“Hardest” problems seem to occur near SAT–UNSAT transition: CSPs: Average-case complexity (5/32)
“Hardest” problems seem to occur near SAT–UNSAT transition: B . S e h n u n e t a l . / A r t $ i c i a l I n t e l l i g e n c e 8 1 ( 1 9 9 6 ) 1 7 - 2 9 2 1 random 3-SAT [SML ’96] 2 0 - v a r i a b l e f o r m u l a s * 4 0 - v a r i a b l e f o r m u l a s - I - 5 0 - v a r i a b l e f o r m u l a s - S k computation 2 5 0 0 time (DPLL) N u m b e r 2 0 0 0 g f I J c a ll s 1 5 0 0 1 0 0 0 5 0 0 0 2 3 4 5 6 7 8 R a t i o o f c l a u s e s - t o - v a r iab le s Fig. 2. Median number of recursive DP calls for random 3-SAT formulas, as a function of the ratio of clauses to variables. s u c h “outliers” [ 11, it appears to be a more informative statistic for current purposes. 3 I n F i g . 2 , w e s e e t h e f o l l o w i n g p a t t e r n : F o r f o r m u l a s t h a t a r e e i t h e r r e l a t i v e l y s h o r t o r r e l a t i v e l y l o n g , D P f i n i s h e s q u i c k l y , b u t t h e f o r m u l a s o f m e d i u m l e n g t h t a k e m u c h l o n g e r . S i n c e f o r m u l a s w i t h f e w c l a u s e s a r e u n d e r - c o n s r r u i n e d a n d h a v e m a n y s a t i s f y i n g a s s i g n m e n t s , a n a s s i g n m e n t i s l i k e l y t o b e f o u n d e a r l y i n t h e s e a r c h . F o r m u l a s w i t h v e r y m a n y c l a u s e s a r e o v e r - c o n s t r a i n e d ( a n d u s u a l l y u n s a t i s f i a b l e ) , s o c o n t r a d i c t i o n s a r e f o u n d e a s i l y , a n d a f u l l s e a r c h c a n b e c o m p l e t e d q u i c k l y . F i n a l l y , f o r m u l a s i n b e t w e e n a r e m u c h h a r d e r b e c a u s e t h e y h a v e r e l a t i v e l y f e w ( i f a n y ) s a t i s f y i n g a s s i g n m e n t s , b u t t h e e m p t y c l a u s e w i l l o n l y b e g e n e r a t e d a f t e r a s s i g n i n g v a l u e s t o m a n y v a r i a b l e s , r e s u l t i n g CSPs: Average-case complexity (5/32) i n a d e e p s e a r c h t r e e . S i m i l a r u n d e r - a n d o v e r - c o n s t r a i n e d a r e a s h a v e b e e n f o u n d f o r r a n d o m i n s t a n c e s o f o t h e r N P - c o m p l e t e p r o b l e m s [ 8 , 3 6 1 . T h e c u r v e s i n F i g . 2 a r e f o r a l l f o r m u l a s o f a g i v e n s i z e , t h a t i s t h e y a r e c o m p o s i t e s o f s a t i s f i a b l e a n d u n s a t i s f i a b l e s u b s e t s . I n F i g . 3 t h e m e d i a n n u m b e r o f c a l l s f o r 5 0 - v a r i a b l e f o r m u l a s i s f a c t o r e d i n t o s a t i s f i a b l e a n d u n s a t i s f i a b l e c a s e s , s h o w i n g t h a t t h e t w o s e t s a r e q u i t e d i f f e r e n t . T h e e x t r e m e l y r a r e u n s a t i s f i a b l e s h o r t f o r m u l a s a r e v e r y h a r d , w h e r e a s t h e r a r e l o n g s a t i s f i a b l e f o r m u l a s r e m a i n m o d e r a t e l y d i f f i c u l t . T h u s , t h e e a s y p a r t s o f t h e c o m p o s i t e d i s t r i b u t i o n a p p e a r t o b e a c o n s e q u e n c e o f a r e l a t i v e a b u n d a n c e o f s h o r t s a t i s f i a b l e f o r m u l a s o r l o n g u n s a t i s f i a b l e o n e s . T o u n d e r s t a n d t h e h a r d a r e a i n t e r m s o f t h e l i k e l i h o o d o f s a t i s f i a b i l i t y , w e e x p e r i m e n - t a l l y d e t e r m i n e d t h e p r o b a b i l i t y t h a t a r a n d o m 5 0 - v a r i a b l e i n s t a n c e i s s a t i s f i a b l e ( F i g . 3 A reasonable question to ask is how big a sample would be required to get a good estimate of the mean. Because of the potentially exponential nature of the problem, as we increase the s a m p le s i z e , w e m a y co n t in u e t o find ever larger (but ever rarer) samples that could place the mean anywhere [ 18,331.
“Hardest” problems seem to occur near SAT–UNSAT transition: B . S e h n u n e t a l . / A r t $ i c i a l I n t e l l i g e n c e 8 1 ( 1 9 9 6 ) 1 7 - 2 9 2 1 random 3-SAT [SML ’96] 2 0 - v a r i a b l e f o r m u l a s * 4 0 - v a r i a b l e f o r m u l a s - I - 5 0 - v a r i a b l e f o r m u l a s - S k computation 2 5 0 0 time (DPLL) N u m b e r 2 0 0 0 g f I J c a ll s 1 5 0 0 1 0 0 0 5 0 0 0 2 3 4 5 6 7 8 R a t i o o f c l a u s e s - t o - v a r iab le s Fig. 2. Median number of recursive DP calls for random 3-SAT formulas, as a function of the ratio of clauses to variables. Understanding the SAT–UNSAT transition seems possibly a s u c h “outliers” [ 11, it appears to be a more informative statistic for current purposes. 3 precursor to addressing the complexity behavior of random k -SAT I n F i g . 2 , w e s e e t h e f o l l o w i n g p a t t e r n : F o r f o r m u l a s t h a t a r e e i t h e r r e l a t i v e l y s h o r t o r r e l a t i v e l y l o n g , D P f i n i s h e s q u i c k l y , b u t t h e f o r m u l a s o f m e d i u m l e n g t h t a k e m u c h l o n g e r . S i n c e f o r m u l a s w i t h f e w c l a u s e s a r e u n d e r - c o n s r r u i n e d a n d h a v e m a n y s a t i s f y i n g a s s i g n m e n t s , a n a s s i g n m e n t i s l i k e l y t o b e f o u n d e a r l y i n t h e s e a r c h . F o r m u l a s w i t h v e r y m a n y c l a u s e s a r e o v e r - c o n s t r a i n e d ( a n d u s u a l l y u n s a t i s f i a b l e ) , s o c o n t r a d i c t i o n s a r e f o u n d e a s i l y , a n d a f u l l s e a r c h c a n b e c o m p l e t e d q u i c k l y . F i n a l l y , f o r m u l a s i n b e t w e e n a r e m u c h h a r d e r b e c a u s e t h e y h a v e r e l a t i v e l y f e w ( i f a n y ) s a t i s f y i n g a s s i g n m e n t s , b u t t h e e m p t y c l a u s e w i l l o n l y b e g e n e r a t e d a f t e r a s s i g n i n g v a l u e s t o m a n y v a r i a b l e s , r e s u l t i n g CSPs: Average-case complexity (5/32) i n a d e e p s e a r c h t r e e . S i m i l a r u n d e r - a n d o v e r - c o n s t r a i n e d a r e a s h a v e b e e n f o u n d f o r r a n d o m i n s t a n c e s o f o t h e r N P - c o m p l e t e p r o b l e m s [ 8 , 3 6 1 . T h e c u r v e s i n F i g . 2 a r e f o r a l l f o r m u l a s o f a g i v e n s i z e , t h a t i s t h e y a r e c o m p o s i t e s o f s a t i s f i a b l e a n d u n s a t i s f i a b l e s u b s e t s . I n F i g . 3 t h e m e d i a n n u m b e r o f c a l l s f o r 5 0 - v a r i a b l e f o r m u l a s i s f a c t o r e d i n t o s a t i s f i a b l e a n d u n s a t i s f i a b l e c a s e s , s h o w i n g t h a t t h e t w o s e t s a r e q u i t e d i f f e r e n t . T h e e x t r e m e l y r a r e u n s a t i s f i a b l e s h o r t f o r m u l a s a r e v e r y h a r d , w h e r e a s t h e r a r e l o n g s a t i s f i a b l e f o r m u l a s r e m a i n m o d e r a t e l y d i f f i c u l t . T h u s , t h e e a s y p a r t s o f t h e c o m p o s i t e d i s t r i b u t i o n a p p e a r t o b e a c o n s e q u e n c e o f a r e l a t i v e a b u n d a n c e o f s h o r t s a t i s f i a b l e f o r m u l a s o r l o n g u n s a t i s f i a b l e o n e s . T o u n d e r s t a n d t h e h a r d a r e a i n t e r m s o f t h e l i k e l i h o o d o f s a t i s f i a b i l i t y , w e e x p e r i m e n - t a l l y d e t e r m i n e d t h e p r o b a b i l i t y t h a t a r a n d o m 5 0 - v a r i a b l e i n s t a n c e i s s a t i s f i a b l e ( F i g . 3 A reasonable question to ask is how big a sample would be required to get a good estimate of the mean. Because of the potentially exponential nature of the problem, as we increase the s a m p le s i z e , w e m a y co n t in u e t o find ever larger (but ever rarer) samples that could place the mean anywhere [ 18,331.
RSB: Statistical physics of (random) CSPs (6/32)
A major advance in the investigation of (random) CSPs was the realization that they may be regarded in the spin glass framework M´ ezard–Parisi ’85 (weighted matching), ’86 (traveling salesman), Fu–Anderson ’86 (graph partitioning) — since these pioneering works, the study of CSPs as models of disordered systems has developed into a rich theory, yielding deep insights as well as novel algorithmic ideas e.g. survey propagation [M´ ezard–Parisi–Zecchina ’02] RSB: Statistical physics of (random) CSPs (6/32)
A major advance in the investigation of (random) CSPs was the realization that they may be regarded in the spin glass framework M´ ezard–Parisi ’85 (weighted matching), ’86 (traveling salesman), Fu–Anderson ’86 (graph partitioning) — since these pioneering works, the study of CSPs as models of disordered systems has developed into a rich theory, yielding deep insights as well as novel algorithmic ideas e.g. survey propagation [M´ ezard–Parisi–Zecchina ’02] A notable consequence of the spin glass connection is an abundance of exact mathematical predictions for random CSPs (concerning threshold phenomena, solution space geometry, . . . ) RSB: Statistical physics of (random) CSPs (6/32)
A major advance in the investigation of (random) CSPs was the realization that they may be regarded in the spin glass framework M´ ezard–Parisi ’85 (weighted matching), ’86 (traveling salesman), Fu–Anderson ’86 (graph partitioning) — since these pioneering works, the study of CSPs as models of disordered systems has developed into a rich theory, yielding deep insights as well as novel algorithmic ideas e.g. survey propagation [M´ ezard–Parisi–Zecchina ’02] A notable consequence of the spin glass connection is an abundance of exact mathematical predictions for random CSPs (concerning threshold phenomena, solution space geometry, . . . ) Some predictions for dense graphs have been sucessfully proved; Parisi formula for SK spin-glasses [Parisi ’80 / Guerra ’03, Talagrand ’06] ζ p 2 q limit of random assignments [M´ ezard–Parisi ’87 / Aldous ’00] RSB: Statistical physics of (random) CSPs (6/32)
A major advance in the investigation of (random) CSPs was the realization that they may be regarded in the spin glass framework M´ ezard–Parisi ’85 (weighted matching), ’86 (traveling salesman), Fu–Anderson ’86 (graph partitioning) — since these pioneering works, the study of CSPs as models of disordered systems has developed into a rich theory, yielding deep insights as well as novel algorithmic ideas e.g. survey propagation [M´ ezard–Parisi–Zecchina ’02] A notable consequence of the spin glass connection is an abundance of exact mathematical predictions for random CSPs (concerning threshold phenomena, solution space geometry, . . . ) Some predictions for dense graphs have been sucessfully proved; Parisi formula for SK spin-glasses [Parisi ’80 / Guerra ’03, Talagrand ’06] ζ p 2 q limit of random assignments [M´ ezard–Parisi ’87 / Aldous ’00] rigorous understanding of sparse setting is comparatively lacking RSB: Statistical physics of (random) CSPs (6/32)
RSB: Sparse random CSPs with RSB (7/32)
This talk concerns the class of sparse random CSPs exhibiting (static) replica symmetry breaking ( RSB ) Solution space geometry has been investigated in several works, leading to this conjectural phase diagram: Krz¸ aka� la–Montanari–Ricci-Tersenghi–Semerjian–Zdeborov´ a ’07, Montanari–Ricci-Tersenghi–Semerjian ’08 RSB: Sparse random CSPs with RSB (7/32)
This talk concerns the class of sparse random CSPs exhibiting (static) replica symmetry breaking ( RSB ) Solution space geometry has been investigated in several works, leading to this conjectural phase diagram: Krz¸ aka� la–Montanari–Ricci-Tersenghi–Semerjian–Zdeborov´ a ’07, Montanari–Ricci-Tersenghi–Semerjian ’08 — latest in significant body of literature including Monasson–Zecchina ’96, Biroli–Monasson–Weigt ’00, M´ ezard–Parisi–Zecchina ’02, M´ ezard–Mora–Zecchina ’05, M´ ezard–Palassini–Rivoire ’05, Achlioptas–Ricci-Tersenghi ’06 RSB: Sparse random CSPs with RSB (7/32)
This talk concerns the class of sparse random CSPs exhibiting (static) replica symmetry breaking ( RSB ) Solution space geometry has been investigated in several works, leading to this conjectural phase diagram: Krz¸ aka� la–Montanari–Ricci-Tersenghi–Semerjian–Zdeborov´ a ’07, Montanari–Ricci-Tersenghi–Semerjian ’08 — latest in significant body of literature including Monasson–Zecchina ’96, Biroli–Monasson–Weigt ’00, M´ ezard–Parisi–Zecchina ’02, M´ ezard–Mora–Zecchina ’05, M´ ezard–Palassini–Rivoire ’05, Achlioptas–Ricci-Tersenghi ’06 We are interested in the rigorous computation of sharp satisfiability thresholds for this class of models RSB: Sparse random CSPs with RSB (7/32)
RSB: Prior work for CSPs without RSB (8/32)
Prior rigorous work for sparse CSPs without RSB : the exact satisfiability threshold has been proved for several problems: RSB: Prior work for CSPs without RSB (8/32)
Prior rigorous work for sparse CSPs without RSB : the exact satisfiability threshold has been proved for several problems: ‚ 2-SAT transition Goerdt ’92, ’96, Chv´ atal–Reed ’92, de la Vega ’92 scaling window: Bollob´ as–Borgs–Chayes–Kim–Wilson ’01 ‚ 1-in- k -SAT transition Achlioptas–Chtcherba–Istrate–Moore ’01 ‚ k -XOR-SAT transition Dubois–Mandler ’02, Dietzfelbinger–Goerdt– –Mitzenmacher–Montanari–Pagh–Rink ’10, Pittel–Sorkin ’12 RSB: Prior work for CSPs without RSB (8/32)
RSB: Prior work for CSPs with RSB (9/32)
For sparse CSPs with RSB , threshold behavior long in question Rigorous bounds on the SAT–UNSAT transition include: RSB: Prior work for CSPs with RSB (9/32)
For sparse CSPs with RSB , threshold behavior long in question Rigorous bounds on the SAT–UNSAT transition include: ‚ random regular graph independent set Bollob´ as ’81, McKay ’87, Frieze–� Luczak ’92, Frieze–Suen ’94, Wormald ’95 ‚ random graph coloring Bollob´ as ’88, Achlioptas–Naor ’04, Coja-Oghlan–Vilenchik ’13 ‚ random k -NAE-SAT Achlioptas–Moore ’02, Coja-Oghlan–Zdeborov´ a ’12, Coja-Oghlan–Panagiotou ’12 ‚ random k -SAT Kirousis et al. ’97, Franz–Leone ’03, Achlioptas–Peres ’03, Coja-Oghlan–Panagiotou ’13, Coja-Oghlan ’14 RSB: Prior work for CSPs with RSB (9/32)
For sparse CSPs with RSB , threshold behavior long in question Rigorous bounds on the SAT–UNSAT transition include: ‚ random regular graph independent set Bollob´ as ’81, McKay ’87, Frieze–� Luczak ’92, Frieze–Suen ’94, Wormald ’95 ‚ random graph coloring Bollob´ as ’88, Achlioptas–Naor ’04, Coja-Oghlan–Vilenchik ’13 ‚ random k -NAE-SAT Achlioptas–Moore ’02, Coja-Oghlan–Zdeborov´ a ’12, Coja-Oghlan–Panagiotou ’12 ‚ random k -SAT Kirousis et al. ’97, Franz–Leone ’03, Achlioptas–Peres ’03, Coja-Oghlan–Panagiotou ’13, Coja-Oghlan ’14 (gap remains in all models: threshold existence not implied) RSB: Prior work for CSPs with RSB (9/32)
For sparse CSPs with RSB , threshold behavior long in question Rigorous bounds on the SAT–UNSAT transition include: ‚ random regular graph independent set Bollob´ as ’81, McKay ’87, Frieze–� Luczak ’92, Frieze–Suen ’94, Wormald ’95 ‚ random graph coloring Bollob´ as ’88, Achlioptas–Naor ’04, Coja-Oghlan–Vilenchik ’13 ‚ random k -NAE-SAT Achlioptas–Moore ’02, Coja-Oghlan–Zdeborov´ a ’12, Coja-Oghlan–Panagiotou ’12 ‚ random k -SAT Kirousis et al. ’97, Franz–Leone ’03, Achlioptas–Peres ’03, Coja-Oghlan–Panagiotou ’13, Coja-Oghlan ’14 (gap remains in all models: threshold existence not implied) Existence of threshold sequence (possibly non-convergent) Friedgut ’99 RSB: Prior work for CSPs with RSB (9/32)
For sparse CSPs with RSB , threshold behavior long in question Rigorous bounds on the SAT–UNSAT transition include: ‚ random regular graph independent set Bollob´ as ’81, McKay ’87, Frieze–� Luczak ’92, Frieze–Suen ’94, Wormald ’95 ‚ random graph coloring Bollob´ as ’88, Achlioptas–Naor ’04, Coja-Oghlan–Vilenchik ’13 ‚ random k -NAE-SAT Achlioptas–Moore ’02, Coja-Oghlan–Zdeborov´ a ’12, Coja-Oghlan–Panagiotou ’12 ‚ random k -SAT Kirousis et al. ’97, Franz–Leone ’03, Achlioptas–Peres ’03, Coja-Oghlan–Panagiotou ’13, Coja-Oghlan ’14 (gap remains in all models: threshold existence not implied) Existence of threshold sequence (possibly non-convergent) Friedgut ’99 Existence of sharp threshold Bayati–Gamarnik–Tetali ’10 (cannot determine threshold location; does not cover random SAT) RSB: Prior work for CSPs with RSB (9/32)
RSB: 1-RSB subclass (10/32)
Many problems within this class (including all on previous slide) are believed to be described by the 1-RSB formalism (one-step replica symmetry breaking) M´ ezard–Parisi ’01 RSB: 1-RSB subclass (10/32)
Many problems within this class (including all on previous slide) are believed to be described by the 1-RSB formalism (one-step replica symmetry breaking) M´ ezard–Parisi ’01 For such problems, the 1-RSB cavity method predicts the exact location of the SAT–UNSAT transition M´ ezard–Parisi-Zecchina ’02, Mertens–M´ ezard–Zecchina ’06 (based on assumptions that are difficult to verify mathematically) RSB: 1-RSB subclass (10/32)
Many problems within this class (including all on previous slide) are believed to be described by the 1-RSB formalism (one-step replica symmetry breaking) M´ ezard–Parisi ’01 For such problems, the 1-RSB cavity method predicts the exact location of the SAT–UNSAT transition M´ ezard–Parisi-Zecchina ’02, Mertens–M´ ezard–Zecchina ’06 (based on assumptions that are difficult to verify mathematically) RSB: 1-RSB subclass (10/32)
Many problems within this class (including all on previous slide) are believed to be described by the 1-RSB formalism (one-step replica symmetry breaking) M´ ezard–Parisi ’01 For such problems, the 1-RSB cavity method predicts the exact location of the SAT–UNSAT transition M´ ezard–Parisi-Zecchina ’02, Mertens–M´ ezard–Zecchina ’06 (based on assumptions that are difficult to verify mathematically) In our work we give rigorous verifications of the 1-RSB prediction for the SAT–UNSAT transition, for the following models: RSB: 1-RSB subclass (10/32)
Many problems within this class (including all on previous slide) are believed to be described by the 1-RSB formalism (one-step replica symmetry breaking) M´ ezard–Parisi ’01 For such problems, the 1-RSB cavity method predicts the exact location of the SAT–UNSAT transition M´ ezard–Parisi-Zecchina ’02, Mertens–M´ ezard–Zecchina ’06 (based on assumptions that are difficult to verify mathematically) In our work we give rigorous verifications of the 1-RSB prediction for the SAT–UNSAT transition, for the following models: ‚ random regular k -NAE-SAT (next few slides) RSB: 1-RSB subclass (10/32)
Many problems within this class (including all on previous slide) are believed to be described by the 1-RSB formalism (one-step replica symmetry breaking) M´ ezard–Parisi ’01 For such problems, the 1-RSB cavity method predicts the exact location of the SAT–UNSAT transition M´ ezard–Parisi-Zecchina ’02, Mertens–M´ ezard–Zecchina ’06 (based on assumptions that are difficult to verify mathematically) In our work we give rigorous verifications of the 1-RSB prediction for the SAT–UNSAT transition, for the following models: ‚ random regular k -NAE-SAT (next few slides) ‚ random regular graph independent set (rest of the talk) RSB: 1-RSB subclass (10/32)
boolean satisfiability
Random (Erd˝ os–R´ enyi) k -CNF is uniform measure over all n -variable, m -clause k -CNF’s ( p 2 n q mk formulas; constraint structure is Erd˝ os–R´ enyi hyper-graph) Random regular k -CNF is uniform measure over all n -variable, m -clause k -CNF’s with fixed variable degree d “ mk { n (2 mk p mk q ! {p d ! q n formulas; constraint structure is regular hyper-graph) “Constraint parameter” is clause density α “ m { n Benchmark problem: SAT–UNSAT transition in random k -SAT ( UBD ) Franco–Paull ’83, Kirousis–Kranakis–Krizanc–Stamatiou ’97; ( LBD ) Chao–Franco ’90, Achlioptas–Moore ’02, Achlioptas–Peres ’03, Coja-Oghlan–Panagiotou ’13, Coja-Oghlan ’14 (gap remains in bounds) SAT: Random SAT (11/32)
Random k -SAT threshold is close to 2 k log 2, but the best known algorithmic lower bound is only — 2 k log k { k Coja-Oghlan ’10 First — 2 k LBD for random k -SAT achieved by non-algorithmic analysis of random k -NAE-SAT : Achlioptas–Moore ’02 harder to satisfy, but easier to study, than SAT A NAE-SAT solution is a SAT solution x such that � x is also SAT — eliminates TRUE/FALSE asymmetry of SAT; but believed to exhibit many of the same qualitative phenomena Bounds on SAT–UNSAT in random (Erd˝ os–R´ enyi) k -NAE-SAT: AM ’02, Coja-Oghlan–Zdeborov´ a ’12, Coja-Oghlan–Panagiotou ’12 lower bounds (approx. halves) the SAT transition (gap remains in bounds) SAT: Random NAE-SAT (12/32)
(main result for NAE-SAT) T HEOREM . Ding, Sly, S. [arXiv:1310.4784, STOC ’14] The random regular k-NAE-SAT problem has SAT–UNSAT transition at explicit threshold α ‹ p k q for all k ě k 0 . In simultaneous work, A. Coja-Oghlan [arXiv:1310.2728v1] considered a different symmetrization of random regular k -SAT, establishing a 1-RSB-type formula for a “quasi-satisfiability” threshold SAT: Threshold for random regular NAE-SAT (13/32)
independent sets
IS: Definition (14/32)
In an undirected graph, an independent set IS: Definition (14/32)
In an undirected graph, an independent set is a subset of vertices containing no neighbors (equivalently, the complement is a vertex cover) IS: Definition (14/32)
IS: Random graphs (15/32)
“Constraint parameter” of random SAT is clause density m { n IS: Random graphs (15/32)
“Constraint parameter” of random SAT is clause density m { n “Constraint parameter” of independent set is the set density — IS: Random graphs (15/32)
“Constraint parameter” of random SAT is clause density m { n “Constraint parameter” of independent set is the set density — SAT–UNSAT corresponds to max-density ( independence ratio ) IS: Random graphs (15/32)
“Constraint parameter” of random SAT is clause density m { n “Constraint parameter” of independent set is the set density — SAT–UNSAT corresponds to max-density ( independence ratio ) The independence ratio is NP-hard to compute exactly; Karp ’72 in fact it is hard to approximate even on bounded-degree graphs Papadimitriou–Yannakakis ’91 and PCP theorem IS: Random graphs (15/32)
“Constraint parameter” of random SAT is clause density m { n “Constraint parameter” of independent set is the set density — SAT–UNSAT corresponds to max-density ( independence ratio ) The independence ratio is NP-hard to compute exactly; Karp ’72 in fact it is hard to approximate even on bounded-degree graphs Papadimitriou–Yannakakis ’91 and PCP theorem Randomize the problem by taking a random graph — IS: Random graphs (15/32)
“Constraint parameter” of random SAT is clause density m { n “Constraint parameter” of independent set is the set density — SAT–UNSAT corresponds to max-density ( independence ratio ) The independence ratio is NP-hard to compute exactly; Karp ’72 in fact it is hard to approximate even on bounded-degree graphs Papadimitriou–Yannakakis ’91 and PCP theorem Randomize the problem by taking a random graph — let A n ” MAX-IND-SET size in random graph G n on n vertices: IS: Random graphs (15/32)
“Constraint parameter” of random SAT is clause density m { n “Constraint parameter” of independent set is the set density — SAT–UNSAT corresponds to max-density ( independence ratio ) The independence ratio is NP-hard to compute exactly; Karp ’72 in fact it is hard to approximate even on bounded-degree graphs Papadimitriou–Yannakakis ’91 and PCP theorem Randomize the problem by taking a random graph — let A n ” MAX-IND-SET size in random graph G n on n vertices: for natural ensembles G n , what are the asymptotics of A n ? dense ER graph G n , p , sparse ER graph G n , d { n , (uniform) random regular graph G n , d IS: Random graphs (15/32)
“Constraint parameter” of random SAT is clause density m { n “Constraint parameter” of independent set is the set density — SAT–UNSAT corresponds to max-density ( independence ratio ) The independence ratio is NP-hard to compute exactly; Karp ’72 in fact it is hard to approximate even on bounded-degree graphs Papadimitriou–Yannakakis ’91 and PCP theorem Randomize the problem by taking a random graph — let A n ” MAX-IND-SET size in random graph G n on n vertices: for natural ensembles G n , what are the asymptotics of A n ? dense ER graph G n , p , sparse ER graph G n , d { n , (uniform) random regular graph G n , d Sharpness of the SAT–UNSAT transition corresponds to concentration of the random variable A n IS: Random graphs (15/32)
IS: Previous work (16/32)
Previous work on random graph independent sets: IS: Previous work (16/32)
Previous work on random graph independent sets: Dense Erd˝ os–R´ enyi ensemble G n , p Grimmett–McDiarmid ’75 IS: Previous work (16/32)
Previous work on random graph independent sets: Dense Erd˝ os–R´ enyi ensemble G n , p Grimmett–McDiarmid ’75 IS: Previous work (16/32)
Previous work on random graph independent sets: Dense Erd˝ os–R´ enyi ensemble G n , p Grimmett–McDiarmid ’75 Sparse Erd˝ os–R´ enyi G n , d { n ; random d -regular G n , d IS: Previous work (16/32)
Previous work on random graph independent sets: Dense Erd˝ os–R´ enyi ensemble G n , p Grimmett–McDiarmid ’75 Sparse Erd˝ os–R´ enyi G n , d { n ; random d -regular G n , d ( UBD ) Bollob´ as ’81, McKay ’87; ( LBD ) Frieze–� Luczak ’92, Frieze–Suen ’94, Wormald ’95 (threshold around 2 p log d q{ d , but gap remains) IS: Previous work (16/32)
Previous work on random graph independent sets: Dense Erd˝ os–R´ enyi ensemble G n , p Grimmett–McDiarmid ’75 Sparse Erd˝ os–R´ enyi G n , d { n ; random d -regular G n , d ( UBD ) Bollob´ as ’81, McKay ’87; ( LBD ) Frieze–� Luczak ’92, Frieze–Suen ’94, Wormald ’95 (threshold around 2 p log d q{ d , but gap remains) Classical argument with martingale bound (’80s) implies the transition sharpens: A n has O p n 1 { 2 q fluctuations about E A n IS: Previous work (16/32)
Previous work on random graph independent sets: Dense Erd˝ os–R´ enyi ensemble G n , p Grimmett–McDiarmid ’75 Sparse Erd˝ os–R´ enyi G n , d { n ; random d -regular G n , d ( UBD ) Bollob´ as ’81, McKay ’87; ( LBD ) Frieze–� Luczak ’92, Frieze–Suen ’94, Wormald ’95 (threshold around 2 p log d q{ d , but gap remains) Classical argument with martingale bound (’80s) implies the transition sharpens: A n has O p n 1 { 2 q fluctuations about E A n Existence of limiting threshold location A n { n Ñ α ‹ proved, but with no information on the actual value Bayati–Gamarnik–Tetali ’10 IS: Previous work (16/32)
(main result for MAX-IND-SET) IS: Threshold for random regular MAX-IND-SET (17/32)
(main result for MAX-IND-SET) T HEOREM . Ding, Sly, S. [arXiv:1310.4787] The maximum independent set size A n in the (uniformly) random d-regular graph G n , d IS: Threshold for random regular MAX-IND-SET (17/32)
(main result for MAX-IND-SET) T HEOREM . Ding, Sly, S. [arXiv:1310.4787] The maximum independent set size A n in the (uniformly) random d-regular graph G n , d has O p 1 q fluctuations around n α ‹ ´ c ‹ log n for explicit α ‹ p d q and c ‹ p d q , provided d ě d 0 . IS: Threshold for random regular MAX-IND-SET (17/32)
IS: Explicit formula (18/32)
Explicit formula for independent set threshold: IS: Explicit formula (18/32)
Explicit formula for independent set threshold: first define φ p q q ” ´ log r 1 ´ q p 1 ´ 1 { λ qs ´ p d { 2 ´ 1 q log r 1 ´ q 2 p 1 ´ 1 { λ qs ´ α log λ IS: Explicit formula (18/32)
Explicit formula for independent set threshold: first define φ p q q ” ´ log r 1 ´ q p 1 ´ 1 { λ qs ´ p d { 2 ´ 1 q log r 1 ´ q 2 p 1 ´ 1 { λ qs ´ α log λ with λ p q q ” q 1 ´ p 1 ´ q q d ´ 1 p 1 ´ q q d and α p q q ” q 1 ´ q ` dq {r 2 λ p q qs 1 ´ q 2 p 1 ´ 1 { λ p q qq IS: Explicit formula (18/32)
Explicit formula for independent set threshold: first define φ p q q ” ´ log r 1 ´ q p 1 ´ 1 { λ qs ´ p d { 2 ´ 1 q log r 1 ´ q 2 p 1 ´ 1 { λ qs ´ α log λ with λ p q q ” q 1 ´ p 1 ´ q q d ´ 1 p 1 ´ q q d and α p q q ” q 1 ´ q ` dq {r 2 λ p q qs 1 ´ q 2 p 1 ´ 1 { λ p q qq Solve for the largest zero q ‹ ď 2 p log d q{ d of φ p q q : IS: Explicit formula (18/32)
Explicit formula for independent set threshold: first define φ p q q ” ´ log r 1 ´ q p 1 ´ 1 { λ qs ´ p d { 2 ´ 1 q log r 1 ´ q 2 p 1 ´ 1 { λ qs ´ α log λ with λ p q q ” q 1 ´ p 1 ´ q q d ´ 1 p 1 ´ q q d and α p q q ” q 1 ´ q ` dq {r 2 λ p q qs 1 ´ q 2 p 1 ´ 1 { λ p q qq Solve for the largest zero q ‹ ď 2 p log d q{ d of φ p q q : then A n ´ n α ‹ ´ c ‹ log n is a tight random variable with α ‹ “ α p q ‹ q and c ‹ “ p 2 log λ p q ‹ qq ´ 1 IS: Explicit formula (18/32)
the function φ p q q for d “ 100 0 . 05 φ 0 − 0 . 05 (log d ) /d q ⋆ 2(log d ) /d 0 IS: Explicit rate function (19/32)
(some remarks) Remarks (20/32)
(some remarks) Our thresholds match the 1-RSB predictions made by physicists (NAE-SAT) Castellani–Napolano–Ricci-Tersenghi–Zecchina ’03, Dall’Asta–Ramezanpour–Zecchina ’08; (independent set) Rivoire ’05, Hartmann–Weigt ’05, Barbier–Krz¸ aka� la–Zdeborov´ a–Zhang ’13 Remarks (20/32)
(some remarks) Our thresholds match the 1-RSB predictions made by physicists (NAE-SAT) Castellani–Napolano–Ricci-Tersenghi–Zecchina ’03, Dall’Asta–Ramezanpour–Zecchina ’08; (independent set) Rivoire ’05, Hartmann–Weigt ’05, Barbier–Krz¸ aka� la–Zdeborov´ a–Zhang ’13 These predictions were derived with the survey propagation ( SP ) method introduced by M´ ezard–Parisi–Zecchina ’02, ’05 see also Braunstein–M´ ezard–Zecchina ’05, Maneva–Mossel–Wainwright ’07 Remarks (20/32)
(some remarks) Our thresholds match the 1-RSB predictions made by physicists (NAE-SAT) Castellani–Napolano–Ricci-Tersenghi–Zecchina ’03, Dall’Asta–Ramezanpour–Zecchina ’08; (independent set) Rivoire ’05, Hartmann–Weigt ’05, Barbier–Krz¸ aka� la–Zdeborov´ a–Zhang ’13 These predictions were derived with the survey propagation ( SP ) method introduced by M´ ezard–Parisi–Zecchina ’02, ’05 see also Braunstein–M´ ezard–Zecchina ’05, Maneva–Mossel–Wainwright ’07 Our method of proof gives some rigorous validation to the 1-RSB & SP heuristics for these models Remarks (20/32)
RSB and moment method
Moments: First and second moment method (21/32)
(probabilistic methods for rigorously bounding the SAT–UNSAT transition) The SAT–UNSAT transition is the threshold for positivity of the random variable Z α ” # solutions at constraint level α (# independent sets of density α in G n , d ) Moments: First and second moment method (21/32)
(probabilistic methods for rigorously bounding the SAT–UNSAT transition) The SAT–UNSAT transition is the threshold for positivity of the random variable Z α ” # solutions at constraint level α (# independent sets of density α in G n , d ) Upper bound is given by the 1 st moment threshold α 1 where E Z α crosses from exponentially large to exponentially small Moments: First and second moment method (21/32)
(probabilistic methods for rigorously bounding the SAT–UNSAT transition) The SAT–UNSAT transition is the threshold for positivity of the random variable Z α ” # solutions at constraint level α (# independent sets of density α in G n , d ) Upper bound is given by the 1 st moment threshold α 1 where E Z α crosses from exponentially large to exponentially small Lower bound : algorithmic analysis meets with barriers; and the (non-constructive) 2 nd moment approach often does much better: e.g. Achlioptas–Moore ’02 Moments: First and second moment method (21/32)
(probabilistic methods for rigorously bounding the SAT–UNSAT transition) The SAT–UNSAT transition is the threshold for positivity of the random variable Z α ” # solutions at constraint level α (# independent sets of density α in G n , d ) Upper bound is given by the 1 st moment threshold α 1 where E Z α crosses from exponentially large to exponentially small Lower bound : algorithmic analysis meets with barriers; and the (non-constructive) 2 nd moment approach often does much better: e.g. Achlioptas–Moore ’02 P p Z ą 0 q ě p E Z q 2 2 nd moment LBD : (apply with Z “ Z α ) E r Z 2 s Moments: First and second moment method (21/32)
P p Z ą 0 q ě p E Z q 2 E r Z 2 s Moments: Second moment lower bound (22/32)
ř ř P p Z ą 0 q ě p E Z q 2 τ P p σ valid q ˆ P p τ valid q σ E r Z 2 s “ ř ř τ P p σ valid AND τ valid q σ Moments: Second moment lower bound (22/32)
ř ř P p Z ą 0 q ě p E Z q 2 τ P p σ valid q ˆ P p τ valid q σ E r Z 2 s “ ř ř τ P p σ valid AND τ valid q σ E r Z 2 s has contribution E Z from exactly-identical pairs σ “ τ ; so contribution from near-identical pairs is clearly at least E Z Moments: Second moment lower bound (22/32)
ř ř P p Z ą 0 q ě p E Z q 2 τ P p σ valid q ˆ P p τ valid q σ E r Z 2 s “ ř ř τ P p σ valid AND τ valid q σ E r Z 2 s has contribution E Z from exactly-identical pairs σ “ τ ; so contribution from near-identical pairs is clearly at least E Z In a sparse CSPs, a typical solution has — n unforced variables, indicating exponential-size clusters of near-identical solutions: near-identical contribution to E r Z 2 s is « p E Z q ˆ p avg. cluster size q Moments: Second moment lower bound (22/32)
ř ř P p Z ą 0 q ě p E Z q 2 τ P p σ valid q ˆ P p τ valid q σ E r Z 2 s “ ř ř τ P p σ valid AND τ valid q σ E r Z 2 s has contribution E Z from exactly-identical pairs σ “ τ ; so contribution from near-identical pairs is clearly at least E Z In a sparse CSPs, a typical solution has — n unforced variables, indicating exponential-size clusters of near-identical solutions: near-identical contribution to E r Z 2 s is « p E Z q ˆ p avg. cluster size q If p avg. cluster size q ≫ E Z then 2 nd moment method fails — occurs if avg. cluster size does not decrease fast enough as α increases towards the 1 st moment threshold Moments: Second moment lower bound (22/32)
Moments: Clustering in independent sets (23/32)
An independent set at density α P p 0 , 1 q must have a positive fraction π of unoccupied vertices with a single occupied neighbor Moments: Clustering in independent sets (23/32)
An independent set at density α P p 0 , 1 q must have a positive fraction π of unoccupied vertices with a single occupied neighbor Such vertices are unforced , indicating a cluster of size ě 2 n π Moments: Clustering in independent sets (23/32)
An independent set at density α P p 0 , 1 q must have a positive fraction π of unoccupied vertices with a single occupied neighbor Such vertices are unforced , indicating a cluster of size ě 2 n π Issue is that π stays positive even above 1 st moment threshold — 2 nd moment begins to fail strictly below the 1 st moment threshold Moments: Clustering in independent sets (23/32)
An independent set at density α P p 0 , 1 q must have a positive fraction π of unoccupied vertices with a single occupied neighbor Such vertices are unforced , indicating a cluster of size ě 2 n π Issue is that π stays positive even above 1 st moment threshold — 2 nd moment begins to fail strictly below the 1 st moment threshold In regime p α 2 , α 1 q , E Z ≫ 1 but E r Z 2 s ≫ p E Z q 2 — that is to say, Z is highly non-concentrated, and the 1 st /2 nd moment method yields no information about its typical behavior Moments: Clustering in independent sets (23/32)
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