Combinatorial optimization and the mean field model

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Combinatorial optimization and the mean field
model

• Johan Wästlund
• Chalmers University of Technology
• Sweden

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Random instances of optimization problems
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Random instances of optimization problems
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Random instances of optimization problems

• Typical distance between nearby points is of
  order n-1/2

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Random instances of optimization problems

• A tour consists of n links, therefore we expect
  the total length of the minimum tour to scale
  like n1/2
• Beardwood-Halton-Hammersley (1959)

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Mean field model of distance

• Distances Xij chosen as i.i.d. variables
• Given n and the distribution of distances, study
  the random variable Ln
• If the distribution models distances in d
  dimensions, we expect Ln to scale like n1-1/d
• In particular, pseudo-dimension 1 means Ln is
  asymptotically independent of n

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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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Mean field model of distance
Convergence in probability to a constant?
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Matching

• Set of edges that gives a pairing of all points

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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 1980s
• Limit of p2/12 for minimum matching on the
  complete graph (Aldous 2000)
• Limit 2.0415 for the TSP (Wästlund 2006)

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• A. Frieze (2004) Up to now there has been
  almost no progress analysing this random model of
  the travelling salesman problem.
• N. J. Cerf et al (1997) Researchers outside
  physics remain largely unaware of the analytical
  progress made on the random link TSP.

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Non-rigorous derivation of the p2/12 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 p2/12 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 p2/12 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 p2/12 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 p2/12 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 xi Xi, where Xi is distributed like X, and
  xi 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 p2/12 limit

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

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Non-rigorous derivation of the p2/12 limit

• Let

-u
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Non-rigorous derivation of the p2/12 limit

• Then if ugt-c,

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Non-rigorous derivation of the p2/12 limit
Hence
is constant
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Non-rigorous derivation of the p2/12 limit
f(-u)

• The constant depends on c and holds when
• cltultc

f(u)
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Non-rigorous derivation of the p2/12 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 p2/12 limit
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Non-rigorous derivation of the p2/12 limit

• What about the cost of the minimum matching?

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Non-rigorous derivation of the p2/12 limit
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Non-rigorous derivation of the p2/12 limit
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Non-rigorous derivation of the p2/12 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
• p2/12.

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• The equation has the explicit solution
• This gives the cost

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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 in 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,

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• For the TSP, the replica-cavity approach gives

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• It follows that
• is constant, and 1 by boundary conditions
• Replica-cavity prediction agrees with the
  rigorous result (Parisi 2006)

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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
• Reprove results of one method with the other
• Find the variance with the replica method
• Find rigorously the distribution of edge costs
  participating in the solution (there is an exact
  conjecture)

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