An instance is a structure (versus language)
• Queries are universal
• e.g. Graph k-coloring
• Instance A graph G and integer k
• Query Is G k-colorable?
• e.g. MST union TSP
• Instance A weighted graph G, integers B1,B2
• Query Does G have a spanning tree of weight B1
  or a tour of weight B2? (Set one to 0)

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Measuring Resources

• Time
• Instance I
• n Size(I) Formally, the number of bits in an
  encoding (logarithmic model) but mostly we will
  use the uniform model, e.g. the number of
  vertices and edges of a graph.
• T(I) number of steps in computation to answer
  the query. Formally, on a model of computation,
  typically a RAM.

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Measuring Resources

• Time (continued)
• Worst Case
• T(n) max T(I) size(I) n
• Average Case
• A(n) sum T(I) Prob(I) size(I) n
• Note distribution dependent!
• Typically all instances of size n equally likely,
  but this may not be representative of the real
  world

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Measuring Resources

• SPACE
• Similar, but measures number of memory cells
  required.
• Formally, each cell must be of fixed size, e.g.
  number of bits fixed.
• For certain purposes, e.g. O(log n) memory
  requirements, we exclude size of input memory.
  (Likely not relevant to this course)

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Asymptotic Notation
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Types of Problems

• Optimization
• Given G find the minimum number of colors
  required to color it.
• Functional
• Given G find an optimal coloring
• Given G and k find a k-coloring if one exists
• Enumeration and Generation (listing)
• Decision (Yes/No)
• Given G and k is there a k-coloring of G?

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The Class P

• P is the set of decision problems for which there
  exists a polynomial time algorithm.
• i.e. O(nk) for some fixed k.
• n is the size of the input.
• See Garey and Johnson, Brassard and Bratley,
  Sipser, Cormen et al.

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The Class NP

• NP is the set of decision problems for which
  there exists a non-deterministic algorithm that
  solves the yes instances of the problem in
  polynomial time.
• (GJ, BB) The algorithm non-deterministically
  writes out a solution (certificate) and in time
  polynomial in the input size verifies that the
  answer is yes.
• E.g. Coloring
• Certificate cV -gt 1..k
• verify for each edge e (u,v), c(u) not c(v)

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Class CO-NP

• The set of problems whose NO instances are solved
  by a poly-time non-deterministic algorithm the
  complement of NP wrt Yes/No
• In Language Terms
• Note that P is in the intersection of NP and
  CO-NP
• Why?

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• We no of know certificate we can use to verify
  that G is not k-colorable in poly-time.
• The Questions
• P ? NP ? CO-NP ?
• The first asks if there is a polynomial time
  deterministic algorithm to solve all NP problems
• The second asks is there a polynomial time
  non-deterministic algorithm to solve all of CO-NP

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Reductions

• Polynomial time many one reduction
  (transformation) from problem P to problem Q
• fP-gtQ Maps instances
• x in Y(P) iff f(x) in Y(Q)
• f is computable by a polynomial time
  deterministic algorithm.

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Reductions
Example Graph 3-coloring to 3-SAT
Can also add 2-clauses restricting each vertex
to at most one color
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Class NPc

• A subclass of NP, Q in NPc iff for every problem
  P in NP there is a polynomial time many reduction
  from P to Q.
• Cooks theorem showed that SAT is in NPc.
• Create a set of clauses which model the possible
  execution of the NDTM that solves the problem.

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Showing a Problem is in NPc

• To show a problem Q is NP-complete
• Show Q is in NP, that is, specify a certificate
  that can be verified in polynomial time
• Choose a problem P in NPc and provide a
  polynomial time reduction from P to Q.
• The second alone shows it is NP-hard

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• Only applies to worst case