The Symmetric Traveling Salesman Polytope: a guided Tour

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The Symmetric Traveling Salesman Polytope a
guided Tour

• Denis Naddef
• Yves Pochet

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The double index integer lp formulation
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The double index lp formulation
Huge set of linear inequalities
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About the Huge Set

• It is known only for V10
• Unlikely to be in a near future known for all V
• We will concentrate here on those inequalities in
  closed set or near closed set from.

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Closed Set Form
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Particular type of set ??

• Consists of a set of Handles H1 Hh
• Integers ?? ah
• And consists of a set of Teeth T1 Tt
• Numbers ?? ?t

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Plain Closed Set Form Facets
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EXAMPLES

• Comb inequalities
• Clique tree and bipartition inequalities
• Path inequalities

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COMB INEQUALITIES
t odd
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Algebraic proof of validity
Ai Ti ? H, Bi Ti \ H LHS ? 1/2 (S i x(d(Ai))
(S i x(d(Bi)) (S i x(d(Ti))) ? 3t
LHS is EVEN and 3t is ODD so one can increase it
to 3t1
WHAT DOES THIS MEAN IN TERM OF HAMILTONIAN
CYCLES?
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All teeth TIGHT
There are at least t red edges, t odd -gt one more
edge in ?(H)
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At least one tooth not tight
Any extra 2 edges in some ?(T) cannot decrease
?(H) by more than 2
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Star and Path Inequalities

• H1 ? H2 ?????? Hh
• Ti intersects properly H1for all i1,..t
• Ti ? Tj ? for i?j
• t odd
• ?i integer far all i1,.., h
• ?j integer far all j1,.., t

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The Interval Condition

• Interval I relative to Tj set of consecutive
  handles with same intersection with Tj
• Weight of I S?? with i in I
• The interval condition ?? max weight of an
  interval

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4 edges in ?(Tj) yields an increase of 2?j
compensated by a saving of twice the weight of
the interval
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Closed Set Form with negative Correcting Term
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Correcting term?

• The correcting term transforms some valid closed
  set form inequalities into facet inducing
  inequalities. Some edges may not belong to any
  tight tour of the valid inequality, the
  correcting term corrects this
• It may not be uniquely determined since it is in
  general the result of a sequential procedure
• Example the ladder inequalities

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Ladder Inequalities

• Two handles H1?and H2 , H1? H2 ?
• ????? T1, T2 , Tt, t2k, k odd,
• Ti? Tj ???i?j
• T1? H1 ? ? T1? H2 ?
• T2? H1 ? T2? H2 ? ?
• Ti? H1 ? ?, Ti? H2 ? ? , i3
• ?1 ?2 1,
• ?i 1 if Ti\(H1? H2)? ??? ?i 2 else

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Closed Set Form with positive Correcting Term

• This correcting term aims at making valid a
  closed set form inequality which would not be
  valid without it

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The Domino Configurations

• A handle H
• A nested set of teeth T1 Tt, t odd
• Ti Ai ?Bi?V, Ai ?Bi ? for every non minimal
  tooth Ti
• A tooth is odd if it is minimal or strictly
  contains an even number of teeth
• There are at least three odd maximal teeth
• Each Ai ????Bi contains at least two odd teeth

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The Domino Inequality

• Let (Ai Bi)e???????a? Ai ? b? Bi and

Ethe edges going from one half domino to the
other without crossing the border of the handle
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Domino Parity Inequality

• Adam Letchford has defined a class of
  inequalities known as the Domino Parity
  Inequalities. He describes a polynomial time
  algorithm to separate them provided the support
  graph of the fractional solution to be separated
  is planar. We conjecture that the domino
  inequalities contain all the facet inducing such
  inequalities.

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Validity

• Since t is odd, and LHS even one can raise to
  3t1. Exactly the same proof as for the comb
  inequalities

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Not valid without the correcting term 20 instead
of 22
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Clique Tree Bipartition
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Two extra edges in d(T) two less in the ?d(Hi)
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Two more extra edges in d(T) FOUR less in the
?d(Hi) from the tight Four more edges in d(T)
case SIX less in the ?d(Hi)
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Bibliography

• D. Naddef, Y. Pochet The traveling Salesman
  Polytope revisited, Math of OR
• D. Naddef, E. Wild The domino Inequalities
  facets for the STSP, to appear Math Prog B
  dedicated to E. Balas
• D. Naddef The traveling Salesman Polytope ,
  Chap 2 of  The Traveling Salesman Problem and
  its variations  Kluwert (2002)

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HAPPY BIRTHDAY JEAN-FRANCOIS