Random matching and traveling salesman problems

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Random matching and traveling salesman problems

• Johan Wästlund
• Chalmers University of Technology
• Sweden

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Mean field model of distance

• The edges of a complete graph on n vertices are
  given i. i. d. nonnegative costs
• Exponential(1) distribution.

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Mean field model of distance

• We are interested in the cost of the minimum
  matching, minimum traveling salesman tour etc,
  for large n.

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Matching

• Set of edges giving a pairing of all points

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Traveling salesman

• Tour visiting all points

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Walkups theorem

• Theorem (Walkup 1979) The expected cost of the
  minimum matching is bounded
• Bipartite model

R
L
n
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Walkups theorem

• cost of the minimum assignment.
• Modify the graph model Multiple edges with costs
  given by a Poisson process
• This obviously doesnt change the minimum
  assignment

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Walkups theorem

• Give each edge a random direction
• Choose the five cheapest edges from each vertex.
• We show that whp this set contains a perfect
  matching

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Halls criterion

• An edge set contains a perfect matching iff for
  every subset S of L,

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Halls criterion

• If Halls criterion holds, an incomplete matching
  can always be extended.

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Halls criterion

• If Halls criterion fails for S, then it also
  fails for

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Halls criterion

• Here we can take S T n1
• If Halls criterion fails, then it fails for some
  S (in L or in R) with

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Walkups theorem
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Walkups theorem
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Walkups theorem

• The directed edges from a given vertex have costs
  from a rate n/2 Poisson process
• The 5 cheapest edges have expected costs 2/n,
  4/n, 6/n, 8/n, 10/n.
• The average cost in this set is 6/n, and there
  are n edges in a perfect matching

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Walkups theorem

• If Halls criterion holds, there is a perfect
  matching of expected cost at most 6.
• What about the cases of failure?

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Walkups theorem

• Randomly color the edges
• Red p
• Blue 1-p
• Take the 5 cheapest blue edges from each vertex.
  If Halls criterion holds, this gives a matching
  of cost 6/(1-p)?
• Otherwise the red edges 1-1, 2-2 etc give a
  matching of cost n/p.

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Walkups theorem

• Total expected cost
• Take p 1/n for instance.
• For large n, the expected cost is lt 6 o(1)?
• This completes the proof.

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Walkups theorem

• Actually
• but we return to this

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Walkups theorem

• Walkups theorem obviously carries over to the
  complete graph (for even n)?
• The method also works for the TSP, minimum
  spanning tree, and other related problems
• Natural conjecture E(cost) converges in
  probability to some constant.

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Statistical physics

• The typical edge in the optimum solution has cost
  of order 1/n, and the number of edges in a
  solution is of order n.
• Analogous to spin systems of statistical physics

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Disordered Systems

• Spin glasses
• AuFe random alloy
• Fe atoms interact

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Statistical physics

• Each particle essentially interacts only with
  its close neighbors
• Macroscopic observables (magnetic field) arise
  as sums of many small terms, and are essentially
  independent of individual particles

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Statistical physics
Convergence in probability to a constant?
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Statistical Physics / C-S

• Spin configuration
• Hamiltonian
• Ground state energy
• Temperature
• Gibbs measure
• Thermodynamic limit

• Feasible solution
• Cost of solution
• Cost of minimal solution
• Artificial parameter T
• Gibbs measure
• n?8

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Statistical physics

• Replica-cavity method of statistical mechanics
  has given spectacular predictions for random
  optimization problems
• M. Mézard, G. Parisi, W. Krauth, 1980s
• Limit of ??/12 for minimum matching on the
  complete graph (Aldous 2000)?
• Limit 2.0415 for the TSP (Wästlund 2006)?

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Non-rigorous derivation of the ??/12 limit

• Matching problem on Kn for large n.
• In principle, this requires even n, but we shall
  consider a relaxation
• Let the edges be exponential of mean n, so that
  the sequence of ordered edge costs from a given
  vertex is approximately a Poisson process of rate
  1.

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Non-rigorous derivation of the ??/12 limit

• The total cost of the minimum matching is of
  order n.
• Introduce a punishment cgt0 for not using a
  particular vertex.
• This makes the problem well-defined also for odd
  n.
• For fixed c, let n tend to infinity.
• As c tends to infinity, we expect to recover the
  behavior of the original problem.

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Non-rigorous derivation of the ??/12 limit

• For large n, suppose that the problem behaves in
  the same way for n-1 vertices.
• Choose an arbitrary vertex to be the root
• What does the graph look like locally around the
  root?
• When only edges of cost lt2c are considered, the
  graph becomes locally tree-like

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Non-rigorous derivation of the ??/12 limit

• Non-rigorous replica-cavity method
• Aldous derived equivalent equations with the
  Poisson-Weighted Infinite Tree (PWIT)?

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Non-rigorous derivation of the ??/12 limit

• Let X be the difference in cost between the
  original problem and that with the root removed.
• If the root is not matched, then X c. Otherwise
  X ?i Xi, where Xi is distributed like X, and
  ?i is the cost of the ith edge from the root.
• The Xis are assumed to be independent.

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Non-rigorous derivation of the ??/12 limit

• It remains to do some calculations.
• We have
• where Xi is distributed like X

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Non-rigorous derivation of the ??/12 limit

• Let

-u
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Non-rigorous derivation of the ??/12 limit

• Then if ugt-c,

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Non-rigorous derivation of the ??/12 limit
Hence
is constant
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Non-rigorous derivation of the ??/12 limit
f(-u)?

• The constant depends on c and holds when
• cltultc

f(u)?
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Non-rigorous derivation of the ??/12 limit

• From definition, exp(-f(c)) P(Xc) proportion
  of vertices that are not matched, and exp(-f(-c))
  exp(0) 1
• e-f(u) e-f(-u) 2 proportion of vertices
  that are matched 1 when c infinity.

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Non-rigorous derivation of the ??/12 limit
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Non-rigorous derivation of the ??/12 limit

• What about the cost of the minimum matching?

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Non-rigorous derivation of the ??/12 limit
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Non-rigorous derivation of the ??/12 limit
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Non-rigorous derivation of the ??/12 limit

• Hence J area under the curve when f(u) is
  plotted against f(-u)!
• Expected cost n/2 times this area
• In the original setting ½ times the area
• ??/12.

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K-L matching
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K-L matching

• Similarly, the K-L matching problem leads to the
  equations

• ? has rate K and ? has rate L
• minK stands for Kth smallest

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K-L matching

• Shown by Parisi (2006) that this system has an
  essentially unique solution
• The ground state energy is given by
• where x and y satisfy an explicit equation
• For K L 2 (equivalent to the TSP), this
  equation is

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The exponential bipartite assignment problem
n
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The exponential bipartite assignment problem

• Exact formula conjectured by Parisi (1998)?
• Suggests proof by induction
• Researchers in discrete math, combinatorics and
  graph theory became interested
• Generalizations

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Generalizations

• by Coppersmith Sorkin to incomplete matchings
• Remarkable paper by M. Buck, C. Chan D. Robbins
  (2000)
• Introduces weighted vertices
• Extremely close to proving Parisis conjecture!

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Incomplete matchings
n
m
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Weighted assignment problems

• Weights ?1,,?m, ?1,, ?n on vertices
• Edge cost exponential of rate ?i?j
• Conjectured formula for the expected cost of
  minimum assignment
• Formula for the probability that a vertex
  participates in solution (trivial for less
  general setting!)?

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The Buck-Chan-Robbins urn process

• Balls are drawn with probabilities proportional
  to weight

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Proofs of the conjectures

• Two independent proofs of the Parisi and
  Coppersmith-Sorkin conjectures were announced on
  March 17, 2003 (Nair, Prabhakar, Sharma and
  Linusson, Wästlund)?

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Rigorous method

• Relax by introducing an extra vertex
• Let the weight of the extra vertex go to zero
• Example Assignment problem with
• ?1?m1, ?1?n1, and ?m1 ?
• p P(extra vertex participates)
• p/n P(edge (m1,n) participates)

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Rigorous method

• p/n P(edge (m1,n) participates)?
• When ??0, this is
• Hence
• By Buck-Chan-Robbins urn theorem,

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Rigorous method

• Hence
• Inductively this establishes the
  Coppersmith-Sorkin formula

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Rigorous results

• Much simpler proofs of Parisi, Coppersmith-Sorkin,
  Buck-Chan-Robbins formulas
• Exact results for higher moments
• Exact results and limits for optimization
  problems on the complete graph

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The 2-dimensional urn process

• 2-dimensional time until k balls have been drawn

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Limit shape as n?8

• Matching
• TSP/2-factor

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Mean field TSP

• If the edge costs are i.i.d and satisfy
  P(lltt)/t?1 as t?0 (pseudodimension 1), then as n
  ?8,
• A. Frieze proved that whp a 2-factor can be
  patched to a tour at small cost

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Further exact formulas
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LP-relaxation of matching in the complete graph Kn
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Future work

• Explain why the cavity method gives the same
  equation as the limit shape in the urn process
• Establish more detailed cavity predictions
• Use proof method of Nair-Prabhakar-Sharma in more
  general settings

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Thank you!