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Optimization in mean field random models

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Spanning tree: (3) = 1 1/8 1/27 ... 1.202 (Frieze 1985) Traveling salesman: 2.0415... A. Frieze proved that whp a 2-factor can be patched to a tour at small cost ... – PowerPoint PPT presentation

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Title: Optimization in mean field random models


1
Optimization in mean field random models
  • Johan Wästlund
  • Linköping University
  • Sweden

2
Statistical Mechanics
  • Each particle has a spin
  • Energy Hamiltonian depends on spins of
    interacting particles
  • Ising model Spins 1, H interacting pairs of
    opposite spin

3
Statistical Mechanics
  • Spin configuration ? has energy H(?)
  • Gibbs measure depends on temperature T
  • T?8 random state
  • T?0 ground state, i.e. minimizing H(?)

4
Statistical Mechanics
  • Thermodynamic limit N ?8
  • Average energy? (suitably normalized)

5
Disordered Systems
  • Spin glasses
  • AuFe random alloy
  • Fe atoms interact

6
Disordered Systems
  • Random interactions between Fe atoms
  • Sherrington-Kirkpatrick model

7
Disordered Systems
  • Quenched random variables gi,j
  • S-K is a mean field model No correlation
    betweeen quenched variables
  • NP hard to find ground state given gi,j

8
Computer Science
  • Test / evaluate heuristics for NP-hard problems
  • Average case analysis
  • Random problem instances

9
Combinatorial Optimization
  • Minimum Matching / Assignment
  • Minimum Spanning Tree
  • Traveling Salesman
  • Shortest Path
  • Points with given distances, minimize total
    length of configuration

10
Statistical Physics / Computer Science
  • 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

11
Mean field models
  • Replica-cavity method has given good results for
    mean field models
  • Parisi solution of S-K model
  • The same methods can be applied to combinatorial
    optimization problems in mean field models

12
Mean field models of distance
  • N points
  • Abstract geometry
  • Inter-point distances given by i. i. d. random
    variables
  • Exponential distribution easiest to analyze
    (pseudodimension 1)

13
Matching
  • Set of edges giving a pairing of all points

14
Spanning tree
  • Network connecting all points

15
Traveling salesman
  • Tour visiting all points

16
Mean field limits
  • No normalization needed! (pseudodimension 1)
  • Matching ?2/120.822 (Mézard Parisi 1985,
    rigorous proof by Aldous 2000)
  • Spanning tree ?(3) 11/81/27 1.202 (Frieze
    1985)
  • Traveling salesman 2.0415 (Krauth-Mézard-Parisi
    1989), now established rigorously!

17
Cavity results
  • Non-rigorous method
  • Aldous derived equivalent equations with the
    Poisson-Weighted Infinite Tree (PWIT)

18
Cavity results
  • Non-rigorous quantity X cost of minimal
    solution cost of minimal solution with the root
    removed
  • Define X1, X2, X3, similarly on sub-trees
  • Leads to the equation
  • Xi distributed like X, ?i are times of events in
    rate 1 Poisson process

19
Cavity results
  • Analytically, this is equivalent to

where
20
Cavity results
  • Explicit solution
  • Ground state energy

21
Cavity results
  • Note that the integral is equal to the area under
    the curve when f(u) is plotted against f(-u)
  • In this case, f satisfies the equation

22
Cavity results
23
K-L matching
24
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

25
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 KL2, this equation is
  • Unfortunately the cavity method is not rigorous

26
The exponential bipartite assignment problem
n
27
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

28
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!

29
Incomplete matchings
n
m
30
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!)

31
The Buck-Chan-Robbins urn process
  • Balls are drawn with probabilities proportional
    to weight

32
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)

33
Annealing
  • Powerful idea Let T?0, forcing the system to
    converge to its ground state
  • Replica-cavity approach
  • Simulated annealing meta-algorithm (optimization
    by random local moves)

34
In the mean field model
Underlying rate 1 variables Yi
  • ri plays the same role as T
  • Local temperature
  • Associate weight to vertices rather than edges

35
Cavity/annealing 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)

36
Annealing
  • p/n P(edge (m1,n) participates)
  • When ??0, this is
  • Hence
  • By Buck-Chan-Robbins urn theorem,

37
Annealing
  • Hence
  • Inductively this establishes the
    Coppersmith-Sorkin formula

38
Results with annealing
  • 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

39
The 2-dimensional urn process
  • 2-dimensional time until k balls have been drawn

40
Limit shape as n?8
  • Matching
  • TSP/2-factor

41
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

42
Further exact formulas
43
LP-relaxation of matching in the complete graph Kn
44
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

45
Thank you!
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