1. Counting and Sampling in Lattices: The Computer Science Perspective Dana Randall Advance Professor of Computing Georgia Institute of Technology

2. Counting and Random Sampling Question: In polynomial time, can we: decide if there is one? find one? count them? sample one at random? In general: Y N ?/Y Eg., Perfect Matchings

3. What about matchings on lattices? On the lattice

4. What about matchings on lattices? On the lattice

5. “Domino tilings” What about matchings on lattices? On the lattice

6. What about matchings on lattices? Q: How many? “Domino tilings” On the lattice

7. What about matchings on lattices? ≥ “Domino tilings” On the lattice Q: How many?

8. ≥ 2 (Area/4) total tilings. What about matchings on lattices? ≥ “Domino tilings” On the lattice Q: How many?

9. What about matchings on lattices? Q2: What does a typical one look like? “Domino tilings” “Lozenge tilings” On the lattice Q: How many?

10. Counting and Random Sampling Question: In polynomial time, can we: decide if there is one? find one? count them? sample one at random? In general: Y N ?/Y Eg., Perfect Matchings On lattices: Y

11. Other Computational Problems in the Sciences Some (unrelated?) problems: Nanotechnology - self-assembly Computer sciences - sampling/counting (e.g., image segmentation) Physics - phase transitions Chemistry - colloids

12. A A A C GG G T A A Nanotechnology Universal model of computation using “Wang tiles” DNA-Based self-assembly [Seeman, Winfree,…….] Construction based on Watson-Crick complementarity A B GCATTGCATT CGTAACGTAA GCATTGCATT AATTCAATTC C C A T T T A A T C

13. Physics Phase transitions: Macroscopic changes to the system due to a microscopic change to some parameter. e.g.: gas/liquid/solid, spontaneous magnetization High temperature Criticality Low temperature Simulations of the Ising model

14. Chemistry Colloids: mixtures of two types of molecules. Must not overlap. Low densityHigh density ? (See poster by Sarah Miracle and Amanda Streib!)

15. Computational Problems in the Sciences Some (unrelated?) problems: Nanotechnology - self-assembly Computer sciences - sampling/counting (e.g., image segmentation) Physics - phase transitions Chemistry - colloids … Simulating the Ising Model (and other spin systems)

16. Lattice models Colorings (Potts Model) Matchings Independent Sets The Ising Model Goals: Efficiently sample and approximately count.

17. Counting and Sampling An FPRAS* for f takes input x, ε, δ, and produces A s.t. Pr [ (1−ε)f(x) ≤ A ≤ (1+ε)f(x) ] ≥ 1−δ and runs in time polynomial in |x|, ε−1 and log(1/δ). (*Fully Polynomial Randomized Approximation Scheme) An FPAUS generates samples from some distribution μ with probability π s.t. ||μ,π|| ≤ δ, and runs in time polynomial in |x| and log(1/δ). (*Fully Polynomial Almost Uniform Sampler) Exact Counting ⇒ Exact Sampling ⇓ Approximate Counting ⇐⇒ Approximate Sampling (FPRUS) (FPAUS) “self-reducible”

18. Main Questions Is the problem efficiently computable (in polynomial time)? Which problems are “intractable”? Does the “natural” sampling method work? Give me *any* fast solution! Is *this* sampling algorithm efficient?

19. E.g., if Ω = indep. sets of a graph G, connect I and I’ iff |I I’| = 1. Markov chains Step 1. Connect the state space. State space Ω ( |Ω| ~ c n ) ~

20. Basics of Markov chains Starting at x: - Pick a neighbor y. - Move to y with prob. P(x,y) = 1/∆. - With all remaining prob. stay at x. Transitions P: Random walk on H (max deg in H) Def’n: A MC is ergodic if it is: Irreducible - for all x,y  Ω,  t: P t (x,y) > 0; (connected) Aperiodic - g.c.d. { t: P t (x,y) > 0 } =1. (not bipartite) (The “t step” transition prob.) H x y

21. The stationary distribution  Thm: Any finite, ergodic MC converges to a unique stationary distribution π. Thm: The stationary distribution π of a reversible chain satisfies the detailed balance condition: π(x) P(x,y) = π(y) P(y,x). 

22. E.g., For  > 0, sample independent set I with prob.: π(I) =, where Z = ∑ J |J|.   0 2 1 |I| Z Q: What if we want to sample from some other distribution? Sampling from non-uniform distributions Step 2. Carefully define the transition probabilities.

23. The Metropolis Algorithm Propose a move from x to y as before, but accept with prob. min (1, π(y)/π(x)), (and with all remaining probability stay at x). [MRRTT ’53] π(y)/∆π(x) ( if π(x) ≥ π(y) ) 1/∆ For independent sets: min(1, ) I   I {v} min(1, -1 ) π(y) (|I|+1) /Z π(x) (|I|) /Z = == 1 π(y) π(x) x y π(x) P(x,y) = π(y) P(y,x)

24. Q: But for how long do we walk? Basics continued… Step 1. Connect the state space. Step 2. Carefully define the transition probabilities. Starting at any state x 0, take a random walk for some number of steps... and output the final state (from  ?). Step 3. Bound the mixing time. This tells us the number of steps to take.

25. The mixing rate Def’n: The total variation distance is ||P t,π|| = max __ ∑ |P t (x,y) - π(x)|. x  Ω y   Ω 2 1 A Markov chain is rapidly mixing if  (  ) is poly(n, log(  -1 )). (or polynomially mixing) Def’n Given , the mixing time is  = min { t: ||P t’,π|| < , t’ ≥ t }. A

26. Spectral gap Let   >    ≥  …  ≥   Ω   be the eigenvalues of P. Gap(P) = 1 - | 2 | is the spectral gap. Thm: (Alon, Alon-Milman, Sinclair)   ≤ log ( )  ≥ log ( ). Gap(P) 1 2 Gap(P) | 2 | 1 π*π* 1 2 

27. Bounding Convergence Time Techniques: Coupling Flows and paths Indirect methods Insights from physics Problems: Colorings Matchings Independent sets Ising model

28. Ex 1: Colorings Given: A graph G (max deg d), k > 1. Goal: Find a random k-coloring of G. MC COL : (Single point replacement) Starting at some k-coloring C 0 Repeat: - With prob 1/2 do nothing. - Pick v  V, c  [k]; - Recolor v with c, if possible. The “lazy” chain If k ≥ d + 2, then the state space is connected. (Therefore π is uniform.) Note: k ≥ d + 1 colorings exist. (Greedy) 

29. Coupling Once they agree, they move in sync (x t =y t x t+1 =y t+1 ) Couple moves, but each simulates the MC y0y0 Start at any x 0 and y 0 x0x0 Simulate 2 processes:

30. Def’n: A coupling is a MC on Ω x Ω: 1)Each process {X t }, {Y t } is a faithful copy of the original MC, 2)If X t = Y t, then X t+1 = Y t+1. Coupling T = max ( E [ T x,y ] ), where T x,y = min {t: X t =Y t | X 0 =x, Y 0 =y}. x,y The coupling time T is: Thm:  (  ) ≤ T e ln  -1. [Aldous’81]

31. Consider a shortest path: x = z 0, z 1, z 2,..., z r = y, dist(z i,z i+1 ) = 1, dist(x,y) = r. Path Coupling Coupling: Show for all x,y , E[  (dist(x,y)) ] ≤ 0. Path coupling: Show for all u,v s.t. dist(u,v)=1, that E[  (dist(u,v)) ] ≤ 0. [Bubley, Dyer, Greenhill’97-8]  E[  (dist(x,y)) ] ≤  i E[  (dist(z i,z i+1 )) ] ≤ 0. 

32. Path coupling for MC COL Thm: MC COL is rapidly mixing if k ≥ 3d. [Jerrum ’95] Pf: Use path coupling: dist(x,y) = 1. x y ww E[∆dist] ≤ ( (k-d)(-1) + 2d(+1) ) = (3d - k) ≤ 0. 1 2nk 1   v = w, c  C \ {,, }: ∆dist = -1, Cases:  v  N(w), c  {, }: ∆dist = +1 (or 0)  o.w.: ∆dist = 0. 2d

33. k-colorings On Z 2: MC col is fast when: k ≥ 8 [Jerrum] k ≥ 6 [BDG, AMMV] k = 3 [LRS] On Z d: MC col is slow for k=3 for large d [GKRS, Peled]

34. Ex 2: Sampling matchings MC MATCH :  Starting at M 0, repeat :  Pick e = (u,v)  E - If e  M, remove e; - If u and v unmatched in M, add e; - If u matched (by e’) and v unmatched (or vice versa), add e and remove e’; - Otherwise do nothing. e u v u v e e’ e u v

35. Sampling Matchings MC MATCH is rapidly mixing if # NPM < poly # PM. [Jerrum, Sinclair] There is an FPRAS (and FPAUG) for matchings on any bipartite graph. [Jerrum, Sinclair, Vigoda]

36. Conductance and flows Ω [Jerrum-Sinclair, Lawler-Sokal]  = min  (S). S  Ω, π(S)≤1/2 S S C  (S) = ∑ π(s) P(s,s’) ∑ π(s) s  S, s’  S C sS sS Thm: ≤ Gap(P) ≤ 2  22 2 (Thm: Coupling won’t work! [Kumar-Ramesh’99])

37. Matchings on Lattices v

38. Markov chain for Lozenge Tilings Repeat:  Pick v in the lattice region;  Add / remove the ``cube’’. at v w.p. ½, if possible. v v

39. Markov chain for Lozenge Tilings The state space is connected. The stationary distribution is uniform over tilings. Thm: The lozenge Markov chain is rapidly mixing. [Luby, R., Sinclair], [Wilson], [R.,Tetali] v v

40. v v Markov chain for Lozenge Tilings

42. Repeat:  Pick v in the lattice region;  Add/remove the “tower of height h” at v. w.p. 1/2h, if possible. Tower chain for Lozenge Tilings

43. To couple: Choose corresponding points and the same direction. 1 2 Do nothingMove w/ prob 1/4

44. Comparison [Diaconis,Saloff-Coste’93] unknown P known P _ w z For each edge (x,y)  P, make a path  x,y using edges in P. Let  (z,w) be the set of paths  x,y using (z,w). _ x y Thm: Gap(P) ≥ Gap(P). _ 1 A A = max { ∑ |  x,y | π(x)P(x,y) } 1 Q(e) e   xy e _ 

45. What about other models? Potts model Dimer model Domino tilings 3-colorings

46. What about other models? Potts model Dimer model Domino tilings 3-colorings

47. What about other models? Potts model Dimer model Domino tilings 3-colorings Pick a 2 x 2 square;

48. What about other models? Potts model Dimer model Domino tilings 3-colorings Pick a 2 x 2 square; Rotate, if possible;

49. What about other models? Potts model Dimer model Domino tilings 3-colorings Pick a 2 x 2 square; Rotate, if possible; Otherwise do nothing.

50. What about other models? Potts model Dimer model Domino tilings 3-colorings Pick a vtx and a color;

51. What about other models? Potts model Dimer model Domino tilings 3-colorings Pick a vtx and a color; Recolor, if possible;

52. What about other models? Potts model Dimer model Domino tilings 3-colorings Pick a vtx and a color; Recolor, if possible; Otherwise do nothing. These local chains are also rapidly mixing on domino tilings and 3-colorings.

53. Goal: Given, sample ind. set I with prob: π(I) = |I| /Z, where Z = ∑ J |J|. Ex 3: Independent Sets MC IND : Starting at I 0, Repeat: - Pick v  V and b  {0,1}; - If v  I, b=0, remove v w.p. min (1, -1 ) - If v  I, b=1, add v w.p. min (1, ) if possible; - O.w. do nothing.

54. When  is small (sparse case) Conjecture: Fast for < 3.79 G = Z 2, (fixed or toroidal boundary).   ≤ 1 [Luby, Vigoda]   ≤ 1.68 [Weitz]   ≤ 1.24 [van den Berg, Steif] MC IND is fast on Z 2 when:   ≤ 2.38 [RSTVY] (strong spatial mixing)

55. When is large (dense case)  > 80 [BCFKTVV]  > 6.19 … [R] MC IND is slow on Z 2 when: Conjecture: Slow for > 3.79 G = Z 2 (with toroidal boundary).

56. large  there is a “bad cut,”... so MC IND is slowly mixing.  Slow mixing of MC IND on Z 2 (large ) (Even) (Odd) 1 0 ∞ SSCSC #R/#B n     (n   n/2) 

57. Ex 4: Ising Configurations c The local chain: Pick a site and a spin and update with the appropriate Metropolis probability. SlowFast Fast [Lubetzky, Sly] ?

58. Tempering: w.p. 1/2: Do a LEVEL move: Fix i ; update  w.p. 1/2: Do a TEMP move: Fix  ; update i  =  x [M+1]  i =   (( ,i)) =  i (  ) / (M+1) iMiM ^ ^ Alternative: Simulated Tempering?  M =   M-1  M-2  0 =0 Thm: Tempering is fast for Ising on K n, for all . [Madras/Zheng] … But not for Potts. [Bhatnagar, R.]

59. Other approaches… Z = ∑  e -  H(  ) = ∑ H  G … Thm: There is an FPRAS for the Ising model that estimates Z (for all , all G). [Jerrum, Sinclair] based on the “high temperature expansion” Thm: There is an FPAUS for the Ising model to sample from  (all , all G).  [R., Wilson] based on the “random cluster respresentation” + JS

60. Conclusions Techniques: Coupling: can be easy when it works Flows: requires global knowledge of chain; very useful for slow mixing Connection to physics: can offer tremendous insights Open problems:... Indirect methods: top down approach; often increases complexity

61. 6vtx / 8vtx: Consider MC 8vtx where  (x) = {#sources+sinks} /Z. Open Problems 3-Colorings: MC col is fast on Z 2 when k=3 or k ≥ 6. What about k=4 or 5? SAWs: Is there an FPRAS / FPAUG? There is an efficient “testable algorithm.” [R., Sinclair] Fast: = 0,.9 < < 1.1; Slow: large ? Matchings: FPRAS / FPAUG on non-bipartite graphs?

62. Thank you!