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Optimization in mean field random models Johan Wästlund Linköping University Sweden

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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

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Statistical Mechanics Spin configuration has energy H( ) Gibbs measure depends on temperature T: T→∞ random state T→0 ground state, i.e. minimizing H( )

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Statistical Mechanics Thermodynamic limit N →∞ Average energy? (suitably normalized)

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Disordered Systems Spin glasses AuFe random alloy Fe atoms interact

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Disordered Systems Random interactions between Fe atoms Sherrington-Kirkpatrick model:

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

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

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

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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→∞

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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

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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)

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Matching Set of edges giving a pairing of all points

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Spanning tree Network connecting all points

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Traveling salesman Tour visiting all points

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Mean field limits No normalization needed! (pseudodimension 1) Matching: 2 /12≈0.822 (Mézard & Parisi 1985, rigorous proof by Aldous 2000) Spanning tree: (3) = 1+1/8+1/27+… ≈1.202 (Frieze 1985) Traveling salesman: 2.0415… (Krauth- Mézard-Parisi 1989), now established rigorously!

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

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Cavity results Non-rigorous quantity X = cost of minimal solution – cost of minimal solution with the root removed Define X 1, X 2, X 3,… similarly on sub-trees Leads to the equation X i distributed like X, i are times of events in rate 1 Poisson process

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Cavity results Analytically, this is equivalent to where

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Cavity results Explicit solution Ground state energy

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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

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Cavity results

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K-L matching

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Similarly, the K-L matching problem leads to the equations: has rate K and has rate L min[K] stands for K:th smallest

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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, this equation is Unfortunately the cavity method is not rigorous K-L matching

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The exponential bipartite assignment problem n

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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 Parisi’s 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 11 22 33

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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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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)

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In the mean field model: Underlying rate 1 variables Y i r i plays the same role as T Local temperature Associate weight to vertices rather than edges

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Cavity/annealing method Relax by introducing an extra vertex Let the weight of the extra vertex go to zero Example: Assignment problem with 1 =…= m =1, 1 =…= n =1, and m+1 = p = P(extra vertex participates) p/n = P(edge (m+1,n) participates)

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Annealing p/n = P (edge (m+1,n) participates) When →0, this is Hence By Buck-Chan-Robbins urn theorem,

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Annealing Hence Inductively this establishes the Coppersmith- Sorkin formula

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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

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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→∞ Matching: TSP/2-factor:

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Mean field TSP If the edge costs are i.i.d and satisfy P(l

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Further exact formulas

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LP-relaxation of matching in the complete graph K n

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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!

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Energy function: E(S 1,…,S N ) = - ½ Σ W ij S i S j + C (W ii = P/N) (Lyapunov function) Attractors= local minima of energy function. Inverse states Mixture.

Energy function: E(S 1,…,S N ) = - ½ Σ W ij S i S j + C (W ii = P/N) (Lyapunov function) Attractors= local minima of energy function. Inverse states Mixture.

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