Approximation Algorithm of Traveling Salesman Problem

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Approximation Algorithm of Traveling Salesman
Problem

• By Lin, Jr-Shiun Chio, Jae Sung
• Speaker Lin, Jr-Shiun

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What is TSP?

• Design the shortest, or minimal cost, route for a
  salesman who wants to travel EVERY cities ONLY
  ONCE and ,lastly, backs to home city.
• In graph, we need to find a tour that starts at a
  node, visits every other node exactly once, and
  returns to the starting node.

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What is TSP

• Definition Find a path through a weighted graph
  which starts and ends at the same vertex,
  includes every other vertex exactly once, and
  minimizes the total cost of edges. (NIST)
• TSP is classify as NP-complete problem, that
  means no polynomial algorithm can guarantee to
  come within countable(?) times of the shortest
  tour.

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Christofides Algorithm

• Nicos Christofides find a way to slove TSP
  problem (1976)
• 1 find a minimum spanning tree T.
• 2 find a perfect matching M from nodes with odd
  degree.
• 3 combine the edges of M and T to graph G
• 4 find an Euler cycle in G by skipping vertices
  already seen

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Christofides Algorithm
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Even nodes with odd degree??

• How can I know that I always have even number of
  nodes, which have odd degree, for me to do the
  MATCHING?

• From handshaking lemma in any graph, the sum of
  all the vertex-degrees is equal to twice the
  number of edges.

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Even nodes with odd degree??

• let S (d) 2m, where m number of edges.
  Therefore S(d) is even.
• Let Se(d) to be the sum of degrees of the
  vertices which have even degree, Se(d) is also
  even.
• Therefore S(d)-Se(d) 2k, k1,2,, which means
  that the sum of degrees of the vertices which
  have odd degree each is also an even number. Thus
  there are even numbers of vertices which have odd
  number of degree. (Dr. Giri Narasimhan)

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Metric TSP

• To convert general TSP to Metric TSP, we need to
  add weight to each edges.
• Why Metric?
• in metric TSP, it suffices to find a cyclic tour
  that visits each vertex at least once, rather
  than exactly once. Using the triangle inequality,
  we can always shortcut a vertex that is visited
  more than once without increasing the cost of the
  tour.
• Metric TSP can be approximated within a factor of
  3/2.

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3/2-approximation

• Let c(TSP) be the cost of the minimum TSP tour,
  c(MST) be the cost of the MST, and c(M) be the
  cost of the matching M.
• Clearly, c(MST) lt(1 1/n)c(TSP ), because n -
  1 edges of a TSP are a spanning tree of the
  graph. Moreover, c(M) lt(TSP)/2
• Because of the triangle inequality, the minimum
  tour on a elements subset of vertices has cost at
  most c(TSP).

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3/2-approximation (cont.)

• If this subset is even cardinality, then the
  positive minimum cost matching on it has cost at
  most c(TSP)/2, because the minimum tour is then a
  disjoint union of two matchings. The set of
  vertices of odd degree in a spanning tree is of
  even cardinality.

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3/2-approximation (cont.)

• An Euler tour exists on the multiset of edges of
  MST and M because each vertex in the edge induced
  subgraph has even degree. The cost of the Euler
  tour is at most c(MST ) c(M) lt(3/2
  1/n)c(TSP ). (Uriel Feige)
• As n goes to infinity, it becomes 3/2
  apporximation.