The Random Matching Problem
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1 The Random Matching Problem Enrico Maria Malatesta Universitá di Milano October 21st, 2016 Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
2 Outline 1 Disordered Systems Overview of Spin Glasses Replica Method 2 Optimization Problems 3 The Matching Problem Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
3 What is a Spin Glass? quenched disorder = frustration Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
4 What is a Spin Glass? quenched disorder = frustration + : Ferromagnetic Bond (probability 1 p) : Antiferromagnetic Bond (probability p) : Spin up : Spin down +? +?? Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
5 What is a Spin Glass? quenched disorder = frustration + : Ferromagnetic Bond (probability 1 p) : Antiferromagnetic Bond (probability p) : Spin up : Spin down +? +? Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
6 What is a Spin Glass? quenched disorder = frustration + : Ferromagnetic Bond (probability 1 p) : Antiferromagnetic Bond (probability p) : Spin up : Spin down + +? Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
7 The Edwards-Anderson Model (EA) 2D square lattice Edwards-Anderson Model: N Spins σ i = ±1; N P (J ik) = N 2π e 2 J2 ik ; H = N Jik σi σk <i,k> Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
8 The Mean-Field Spin Glass Model lim EA = Sherrington-Kirkpatrick D Model: N Spins σ i = ±1; N P (J ik) = N 2π e 2 J2 ik ; H = N Jik σi σk i<k 1 β f(β) = lim ln Z N N (... ) = djikp (Jik)(... ) i<k The Complete Graph K 16 ln Z difficult! Z easy! Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
9 Replica Method Replica Trick : Z n 1 ln Z n ln Z = lim = lim n 0 n n 0 n Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
10 Replica Method Replica Trick : Z n = 1 α<β n [ A[q] = nβ2 4 + β2 2 Z n 1 ln Z n ln Z = lim = lim n 0 n n 0 n dq αβ ( Nβ 2 2π ) 1/2 ] 1 α<β n q 2 αβ ln {σ} e NA[q] exp Saddle Point Approximation for N : A[q] = 0 q αβ qsp { β 2 1 α<β n min A[q] A[q sp] β f = lim = lim n 0 n n 0 n q αβ σ ασ β } Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
11 Replica Symmetry Breaking (RSB) n = 12 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 } {{ } RS Replica Symmetry (RS) S RS (0) = 1/2π 0.16 Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
12 Replica Symmetry Breaking (RSB) n = 12 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q = q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 } {{ } RS n = 12, m1 = 4 0 q1 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 0 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 0 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 q1 0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 0 q1 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 0 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 0 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 q1 0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 0 q1 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 0 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 0 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q0 q1 0 } {{ } 1RSB Replica Symmetry (RS) S RS (0) = 1/2π 0.16 Replica Symmetry Breaking (RSB) S 1RSB (0) 0.01 Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
13 Replica Symmetry Breaking (RSB) n = 12 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q = q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 q q q q q q q q q q q q 0 } {{ } RS n = 12, m1 = 4 0 q1 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 0 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 0 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 q1 0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 0 q1 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 0 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 0 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 q1 0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 0 q1 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 0 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 0 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q0 q1 0 } {{ } 1RSB = n = 12, m1 = 4, m2 = 2 0 q1 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 0 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 0 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 q1 0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 0 q1 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 0 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 0 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 q1 0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 0 q1 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 0 q1 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 0 q1 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 q1 0 } {{ } 2RSB =... Replica Symmetry (RS) S RS (0) = 1/2π 0.16 Replica Symmetry Breaking (RSB) S 1RSB (0) 0.01 Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
14 Optimization Problems { Space of configurations C instance Cost function E : C IR The task is to find an optimal solution, i.e. an element ξ C E(ξ ) E(ξ), ξ C Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
15 Optimization Problems { Space of configurations C instance Cost function E : C IR The task is to find an optimal solution, i.e. an element ξ C E(ξ ) E(ξ), ξ C OPTIMIZATION Instance Cost Function Optimal Configuration Minimal Cost STATISTICAL PHYSICS Sample Energy Ground State Ground State Energy E(ξ ) = lim β β ln Z Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
16 The Matching Problem Monopartite Matching Problem Given 2N points; Random matrix of distances between points: l ij, find a perfect matching of minimal lenght. Given Bipartite Matching Problem N red points and N blue points; Random matrix of distances between points: l ij, find a perfect matching of minimal lenght. Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
17 The Matching Problem Monopartite Matching Problem Given 2N points; Random matrix of distances between points: l ij, find a perfect matching of minimal lenght. Given Bipartite Matching Problem N red points and N blue points; Random matrix of distances between points: l ij, find a perfect matching of minimal lenght. Why Matching? It is critical; It contains frustration; It belongs to the P computational complexity class; Only the RS ansatz is needed. Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
18 How to extract distances? Euclidean Matching Problem Points x i are extracted indipendently and uniformly in a D dimensional space Ω D IR D. l ij = x i x j p Random Matching Problem We extract distances l ij indipendently with an identical distribution ρ(l ij) = l r ij Γ (r + 1) e l ij, l ij IR + ρ(l ij) = (r + 1) l r ij, l ij [0, 1] Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
19 How to extract distances? Euclidean Matching Problem Points x i are extracted indipendently and uniformly in a D dimensional space Ω D IR D. l ij = x i x j p Random Matching Problem We extract distances l ij indipendently with an identical distribution ρ(l ij) = l r ij Γ (r + 1) e l ij, l ij IR + ρ(l ij) = (r + 1) l r ij, l ij [0, 1] Remarks: No correlations in Random Matching Problem; Fixing D/p = r + 1 and sending p, D we can neglect correlations. Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
20 Partition function of the RBMP Using occupation numbers n ij = 0, or 1 we can write Constraints of the RBMP: i 1,..., N Energy of a configuration: E ({n ji}) = N nij lij i,j=1 Partition function: Z = [ N ( )] [ N N ( N δ 1 n ij δ 1 {n ij=0,1} } {{ } sum over configurations i=1 j=1 i=1 N j=1 nij = N j=1 nji = 1 j=1 n ji )] } {{ } constraints e βe({n ji}) }{{} Boltzmann weight of a configuration Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
21 Mean Length of the Optimal Matching For both distributions the optimal cost in the N limit is the same: E r(t = 0) = (r + 1) 2 1/(r+1) + G r(l) = 2 + l dy (l + y)r Γ (r + 1) e Gr (y) Gr (l) dl G r(l) e In the case r = 0: G 0(l) = ln ( 1 + e 2l) and E 0(T = 0) = π2 6 Gr (l) r = 2 3 Er 2.5 r = 1 2 π 2 /6 r = 0 l 1.5 r Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
22 Finite Size Corrections in the RBMP L min(r) = E r + Fr N f(r), The only result was obtained by G. Parisi, M. Rathiéville, (Eur. Phys. J. B 29, (2002)), for r = 0 L exp min(r = 0) = π2 6 1 N and L flat min(r = 0) = π ξ(3) N Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
23 Finite Size Corrections in the RBMP L min(r) = E r + Fr N f(r), The only result was obtained by G. Parisi, M. Rathiéville, (Eur. Phys. J. B 29, (2002)), for r = 0 L exp min(r = 0) = π2 6 1 N But what happens for r > 0? and L flat min(r = 0) = π ξ(3) N f(r) = 1 for ρ(l ij) = (r + 1) l r ij For the Poisson distribution, instead: f(r) = 1 r + 1 F r = 2 2/(r+1) (r + 1) + l + dl G r(l) Fr du e Gr (u) r Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
24 Conclusions and Outlook SPIN GLASSES COMBINATORIAL OPTIMIZATION Outlook: Evaluate exactly the finite size correction in the r case; Asymptotic expansion around r = of the finite size correction. Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
25 THANKS FOR YOUR ATTENTION! Enrico Maria Malatesta (UniMi) The Random Matching Problem October 21st, / 15
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Index Amit, D., 135-136 Anderson localization, 49-50, 245n9 Anderson, Philip, 5, 49, 79, 127-129, 155, 169-170, 223, 225-228, 248n2, 251n2 Anfinsen, C.B., 163 annealing, 119, 245n7 simulated, see simulated