Shifing the thresold of phase transition in 2-SAT and random graphs - PowerPoint PPT Presentation

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Shifing the thresold of phase transition in 2-SAT and random graphs Sergey Dovgal 1 , 2 , 3 , 4 Vlady Ravelomanana 2 1 Université Paris-13, 2 Université Paris-Diderot 3 Moscow Institute of Physics and Technology 4 Institute for Information Transmission Problems, Moscow Acknowledgements: Élie de Panafieu, Fedor Petrov, ipython+sympy+cpp May 19, 2017 D., Ravelomanana Shifing the phase transition 1 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Outline 2-SAT, phase transitions and degree constraints 1 Lower bound for 2-SAT 2 3 Saddle-point method and analytic lemma 4 Related results D., Ravelomanana Shifing the phase transition 2 / 40

2-SAT, phase transitions and degree constraints Phase transition Lower bound for 2-SAT Shifing the phase transition Saddle-point method and analytic lemma Graphs with degree constraints Related results Experimental results Outline 1 2-SAT, phase transitions and degree constraints 2 Lower bound for 2-SAT 3 Saddle-point method and analytic lemma 4 Related results D., Ravelomanana Shifing the phase transition 3 / 40

2-SAT, phase transitions and degree constraints Phase transition Lower bound for 2-SAT Shifing the phase transition Saddle-point method and analytic lemma Graphs with degree constraints Related results Experimental results Phase transition in Erdős–Rényi random graphs n vertices, m edges, m = 1 2 n ( 1 + µ n − 1 / 3 ) “gas” µ → −∞ : planar graph, trees and unicycles, max 1 component size O ( log n ) . “liquid” | µ | = O ( 1 ) : complex components appear, max 2 component size O ( n 2 / 3 ) . “crystal” µ → + ∞ : non-planar, complex compontnes, max 3 component size linear O ( n ) . D., Ravelomanana Shifing the phase transition 4 / 40

2-SAT, phase transitions and degree constraints Phase transition Lower bound for 2-SAT Shifing the phase transition Saddle-point method and analytic lemma Graphs with degree constraints Related results Experimental results Phase transition in Erdős–Rényi random graphs n vertices, m edges, m = 1 2 n ( 1 + µ n − 1 / 3 ) “gas” µ → −∞ : planar graph, trees and unicycles, max 1 component size O ( log n ) . “liquid” | µ | = O ( 1 ) : complex components appear, max 2 component size O ( n 2 / 3 ) . “crystal” µ → + ∞ : non-planar, complex compontnes, max 3 component size linear O ( n ) . D., Ravelomanana Shifing the phase transition 4 / 40

2-SAT, phase transitions and degree constraints Phase transition Lower bound for 2-SAT Shifing the phase transition Saddle-point method and analytic lemma Graphs with degree constraints Related results Experimental results Phase transition in Erdős–Rényi random graphs n vertices, m edges, m = 1 2 n ( 1 + µ n − 1 / 3 ) “gas” µ → −∞ : planar graph, trees and unicycles, max 1 component size O ( log n ) . “liquid” | µ | = O ( 1 ) : complex components appear, max 2 component size O ( n 2 / 3 ) . “crystal” µ → + ∞ : non-planar, complex compontnes, max 3 component size linear O ( n ) . D., Ravelomanana Shifing the phase transition 4 / 40

2-SAT, phase transitions and degree constraints Phase transition Lower bound for 2-SAT Shifing the phase transition Saddle-point method and analytic lemma Graphs with degree constraints Related results Experimental results Phase transition :: largest component, n = 1000 D., Ravelomanana Shifing the phase transition 5 / 40

2-SAT, phase transitions and degree constraints Phase transition Lower bound for 2-SAT Shifing the phase transition Saddle-point method and analytic lemma Graphs with degree constraints Related results Experimental results Phase transition :: planarity, n = 1000 D., Ravelomanana Shifing the phase transition 6 / 40

2-SAT, phase transitions and degree constraints Phase transition Lower bound for 2-SAT Shifing the phase transition Saddle-point method and analytic lemma Graphs with degree constraints Related results Experimental results Phase transition :: diameter, n = 1000 D., Ravelomanana Shifing the phase transition 7 / 40

2-SAT, phase transitions and degree constraints Phase transition Lower bound for 2-SAT Shifing the phase transition Saddle-point method and analytic lemma Graphs with degree constraints Related results Experimental results Phase transition :: connected components, n = 1000 D., Ravelomanana Shifing the phase transition 8 / 40

2-SAT, phase transitions and degree constraints Phase transition Lower bound for 2-SAT Shifing the phase transition Saddle-point method and analytic lemma Graphs with degree constraints Related results Experimental results 2SAT Transition 1 [Bollobás, Borgs, Chayes, Kim, and Wilson ’99] 2SAT Transition 2 [Coppersmith, Gamarnik, Hajaghayi, Sorkin ’03] MAX 2-SAT Transition 3 [Cooper, Freize, Sorkin ’07] 2SAT with degree sequence constraints D., Ravelomanana Shifing the phase transition 9 / 40

2-SAT, phase transitions and degree constraints Phase transition Lower bound for 2-SAT Shifing the phase transition Saddle-point method and analytic lemma Graphs with degree constraints Related results Experimental results Shifing the phase transition m = 1 2 n ( 1 + µ n − 1 / 3 ) ⇒ m = α n ( 1 + µ n − 1 / 3 ) 1 Achlioptas percolation process ( α = 0 . 535?) 2 Degree sequence models (less detailed information) 3 Degree set constraint :: current talk D., Ravelomanana Shifing the phase transition 10 / 40

2-SAT, phase transitions and degree constraints Phase transition Lower bound for 2-SAT Shifing the phase transition Saddle-point method and analytic lemma Graphs with degree constraints Related results Experimental results Shifing the phase transition m = 1 2 n ( 1 + µ n − 1 / 3 ) ⇒ m = α n ( 1 + µ n − 1 / 3 ) 1 Achlioptas percolation process ( α = 0 . 535?) 2 Degree sequence models (less detailed information) 3 Degree set constraint :: current talk D., Ravelomanana Shifing the phase transition 10 / 40

2-SAT, phase transitions and degree constraints Phase transition Lower bound for 2-SAT Shifing the phase transition Saddle-point method and analytic lemma Graphs with degree constraints Related results Experimental results Shifing the phase transition m = 1 2 n ( 1 + µ n − 1 / 3 ) ⇒ m = α n ( 1 + µ n − 1 / 3 ) 1 Achlioptas percolation process ( α = 0 . 535?) 2 Degree sequence models (less detailed information) 3 Degree set constraint :: current talk D., Ravelomanana Shifing the phase transition 10 / 40

2-SAT, phase transitions and degree constraints Phase transition Lower bound for 2-SAT Shifing the phase transition Saddle-point method and analytic lemma Graphs with degree constraints Related results Experimental results Example of graph with degree constraints 21 9 10 23 13 5 24 12 11 14 3 17 20 25 8 2 7 26 19 1 4 18 16 6 15 22 Figure: Random labeled graph from G 26 , 30 , Ω with the set of degree constraints Ω = { 1 , 2 , 3 , 5 , 7 } . D., Ravelomanana Shifing the phase transition 11 / 40

2-SAT, phase transitions and degree constraints Phase transition Lower bound for 2-SAT Shifing the phase transition Saddle-point method and analytic lemma Graphs with degree constraints Related results Experimental results Constant of phase transition Ω — the set of degree constraints 1 Random graphs m = 1 ? 2 n ( 1 + µ n − 1 / 3 ) ⇒ m = α n ( 1 + µ n − 1 / 3 ) 2 Random 2-CNF ? m = 1 · n ( 1 + µ n − 1 / 3 ) ⇒ m = 2 α n ( 1 + µ n − 1 / 3 ) 3 How to compute α depending on Ω ? D., Ravelomanana Shifing the phase transition 12 / 40

2-SAT, phase transitions and degree constraints Phase transition Lower bound for 2-SAT Shifing the phase transition Saddle-point method and analytic lemma Graphs with degree constraints Related results Experimental results Constant of phase transition Ω — the set of degree constraints 1 Random graphs m = 1 ? 2 n ( 1 + µ n − 1 / 3 ) ⇒ m = α n ( 1 + µ n − 1 / 3 ) 2 Random 2-CNF ? m = 1 · n ( 1 + µ n − 1 / 3 ) ⇒ m = 2 α n ( 1 + µ n − 1 / 3 ) 3 How to compute α depending on Ω ? D., Ravelomanana Shifing the phase transition 12 / 40

2-SAT, phase transitions and degree constraints Phase transition Lower bound for 2-SAT Shifing the phase transition Saddle-point method and analytic lemma Graphs with degree constraints Related results Experimental results Constant of phase transition Ω — the set of degree constraints 1 Random graphs m = 1 ? 2 n ( 1 + µ n − 1 / 3 ) ⇒ m = α n ( 1 + µ n − 1 / 3 ) 2 Random 2-CNF ? m = 1 · n ( 1 + µ n − 1 / 3 ) ⇒ m = 2 α n ( 1 + µ n − 1 / 3 ) 3 How to compute α depending on Ω ? D., Ravelomanana Shifing the phase transition 12 / 40