Planar graphs and the Travelling Salesman Problem

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Planar graphs and the Travelling Salesman Problem

• Nick Pearson
• April 19th 2005

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Overview

• TSP and the Graphical TSP (GTSP)
• Planarity
• Review of Known Valid Inequalities
• The Separation Problem
• Fast Separation for the Planar (G)TSP
• Conclusion

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1. The Symmetric TSP

• The problem is to find a minimum cost Hamiltonian
  cycle in an edge-weighted undirected graph.

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1. The Symmetric TSP
Define a 0-1 variable xij for each e ?
E Minimise ? i,j ?E cijxij Subject to ?j
adj. i xij 2 (? i ? V) (degree
equations) ?i?S, j?S xij ? 2 (? S ?
V) (SECs) xij ? 0, 1E (binary
condition) (Due to Dantzig, Fulkerson Johnson,
1954.)
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1. The Graphical TSP

• A relaxation of the STSP introduced by
  Cornuéjols, Fonlupt Naddef (1985).
• Calls for a minimum cost spanning closed walk in
  G. That is, each vertex must be visited at least
  once, and edges may be traversed more than once.

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1. The Graphical TSP
Again, define a 0-1 variable xij for each e ?
E Minimise ? i,j ?E cijxij Subject to ?j
adj. i xij ? 0 mod 2 (? i ? V) (parity) ?i?S,
j?S xij ? 2 (? S ? V) (SECs) xij ?
ZE (integrality) This formulation is not
an IP, due to the parity conditions, which state
that every vertex must have even degree.
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1. Applications of the (G)TSP

• Sub-problem in many transportation and logistics
  applications
• Drilling of printed circuits boards
• Warehouse material handling
• Stock cutting problems
• Job scheduling

USA 13509

• Ideal platform for the study of general methods
  that can be applied to a wide range of discrete
  optimisation problems

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2. Planarity
Let G (V, E) be an undirected graph. G is
planar if it can be embedded in the plane with no
edges crossing.
planar
non-planar
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2. Properties of planar graphs
We will use the notation n V, m
E. Planar graphs are sparse. Eulers theorem
implies m ? 3(n - 2) n - m f 2
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2. 1st Major Tool Geometric Duality
The dual has f vertices, m edges and n faces. The
dual of the dual is the original graph.
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2. 1st Major Tool Geometric Duality
A cut in the dual is a cycle in the original, and
vice-versa.
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2. 2nd Major Tool Separator Theorem
Lipton Tarjan (1979) showed that there exists a
set of about ?n vertices which, when removed
from the graph cause the graph to become
disconnected, such that each remaining component
contains ? 2n/3 vertices. A separator can be
found in O(n) time.
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2. Applications of planar graphs

• Telecommunications e.g. spanning trees

• Vehicle routing e.g. Roads without underpasses

• VLSI e.g. computer chips

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Why study the Planar TSP and GTSP?

• In some applications, G is naturally planar
  (e.g. road networks, provided there are no over-
  or under-passes)
• Fractional solutions encountered when running
  cutting plane algorithms for the general TSP are
  often planar (Jünger, Cook)
• Interesting from viewpoint of computational
  complexity
• An effective heuristic for planar Euclidean TSPs
  is to extract a planar subgraph and then solve
  the TSP or GTSP on that
• (Stewart, 1997 used the Delaunay triangulation
  resulting tours were within 1.2 of optimal on
  TSPLIB instances.)

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2. Known Results for Planar TSPs

• The Hamitonian circuit problem is NP-complete
  for
• Cubic, 3-connected, planar graphs
• (Garey, Johnson Tarjan, 1976)
• Planar triangulations (Chvátal, 1981)
• Delaunay (internal) triangulations (Dillencourt,
  1994)
• Grid graphs (Itai, Papadimitriou Szwarcfiter,
  1982)

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2. Known Results for Planar TSPs
But there is some evidence that planar TSP and
GTSP are 'relatively easy' Arora et al. (1998)
gave a PTAS for planar GTSP, though general GTSP
is APX-hard Letchford (2000) gave a fast
separation algorithm for comb inequalities in
planar case Deinecko, Klinz Woeginger (2004)
give a O(c?n) algorithm for planar TSP, whereas
for general TSP we have O(n2 2n) (Held Karp,
1962)
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3. Review of known valid inequalities

• For both variants of the problem, the SECs, and
  the non-negativity inequalities xe ? 0, induce
  facets of the associated integer polyhedron under
  mild conditions on G.
• However, for most graphs, further inequalities
  are needed to describe the polyhedron, such as
• 2-matchings
• Combs
• Domino-parity

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3. Review of known valid inequalities
Example The 2-matching inequalities of Edmonds
(1965). H ? Tj Tj \ H 1
for all j,
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4. Cutting plane algorithm

• Solve an initial LP relaxation (usually by the
  primal simplex method).
• Let x be the solution. If it is integer and
  feasible, output the optimal solution and stop.
• Look for one or more violated facet-inducing
  inequalities. If no violated inequalities are
  found, output the final lower bound and stop.
• Add the inequalities to the LP. Resolve the LP
• (usually by the dual simplex method) and go to
  (2).

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4. The Separation Problem

• How do we find violated inequalities?
• We can not test each one explicitly

To use a class of inequalities in a cutting plane
algorithm, we need to solve the following
separation problem (Grötschel, Lovász Schrijver
1988) Given a vector x ? ?? as input,
either find an inequality in the class which is
violated by x, or prove that none exists.
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4. The Separation Problem
For the STSP/GTSP on a general graph, the current
best exact separation algorithms have the
following run-times SECs O(nm n2 log
n) (Nagamochi, Ono Ibaraki, 1994). 2-Matchings
O(n2 m log (n2 /m)) (Letchford, Reinelt
Theis, 2004) Simple combs O(n2 m2 log (n2
/m)) (Fleischer, Letchford Lodi, 2003) We will
show that much better algorithms exist in the
planar case.
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5. Fast Separation
Letchford P. (2004) gave an O(n 3/2 log n)
algorithm for computing minimum odd cuts in
planar graphs. Summary
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6. Conclusion

• There is evidence that the TSP and GTSP will be
  much easier to solve to optimality via
  branch-and-cut in the planar case than in the
  case of general graphs.
• The TSP on the Delaunay Triangulation
• (a special internally triangulated planar graph)
  may very well provide a good heuristic for the
  Euclidean TSP
• We are now implementing this idea.