Physics of Algorithms, Santa Fe 2009 Replica Symmetry and Combinatorial Optimization Johan W¨astlund Physics of Algorithms, Santa Fe 2009 Johan W¨astlund Replica Symmetry and Combinatorial Optimization Physics of Algorithms, Santa Fe 2009 Comments (like this one) have been added in order to make sense of some of the slides. Some of those comments represent what I said, or might have said, in the talk. The talk is based on the paper arXiv:0908.1920. Johan W¨astlund Replica Symmetry and Combinatorial Optimization Physics of Algorithms, Santa Fe 2009 Mean Field Model of Distance Johan W¨astlund Replica Symmetry and Combinatorial Optimization Quenched disorder Physics of Algorithms, Santa Fe 2009 Mean Field Model of Distance i.i.d edge lengths, say uniform [0; 1] Johan W¨astlund Replica Symmetry and Combinatorial Optimization Physics of Algorithms, Santa Fe 2009 Mean Field Model of Distance i.i.d edge lengths, say uniform [0; 1] Quenched disorder Johan W¨astlund Replica Symmetry and Combinatorial Optimization Spanning Tree Matching Traveling Salesman 2-factor Edge Cover Physics of Algorithms, Santa Fe 2009 Optimization problems Minimize total length: Johan W¨astlund Replica Symmetry and Combinatorial Optimization Matching Traveling Salesman 2-factor Edge Cover Physics of Algorithms, Santa Fe 2009 Optimization problems Minimize total length: Spanning Tree Johan W¨astlund Replica Symmetry and Combinatorial Optimization Traveling Salesman 2-factor Edge Cover Physics of Algorithms, Santa Fe 2009 Optimization problems Minimize total length: Spanning Tree Matching Johan W¨astlund Replica Symmetry and Combinatorial Optimization 2-factor Edge Cover Physics of Algorithms, Santa Fe 2009 Optimization problems Minimize total length: Spanning Tree Matching Traveling Salesman Johan W¨astlund Replica Symmetry and Combinatorial Optimization Edge Cover Physics of Algorithms, Santa Fe 2009 Optimization problems Minimize total length: Spanning Tree Matching Traveling Salesman 2-factor Johan W¨astlund Replica Symmetry and Combinatorial Optimization Physics of Algorithms, Santa Fe 2009 Optimization problems Minimize total length: Spanning Tree Matching Traveling Salesman 2-factor Edge Cover Johan W¨astlund Replica Symmetry and Combinatorial Optimization M´ezard-Parisi (1985{87), Matching, TSP Krauth-M´ezard (1989), TSP Predictions tested by N. Sourlas, A. Percus, O. Martin, S. Boettcher,... Success of BP, D. Shah, M. Bayati,... Physics of Algorithms, Santa Fe 2009 Replica/cavity method Replica symmetric prediction of the length of the optimum solution for large N. (Sketch of the background with a certain bias towards people in the audience...) Johan W¨astlund Replica Symmetry and Combinatorial Optimization Krauth-M´ezard (1989), TSP Predictions tested by N. Sourlas, A. Percus, O. Martin, S. Boettcher,... Success of BP, D. Shah, M. Bayati,... Physics of Algorithms, Santa Fe 2009 Replica/cavity method Replica symmetric prediction of the length of the optimum solution for large N. (Sketch of the background with a certain bias towards people in the audience...) M´ezard-Parisi (1985{87), Matching, TSP Johan W¨astlund Replica Symmetry and Combinatorial Optimization Predictions tested by N. Sourlas, A. Percus, O. Martin, S. Boettcher,... Success of BP, D. Shah, M. Bayati,... Physics of Algorithms, Santa Fe 2009 Replica/cavity method Replica symmetric prediction of the length of the optimum solution for large N. (Sketch of the background with a certain bias towards people in the audience...) M´ezard-Parisi (1985{87), Matching, TSP Krauth-M´ezard (1989), TSP Johan W¨astlund Replica Symmetry and Combinatorial Optimization Success of BP, D. Shah, M. Bayati,... Physics of Algorithms, Santa Fe 2009 Replica/cavity method Replica symmetric prediction of the length of the optimum solution for large N. (Sketch of the background with a certain bias towards people in the audience...) M´ezard-Parisi (1985{87), Matching, TSP Krauth-M´ezard (1989), TSP Predictions tested by N. Sourlas, A. Percus, O. Martin, S. Boettcher,... Johan W¨astlund Replica Symmetry and Combinatorial Optimization Physics of Algorithms, Santa Fe 2009 Replica/cavity method Replica symmetric prediction of the length of the optimum solution for large N. (Sketch of the background with a certain bias towards people in the audience...) M´ezard-Parisi (1985{87), Matching, TSP Krauth-M´ezard (1989), TSP Predictions tested by N. Sourlas, A. Percus, O. Martin, S. Boettcher,... Success of BP, D. Shah, M. Bayati,... Johan W¨astlund Replica Symmetry and Combinatorial Optimization Physics of Algorithms, Santa Fe 2009 Giorgio Parisi in Les Houches 1986, having calculated the π2=12 limit for minimum matching. Johan W¨astlund Replica Symmetry and Combinatorial Optimization π2 Matching 12 ≈ 0:822 M´ezard-Parisi, Aldous Spanning tree ζ(3) ≈ 1:202 Frieze R 1 −x −y TSP/2-factor 0 ydx, (1 + x=2)e + (1 + y=2)e = 1 ≈ 2:0415 Krauth-M´ezard-Parisi, W¨astlund(TSP ∼ 2-factor proved by Frieze) 1 2 −x Edge cover 2 min(x + e ) ≈ 0:728 Not yet proved, but the bipartite case is done in joint work with M. Hessler Physics of Algorithms, Santa Fe 2009 Limits in pseudo-dimension 1 Limit costs for uniform [0; 1] edge lengths (no normalization needed!) Johan W¨astlund Replica Symmetry and Combinatorial Optimization Spanning tree ζ(3) ≈ 1:202 Frieze R 1 −x −y TSP/2-factor 0 ydx, (1 + x=2)e + (1 + y=2)e = 1 ≈ 2:0415 Krauth-M´ezard-Parisi, W¨astlund(TSP ∼ 2-factor proved by Frieze) 1 2 −x Edge cover 2 min(x + e ) ≈ 0:728 Not yet proved, but the bipartite case is done in joint work with M. Hessler Physics of Algorithms, Santa Fe 2009 Limits in pseudo-dimension 1 Limit costs for uniform [0; 1] edge lengths (no normalization needed!) π2 Matching 12 ≈ 0:822 M´ezard-Parisi, Aldous Johan W¨astlund Replica Symmetry and Combinatorial Optimization R 1 −x −y TSP/2-factor 0 ydx, (1 + x=2)e + (1 + y=2)e = 1 ≈ 2:0415 Krauth-M´ezard-Parisi, W¨astlund(TSP ∼ 2-factor proved by Frieze) 1 2 −x Edge cover 2 min(x + e ) ≈ 0:728 Not yet proved, but the bipartite case is done in joint work with M. Hessler Physics of Algorithms, Santa Fe 2009 Limits in pseudo-dimension 1 Limit costs for uniform [0; 1] edge lengths (no normalization needed!) π2 Matching 12 ≈ 0:822 M´ezard-Parisi, Aldous Spanning tree ζ(3) ≈ 1:202 Frieze Johan W¨astlund Replica Symmetry and Combinatorial Optimization 1 2 −x Edge cover 2 min(x + e ) ≈ 0:728 Not yet proved, but the bipartite case is done in joint work with M. Hessler Physics of Algorithms, Santa Fe 2009 Limits in pseudo-dimension 1 Limit costs for uniform [0; 1] edge lengths (no normalization needed!) π2 Matching 12 ≈ 0:822 M´ezard-Parisi, Aldous Spanning tree ζ(3) ≈ 1:202 Frieze R 1 −x −y TSP/2-factor 0 ydx, (1 + x=2)e + (1 + y=2)e = 1 ≈ 2:0415 Krauth-M´ezard-Parisi, W¨astlund(TSP ∼ 2-factor proved by Frieze) Johan W¨astlund Replica Symmetry and Combinatorial Optimization Physics of Algorithms, Santa Fe 2009 Limits in pseudo-dimension 1 Limit costs for uniform [0; 1] edge lengths (no normalization needed!) π2 Matching 12 ≈ 0:822 M´ezard-Parisi, Aldous Spanning tree ζ(3) ≈ 1:202 Frieze R 1 −x −y TSP/2-factor 0 ydx, (1 + x=2)e + (1 + y=2)e = 1 ≈ 2:0415 Krauth-M´ezard-Parisi, W¨astlund(TSP ∼ 2-factor proved by Frieze) 1 2 −x Edge cover 2 min(x + e ) ≈ 0:728 Not yet proved, but the bipartite case is done in joint work with M. Hessler Johan W¨astlund Replica Symmetry and Combinatorial Optimization For general d take 1=d lij = (N · Xij ) where Xij is exponential(1). This normalization is different from a moment ago! Now unit of length = distance at which expected number of neighbors equals 1. Theorem For d ≥ 1, Cost[Matching] p −! β (d) N=2 M Cost[TSP] p −! β (d) N TSP Replica symmetric predictions of βM (d) and βTSP (d) are correct Physics of Algorithms, Santa Fe 2009 Pseudo-dimension d Pseudo-dimension d means P(l < r) / r d as r ! 0 Johan W¨astlund Replica Symmetry and Combinatorial Optimization Theorem For d ≥ 1, Cost[Matching] p −! β (d) N=2 M Cost[TSP] p −! β (d) N TSP Replica symmetric predictions of βM (d) and βTSP (d) are correct Physics of Algorithms, Santa Fe 2009 Pseudo-dimension d Pseudo-dimension d means P(l < r) / r d as r ! 0 For general d take 1=d lij = (N · Xij ) where Xij is exponential(1). This normalization is different from a moment ago! Now unit of length = distance at which expected number of neighbors equals 1. Johan W¨astlund Replica Symmetry and Combinatorial Optimization Cost[TSP] p −! β (d) N TSP Replica symmetric predictions of βM (d) and βTSP (d) are correct Physics of Algorithms, Santa Fe 2009 Pseudo-dimension d Pseudo-dimension d means P(l < r) / r d as r ! 0 For general d take 1=d lij = (N · Xij ) where Xij is exponential(1). This normalization is different from a moment ago! Now unit of length = distance at which expected number of neighbors equals 1. Theorem For d ≥ 1, Cost[Matching] p −! β (d) N=2 M Johan W¨astlund Replica Symmetry and Combinatorial Optimization Replica symmetric predictions of βM (d) and βTSP (d) are correct Physics of Algorithms, Santa Fe 2009 Pseudo-dimension d Pseudo-dimension d means P(l < r) / r d as r ! 0 For general d take 1=d lij = (N · Xij ) where Xij is exponential(1). This normalization is different from a moment ago! Now unit of length = distance at which expected number of neighbors equals 1. Theorem For d ≥ 1, Cost[Matching] p −! β (d) N=2 M Cost[TSP] p −! β (d) N TSP Johan W¨astlund Replica Symmetry and Combinatorial Optimization Physics of Algorithms, Santa Fe 2009 Pseudo-dimension d Pseudo-dimension d means P(l < r) / r d as r ! 0 For general d take 1=d lij = (N · Xij ) where Xij is exponential(1).