Approximation Algorithms for the Traveling Salesperson Problem

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Approximation Algorithms for the Traveling
Salesperson Problem
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Approximation Algorithm

• Up to now, the best algorithm for solving an
  NP-complete problem requires exponential time in
  the worst case. It is too time-consuming.
• To reduce the time required for solving a
  problem, we can relax the problem, and obtain a
  feasible solution close to an optimal solution

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Approximation Algorithm
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Error Ratio Bound
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(1?)-approximation algorithm
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An approximation algorithm for Euclidean
traveling salesperson problem (ETSP)

• The ETSP is to find a shortest closed path
  through a set S of n points in the plane.
• The ETSP is NP-hard.

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Approximation Algorithm for ETSP

• Input A set S of n points in the plane.
• OutputAn approximate traveling salesperson tour
  of S.
• Step 1 Find a minimal spanning tree T of S.
• Step 2 Find a minimal Euclidean weighted
  matching M on the set of vertices of odd degrees
  in T. Let GM?T.
• Step 3 Find an Eulerian cycle of G and then
  traverse it to find a Hamiltonian cycle as an
  approximate tour of ETSP by bypassing all
  previously visited vertices.

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Eulerian Cycle

• An Eulerian path (Eulerian trail, Euler walk) in
  a graph is a path that uses each edge precisely
  once. If such a path exists, the graph is called
  traversable.
• An Eulerian cycle (Eulerian circuit, Euler tour)
  in a graph is a cycle with uses each edge
  precisely once. If such a cycle exists, the graph
  is called Eulerian.

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• L. Euler showed that an Eulerian cycle exists if
  and only if all vertices in the graph are of even
  degree and all edges are contained in the same
  component.
• L. Euler also showed an Eulerian path exists, if
  and only if at most two vertices in the graph are
  of odd degree and all edges are contained in the
  same component.

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• Leonhard Euler (April 15, 1707 - September 18,
  1783) (pronounced "oiler") was a Swiss
  mathematician and physicist. He is considered
  (together with Gauss) to be the greatest
  mathematician ever.
• Leonhard Euler stated and solved the problem of
  Seven Bridges of Königsberg in 1736, which is the
  first formally discussed problem in graph theory.

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Eulerian Cycle exists becausedegree(V1)
4 degree(V2) 2 degree(V3) 4 degree(V4) 4.
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This is the Eulerian Cycle.
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• A Hamiltonian path (also called traceable path)
  is a path that visits each vertex exactly once.
• A Hamiltonian cycle (also called Hamiltonian
  circuit, vertex tour or graph cycle) is a cycle
  that visits each vertex exactly once, except for
  the starting vertex.

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Minimal Euclidean Weighted Matching Problem

• Given a set of points in the plane, the minimal
  Euclidean weighted matching problem is to join
  the points in pairs by line segments such that
  the total length is minimum.

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Approximation Algorithm for ETSP

• E.g.
• Step1

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Approximation Algorithm for ETSP

• Step2The number of points with odd degrees must
  be even. , which is even

One edge contributes 2 degrees
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Approximation Algorithm for ETSP

• Step3

P3 and P4 are visited twice. By bypassing P3 and
P4 and connecting P6 to P1 directly, we obtain a
Hamiltonian cycle.
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Approximation Algorithm for ETSP

• Time complexity O(n3)
• Step 1 O(nlogn)
• Step 2 O(n3)
• Step 3 O(n)
• How close the approximate solution to an optimal
  solution?

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How good is the solution ?

• The approximate tour is within 3/2 of the optimal
  one.
• Reasoning
• L optimal ETSP tour, T MST, Lp a path
  derived by removing one edge from L (Lp is also a
  spanning tree)
• ? length(T)?length(Lp)?length(L)
• Let Lpj1i1j2i2j3i2m, where i1, i2,, i2m is
  the set of odd-degree vertices in T where indices
  of vertices in the set are arranged in the same
  order as they are in the optimal ETSP tour L.

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How good is the solution ?

• Let M be the minimal Euclidean weighted matching.
• Consider the two matchingsM1i1,i2,i3,i4,,
  i2m-1,i2m and M2i2,i3,i4,i5,,i2m,i1.
  We have
• length(L)? length(M1) length(M2) (triangular
  inequality) ? 2 length(M )
• ? length(M)? 1/2 length(L )
• G T?M
• ? length(T) length(M) ? length(L) 1/2
  length(L)
• 3/2 length(L)

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The Bottleneck Traveling Salesperson Problem
(BTSP)

• Minimize the longest edge of a tour.
• This is a mini-max problem.
• This problem is NP-hard.
• The input data for this problem fulfill the
  following assumptions
• The graph is a complete graph.
• All edges obey the triangular inequality rule.

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An Algorithm for Finding an Optimal Solution

• Step1 Sort all edges in G (V,E) into a
  nondecreasing sequence e1?e2??em. Let
  G(ei) denote the subgraph obtained from G by
  deleting all edges longer than ei.
• Step2 i?1
• Step3 If there exists a Hamiltonian cycle in
  G(ei), then this cycle is the solution hence,
  stop.
• Step4 i?i1. Go to Step 3.

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An Example for BTSP Algorithm

• e.g.

• There is a Hamiltonian cycle, A-B-D-C-E-F-G-A, in
  G(BD).
• The optimal solution is 13.

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Time complexity of the algorithm

• The Hamiltonian cycle problem is NP-hard.
• The algorithm cannot be a polynomial one.

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Theorem for Hamiltonian Cycles

• Def The t-th power of G(V,E), denoted as
  Gt(V,Et), is a graph that an edge (u,v)?Et if
  there is a path from u to v with at most t edges
  in G.
• Theorem If a graph G is bi-connected, then G2
  has a Hamiltonian cycle.

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An Example for the Theorem
G2
A Hamiltonian cycle A-B-C-D-E-F-G-A
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An Approximation Algorithm for BTSP

• Input A complete graph G(V,E) where all edges
  satisfy triangular inequality.
• Output A tour in G whose longest edge is not
  greater than twice of the value of an optimal
  solution to the special bottleneck traveling
  salesperson problem of G.
• Step 1 Sort the edges into e1?e2??em.
• Step 2 i 1.
• Step 3 If G(ei) is bi-connected, construct
  G(ei)2, find a Hamiltonian cycle in G(ei)2 and
  return this as the output.
• Step 4 i i 1. Go to Step 3.

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An Example
Add some more edges. Then it becomes bi-connected.
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• A Hamiltonian cycle A-G-F-E-D-C-B-A.
• The longest edge 16

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Time complexity

• The algorithm is of polynomial time
• The determination of bi-connectedness can be
  solved by polynomial algorithms.
• If G is bi-connected, then there exists a
  polynomial algorithm to solve the Hamiltonian
  cycle problem of G2.

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How Good is the Solution ?

• The value of an approximate solution is bounded
  by two times that of an optimal solution.
• Reasoning
• A Hamiltonian cycle is bi-connected.
• eop the longest edge of an optimal solution
• G(ei) the first bi-connected graph
• ei?eop
• The length of the longest edge in G(ei)2?2ei
• (triangular inequality)
  ?2eop