Hard or Intractable Problems NPComplete Problems

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Hard or Intractable Problems(NP-Complete
Problems)

• ECE573 Data Structures and Algorithms
• Electrical and Computer Engineering Dept.
• Rutgers University
• http//www.cs.rutgers.edu/vchinni/dsa/

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Polynomial or Intractable?

• Polynomial time complexity
• O(nk) time algorithm, k constant
• Intractable
• No O(nk) algorithm is known
• O(nn) or O(n!) or O(kn) time required

No HW6 PA6 We got to build this lost piece of
art work Can have some additional time
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Similar Problems?

• Eulers Problem
• Is there a path through a graph which traverses
  each edge only once?
• Hamiltons Problem
• Is there a path which visits each vertex only
  once?

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Eulers Problem The Bridges of Königsberg

• Can I make a tour through the park, crossing
  each bridge only once?

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Eulerian Path

• An Eulerian Path exists iff
• It is possible to go from any vertex to any other
  by following the edges (graph must be connected)
• Every vertex must have an even number of edges
  connected to it -with at most two exceptions
  (the start and end points)

Obviously necessary conditions Sufficiency proof
may be found in the literature
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Transform the Bridges Problem

• Nodes land
• Edges bridges
• No path!
• Determined in O(n) time
• Count degree of each vertex
• Hamiltonian Path?

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Hamiltonian Path

• Although a path may be easily found in this case,
  no known efficient algorithm for finding a
  Hamiltonian path
• But if we find a path,
• we can verify that it is a Hamiltonian path in
  polynomial time
• Check that each edge in the path is an edge in
  the graph (use adjacency matrix) - O(E)
• Check that each vertex is visited once only
  -O(n2)

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Classes P and NP

• Class P
• Problems solvable in polynomial time
• eg Eulers problem
• Class NP
• Non-deterministic Polynomial
• eg Hamiltons problem

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Class NP

• Non-deterministic Polynomial
• At each step, guesswhich step to take next(eg
  which vertex to check)
• Have you found a correct solution?
• Need polynomial time only
• Nothing from previous steps guides the next step
• You have to try all possibilities
• O(n!) algorithms

Non-deterministic part You simply guess!
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Winning the Turing Award

• No proof that any problem ? class NP
• Problems (eg Hamiltonian path) are believed to be
  in class NP
• Because no polynomial time algorithm is known
• Proving that a problem ? class NP
  or finding a class P algorithm ?Instant
  fame!

Computings equivalent of the Nobel Prize
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Composite Numbers

• Determine whether a number n can be written as
  the product of two other numbers
• If the divisors can be found, then its simple to
  verify that they are divisors
• But, no efficient method of finding them is known

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Assignment Problem

• You have to assign students to rooms in a college
• You have a compatibility map
• Students are vertices,
• Compatible students are linked by an edge
• If rooms hold 2 students,
• a class P algorithm
• but if rooms hold 3 students,
• a class NP problem!

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Boolean Satisfiability

• Find an assignment of values (true or false) to
  ai so that
• is true
• where op are boolean operators and, or, ...
• Equivalent to circuit satisfiability problem
• Find a set of inputs which produce a true at the
  output of a circuit composed of logic gates
• Solution
• Try all 2n possible assignments

a1 op a2 op a3 ... op an
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Map Coloring

• Color a map using only three colors, so that
  adjoining countries dont have the same color
• Easily answered for 2 colors
• point at which an odd numberof countries meet
  no solution
• Solvable for 4 colors
• proof exists that 4 colors suffice for any map

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Map Coloring - Equivalent Problems

• Graph equivalent
• Vertices countries
• Edges connect countries with common borders
• Work scheduling
• Vertices tasks
• Edges represent common resources
• eg linked tasks require some machine
• 3 color marking of the vertices
• allocation of tasks to 3 shifts in a day

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Reducing Problems

• Map Coloring ? Boolean Satisfiability
• A solution can be constructed by assignment of
  true or false to a set of variables

ar true ? A is red
ab true ? A is blue
ag true ? A is green
br true ? B is red
Solution If we can find a set of values for ax,
bx, ... which make this expression true, weve
found a map coloring
((ar ab ag ) (bb br. ... ) (cb
...
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NP-Complete Problems

• Special class of problems in NP
• The NP-complete problems
• All problems in NP are efficiently reducible to
  them
• Efficiently in polynomial time
• Many problems have been shown to be efficiently
  reducible to boolean satisfiability
• but an efficient solution to this (and thus any
  other NP-complete problem!) has eluded many
  researchers for a long time!
• Its believed that class NP ? class P

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Solving NP-Complete problems

• NP problems are solvable
• Its just a question of patience (or longevity!)
• Non-deterministic algorithms
• Alternately
• Non-deterministic (guessing) step
• Select the next step randomly
• No guidelines any step may be the solution
• Deterministic step
• Determine (in polynomial time) whether you have
  found a solution or not

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The Travelling Salesman Problem

• A salesman has a number of cities which he must
  visit, find the minimum cost tour
• Optimization problem
• Can we reduce it to boolean satisfiability?
• Problem must have a true /false answer
• Recast as
• Does a tour exist with cost less than x?
• Reduce x until the answer is provably no ...
• Optimal tour
• This problem may be proved equivalent to finding
  a Hamiltonian circuit

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The Travelling Salesman Problem

• Many real world problems map to TSP
• Drilling circuit boards
• Inspection tours
• ...
• How do we solve it?
• Heuristics
• Approximate techniques leading to near-optimal
  solutions
• If you find the optimal solution, will you know?

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Travelling Salesman Heuristics

• Start with the Minimum Spanning Tree
• Optimal tour cost lt 2 MST cost
• Heuristics
• Christofides algorithm
• Prune the tour by taking short-cuts
• Tour must be an Eulerian path
• Each vertex of even degree
• Add edges to odd degree vertices
• Christofides algorithm produces a tour lt 1.5
  optimal tour cost

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Travelling Salesman Heuristics

• Banding
• Divide the region into bands
• Number of citiesin each bandsufficiently small
  that the TSPproblem can be solved by
  exhaustion(brute-force or try all solutions)
• Greedy algorithm to join the band solutions

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Travelling Salesman Heuristics

• Genetic algorithms
• Biological evolution analogy
• Reasonably effective
• Trivially parallelizable!
• Simulated annealing
• Model Annealing of solid
• Solution heads towards local minimum
• Temperature allows some probability of climbing
  hills to escape a local minimum

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Games

• Search trees can be huge
• Chess 40 possible choices for each move
• 10-move look-ahead needed to win
• gt4010 positions to explore
• Modern computers just touching this capability
• Pentium _at_ 1000MHz 105 evaluations/second
• Still a rather boring game ... 1 month/move??