Accelerating Random Walks Wei Dept. of Computer Science Cornell University (joint work with Bart Selman)

Introduction  This talk deals with gathering new information to accelerate search and reasoning.  Although the method proposed can to be generalized to other search problems, we focus on SAT problem in this talk.

Boolean Satisfiability Problem  Boolean Satisfiability Problem (SAT) asks if a Boolean expression can be made true by assigning Boolean values to its variables.  The problem is well-studied in AI community with direct application in reasoning, planning, CSP etc.  Does statement s hold in world A (represented by a set of clauses)? A ᅣ s  (¬s) ^ A unsatisfiable

SAT  SAT (even 3SAT) is NP-complete. Best theoretical bound so far is O(1.334 N ) for 3SAT (Schoening 1999)  In practice, there are two different kinds of solvers DPLL (Davis, Logemann and Loveland 1962) Local Search (Selman et al 1992)

DPLL (x 1   x 2  x 3 )  (x 1   x 2   x 3 )  (  x 1   x 2  x 3 )  DPLL was first proposed as a simple depth-first tree search. x1x1 x2x2 FT T null F solution x2x2

DPLL  Recently (since late 90’s), many improvements: Randomization, restart, out- of-order backtracking, and clause learning

Local Search  The idea: Start with a random assignment. And make local changes to the assignment until a solution is reached  Pro: often efficient in practice. Sometimes the only feasible way for some problems  Con: Cannot prove nonexistence of solutions. Hard to analyze theoretically

Local Search  Initially, pure hill-climbing greedy search: Procedure GSAT Start with a random truth assignment Repeat s:= set of neighbors of current assignment that satisfies most clauses pick an assignment in s randomly and move to that new assignment Until a satisfying assignment is found

Local Search, cont.  Later, other local search schemes used: Simulated annealing Tabu search Genetic algorithm Random Walk and its variants  the most successful so far

Unbiased (Pure) Random Walk for SAT Procedure Random-Walk (RW) Start with a random truth assignment Repeat c:= an unsatisfied clause chosen at random x:= a variable in c chosen at random flip the truth value of x Until a satisfying assignment is found

Random Walk algorithms  Random walk algorithm (e.g. Walksat) offer significant improvement on performance over hill-climbing algorithms. IDVarsclausesGSAT+wWalksat Ssa Ssa Ssa Ssa

New approached to speed up random walks  Next, we’ll discuss how to use new knowledge to improve random walk algorithm STAGE – use more state information Long distance link discovery – transform the search space

I) STAGE algorithm Boyan and Moore 1998  Idea: more features of the current state may help local search  Task: to incorporate these features into improved evaluation functions, and help guide search

Method  The algorithm learns function V: the expected outcome of a local search algorithm given an initial state. Can this function be learned successfully? x y x V(x)

Features  State feature vector: problem specific Example: for 3SAT, following features are useful: 1.% of clauses currently unsat (=obj function) 2.% of clauses satisfied by exactly 1 variable 3.% of clauses satisfied by exactly 2 variables 4.% of variables set to their naïve setting

Learner  Fitter: can be any function approximator; polynomial regression is used in practice.  Training data: generated on the fly; every LS trajectory produces a series of new training data.  Restrictions on LS: it must terminate in poly- time; it must be Markovian.

Diagram of STAGE Run LS  to Optimize Obj Train the approximator new training data Hillclimb to Optimize V Generate good start point

Results  Works on many domain, such as bin- packing, channel routing, SAT Standard LS STAGE +LS Binpacking Channel Routing SAT(par32) Table gives average solution quality

Discussion  Is the learned function a good approximation to V? – Somewhat unclear. (“worrisome”: performance is not improved when authors used quadratic regression to replace linear regression. Learning does help however.)  Why not learn a better objective function and search on that function directly (clause weighing, adding clauses)?

Method  The algorithm learns function V: the expected outcome of a local search algorithm given an initial state x y x V(x)

II) Long-distance Link Discovery  Random walk-style methods are successful on hard randomly generated instances, as well as on a number of real-world benchmarks.  However, they are generally less effective in highly structured domains compared to backtrack methods such as DPLL.  Key issue: random walk needs O(N 2 ) flips to propagate dependencies among variables, while in unit-propagation in DPLL takes only O(N).

Overview  Random Walk Strategies - unbiased random walk - biased random walk  Chain Formulas - binary chains - ternary chains  Practical Problems  Conclusion and Future Directions

Unbiased (Pure) Random Walk for SAT Procedure Random-Walk (RW) Start with a random truth assignment Repeat c:= an unsatisfied clause chosen at random x:= a variable in c chosen at random flip the truth value of x Until a satisfying assignment is found

Unbiased RW on any satisfiable 2SAT Formula  Given a satisfiable 2SAT formula with n variables, a satisfying assignment will be reached by Unbiased RW in O(n 2 ) steps with high probability. (Papadimitriou, 1991)  Elegant proof! (next)

Given a satisfiable 2-SAT formula F. RW starts with a random truth assignment A0. Consider an unsatisfied clause: (x_3 or (not x_4)) A0 must have x_3 False and x_4 True (both “wrong”) A satisfying truth assignment, T, must have x_3 True or x_4 False (or both) Now, “flip” truth value of x_3 or x_4. With (at least) 50% chance, Hamming distance to satisfying assignment T is reduced by 1. I.e., we’re moving the right direction! (of course, with 50% (or less) we are moving in the wrong direction… doesn’t matter!)

We have an unbiased random walk with a reflecting barrier at distance N from T (max Hamming distance) and an absorbing barrier (satisfying assignment) at distance 0. We start at a Hamming distance of approx. ½ N. Property of unbiased random walks: after N^2 flips, with high probability, we will hit the origin (the satisfying assignment). (Drunkards walk) So, O(N^2) randomized algorithm (worst-case!) for 2-SAT. TA0 T

Unfortunately, does not work for k-SAT with k>= 3.  Reason: example unsat clause: (x_1 or (not x_4) or x_5) now only 1/3 chance (worst-case) of making the right flip!

Unbiased RW on 3SAT Formulas Random walk takes exponential number of steps to reach 0. T A0

Comments on RW 1)Random Walk is highly “myopic” does not take into account any gradient of the objective function (= number of unsatisfied clauses)! Purely “local” fixes. 2)Can we make RW practical for SAT? Yes --- inject greedy bias into walk  biased Random Walk.

Biased Random Walk (1 st minimal greedy bias) Procedure Random-Walk-with-Freebie (RWF) Start with random truth assignment Repeat c:= an unsatisfied clause chosen at random if there exist a variable x in c with break value = 0 // greedy bias flip the value of x (a “freebie” flip) else x:= a variable in c chosen at random // pure walk flip the value of x Until a satisfying assignment is found break value == # of clauses that become unsatisfied because of flip.

Biased Random Walk (adding more greedy bias) Procedure WalkSat Repeat c:= an unsatisfied clause chosen at random if there exist a variable x in c with break value = 0 // greedy bias flip the value of x (freebie move) else with probability p // pure walk x:= a variable in c chosen at random flip the value of x with probability (1-p) x:= a variable in c with smallest break value // more greedy bias flip the value of x Until a satisfying assignment is found Note: tune parameter p.

Chain Formulas  To better understand the behavior of pure and biased RW procedures on SAT instances, we introduce Chain Formulas.  These formulas have long chains of dependencies between variables.  They effectively demonstrate the extreme properties of RW style algorithms.

Binary Chains  Consider formulas 2-SAT chain, F 2chain x 1  x 2 x 2  x 3 … x n-1  x n x n  x 1 Note: Only two satisfying assignments --- TTTTTT … and FFFFFF…

Binary Chains Walk is exactly balanced.

Binary Chains  We obtain the following theorem Theorem 1. With high probability, the RW procedure takes  n 2 ) steps to find a satisfying assignment of F 2chain.  DPLL algorithm’s unit propagation mechanism finds an assignment for F 2chain in linear time.  Greedy bias does not help in this case: both RWF and WalkSat takes  n 2 ) flips to reach a satisfying assignment on these formulas.

Speeding up Random Walks on Binary Chains Pure binary chain Binary chain with redundancies (implied clauses) Aside: Note small-world flavor (Watts & Strogatz 99, Kleinberg 00).

Results: Speeding up Random Walks on Binary Chains * : empirical results ** : theoretical proof available Pure binary chain Chain with redundancies RW  (n 2 ) ** RWF  (n 2 ) **  (n 1.2 ) * WalkSat  (n 2 ) *  (n 1.1 ) * Becomes almost like unit prop.

Ternary Chains In general, even a small bias in the wrong direction leads to exponential time to reach 0.

Ternary Chains  Consider formulas F 3chain, low(i) x 1 x 2 x 1  x 2  x 3 … x low(i)  x i-1  x i … x low(n)  x n-1  x n Note: Only one satisfying assign.: TTTTT… *These formulas are inspired by Prestwich [2001]

Ternary Chains long link short link medium link Effect of X 1 and X 2 needs to propagate through chain.

Theoretical Results on 3-SAT Chains Function low(i)Expected run time of pure RW i-2 (highly local) ~ Fib(n) (i.e., exp.)  i/2  (interm. reach) O(n. n log n ) (i.e., quasi-poly)  log i  (interm. reach) O(n 2. (log n) 2 ) (i.e., poly) 1 (full back reach) O(n 2 ) low(i) captures how far back the clauses reach.

Proof  The proofs of these claims are quite involved, and are available at  Here, just the intuitions.  Each RW process on these formulas can be decomposed into a series of decoupled, simpler random walks.

Simple case: single “0”  …  / / /3 zizi ?Exp. # steps decomposes: Exp # steps from Exp # steps from Z i is the assignment with all 1’s except for ith position 0.

Recurrence Relations Our formula structure gives us: E(f(z i )) = (E(f(z low(i) ) + E(f(z i ) + 1) * 1/3 + (E(f(z i-1 ) + E(f(z i ) + 1) * 1/3 + 1 * 1/3  E(f(z i )) = E(f(z low(i) ) + E(f(z i-1 ) + 3

Recurrence Relations  Solving this recurrence for different low(i)’s, we get Function low(i)E(f(z i )) i-2  Fib(i)  i/2  i log i  log i  i. (log i) 2 1  i i This leads to the complexity results for the overall RW.

Decompose: multiple “0”s   Start Sat assign.

Results for RW on 3-SAT chains. Function low(i)Expected Running time of pure RW i-2~ Fib(n)  i/2  O(n. n log n )  log i  O(n 2. (log n) 2 ) 1O(n 2 )

Recap Chain Formula Results  Adding implied constraints capturing long- range dependencies speeds random walk on 2-Chain to near linear time.  Certain long-range dependencies in 3-SAT lead to poly-time convergence of random walks.  Can we take advantage of these results on practical problem instances? Yes! (next)

Results on Practical Benchmarks  Idea: Use a formula preprocessor to uncover long- range dependencies and add clauses capturing those dependencies to the formula.  We adapted Brafman’s formula preprocessor to do so. (Brafman 2001)  Experiments on recent verification benchmark. (Velev 1999)

Empirical Results SSS-SAT-1.0 instances (Velev 1999). 100 total.  level of redundancy added (20% near optimal) Formulas (redun. level) <40 sec<400 sec<4000 sec  =  =  =

Optimal Redundancy Rate Time vs Redundancy Rate Flips vs Redundancy Rate WalkSat(noise=50) on dlx2_cc_bug01.cnf from SAT-1.0 Suite

Conclusions  We introduced a method for speeding up random walk style SAT algorithms based on the addition of constraints that capture long range dependencies.  On a binary chain, we showed how by adding implied clauses, biased RW becomes almost as effective as unit-propagation.

Conclusions, Cont.  In our formal analysis of ternary chains, we showed how the performance of RW varies from exponential to polynomial depending on the range of dependency links. We identified the first subclass of 3-SAT problems solvable in poly-time by unbiased RW  We gave a practical validation of our approach.

Future Directions  It seems likely that many other dependency structures could speed up random walk style methods.  It should be possible to develop preprocessor to uncover other dependencies. For example, in graph coloring problem we have:  x 1   x 4,  x 2   x 5,  x 3   x 6, … x 1  x 4  x 7  x 10, …

The end.