Chapter 1' Problems and Algorithms

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Chapter 1. Problems and Algorithms

• 1.1 Two Problems
• The Traveling Salesman Problem
• 47 drilling platforms off the coast of Nigeria.
• Need to visit platforms by a helicopter to
  regulate the rates of flows.
• Starting from the onshore base, what is the
  cheapest (shortest) route to visit each platform
  once and return to the base?
• Euclidean traveling salesman problem.
• Input a set V of points in the Euclidean plane.
• Objective Find a simple circuit passing through
  the points for which the sum of the lengths of
  the edges is minimized.
• Enumeration (n-1)!/2 ways
• No efficient algorithm is known. (NP-hard)

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• Related problem Hamiltonian cycle (circuit) (a
  closed simple path that visits each node of a
  graph G (V, E) exactly once)
• Hamilton (1859) regular dodecahedron ( 12
  pentagonal faces and 20 nodes, visiting cities in
  the globe)

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• There exist graphs for which Hamiltonian cycle
  doesnt exist.

Petersen graph
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• Q Does there exist easily identifiable necessary
  and sufficient conditions to determine whether an
  arbitrary graph G is Hamiltonian?
• Unlikely, the problem is NP-complete.
• Three types of problems in combinatorics
• Existence problems
• Enumeration problems
• Optimization, construction problems

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• The Matching Problem
• Drawing a logic circuit using a plotter, assume
  graph is connected.
• Total drawing time pen-down pen-up
• minimize pen-up time

odd nodes
even nodes
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• Seven bridges in Königsberg Eulers problem

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• Def Euler trail of a connected graph G A trail
  (edge-simple path) that traverses every edge of
  G.
• Def Tour of G closed walk(path in the text)
  that traverses each edge of G at least once.
• Euler tour tour that traverses each edge
  exactly once (a closed Euler trail)
• A graph is called eulerian if it contains an
  Euler tour.

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• Thm A nonempty and connected graph is eulerian
  if and only if it has no nodes of odd degree.
• Pf) ?) clear
• ? ) Induction on E.
• True for E 0. Assume true for m-1 edges.
• A connected graph with m ? 1 edges and each node
  has even degree ? ? edge-simple closed path P.
• G\P gives a graph with one or more components
  and each component is eulerian by induction
  hypothesis. Each component has a node in common
  with P.
• Now splice P and the Euler tours of the
  components. ?
• Note that an algorithm to find an Euler tour can
  be designed using the idea of the proof.

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• Cor A connected graph has an Euler trail if and
  only if it has at most two nodes of odd degree.
• Pf) the number of nodes of odd degree is even
  ( ? v ? V d(v) 2E )
• ?) clear
• ?) If all nodes even degree, true from the
  theorem.
• If two nodes of odd degree ( say u, v ), add
  edge (u, v) to G, get an eulerian graph. Now
  delete (u, v) in a Euler tour starting from u. ?

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• Back to circuit drawing
• If the circuit diagram is eulerian, we can draw
  the diagram without lifting the pen.
• If the diagram is not eulerian, add some edges
  (pen-up movement) to make the diagram eulerian.
  Which edges to add to minimize the pen-up
  movement?
• t(p, q) time to move (up or down) from point p
  to q.
• Assume t satisfies triangle inequality for any
  points p, q, r, we have t(p, r) ? t(p, q)
  t(q,r).
• Then extra edges added only pair up the odd nodes

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• Euclidean Matching Problem
• Input A set V of points in the Euclidean plane.
• Objective Find a set of lines, such that each
  point is an end of exactly one line (perfect
  matching), and such that the sum of the lengths
  of the lines is minimized.
• Efficient algorithm exists. (Edmond, 1965)

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• A problem similar to the circuit drawing problem
• Chinese Postman Problem
• Given a connected graph G (V, E), edge cost ce
  ? 0, e ? E.
• Find a minimum cost closed path (walk)
  traversing every edge at least once.
• approach) Duplicate an edge with the same cost
  if postman needs to traverse the edge again
• ? make the graph eulerian by duplicating edges
  while the sum of costs of additional edges is
  minimized.
• Note that we do not need to duplicate an edge
  more than once.

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• Example

• Note the difference between the circuit drawing
  and the chinese postman problem.
• Other variations directed graph, mixed graph
• More on this problem later when we study T-join.

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• 1.2 Measuring Running Times
• Efficiency of an algorithm measured by giving
  upper bounds on the number of steps required to
  solve a problem of a given size.
• Arithmetic model each elementary operation has
  unit cost, regardless of the length of the
  numbers
• Bit model numbers represented in binary
  notation and arithmetic operation is carried out
  bit by bit.
• Def f, g Z ? R
• We say f(n) O(g(n)) if ? c gt 0 and n0 such
  that f(n) ? c?g(n) ? n ? n0.
• e.g.) 5n2 3n is O(n2), 35 ?2n n3 is O(2n).
• Def We say that an algorithm is a polynomial
  time algorithm if the running time bound is O(nk)
  for some fixed k, where n is the problem size (
  in arithmetic or bit model )

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• If an algorithm is polynomial time in arithmetic
  model and the size of the numbers appearing in
  the algorithm do not grow too fast ( i.e., if t
  is the number of bits to represent problem data,
  the length of encoding of numbers occurring in
  the algorithm is O(tk) for some fixed k), it is
  polynomial time algorithm in bit model too. (
  most cases)