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1 Backdoor Sets in SAT Instances Ryan Williams Carnegie Mellon University Joint work in IJCAI03 with: Carla Gomes and Bart Selman Cornell University

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2 Significant progress in Complete search methods! Software and hardware verification – complete methods are critical - e.g. for verifying the correctness of chip design, using SAT encodings Current methods can verify automatically the correctness of a large fraction of a Pentium IV. (Complete = always returns SAT or unSAT)

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3 A “real world” example (Thanks to: Oliver Kullmann)

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4 i.e. (( x 1 ) or x 7 ) and (( x 1 ) or x 6 ) and … etc. Bounded Model Checking instance:

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5 (x 177 or x 169 or x 161 or x 153 … or x 17 or x 9 or x 1 or ( x 185 )) clauses / constraints are getting more interesting… 10 pages later: …

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6 4000 pages later: … ?!! a 59-cnf clause…

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7 Finally, 15,000 pages later: The MiniSat solver (Een&Sorensson) solves this instance in 2 seconds. Note that:… !!!

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8 Gap between Theory and Practice The good scaling behavior of state-of-the art SAT solvers seems to defy our complexity-theoretic intuition that SAT is NP-complete! How can we explain this gap between theory and practice? What makes this possible? Our answer: Hidden tractable substructure in real-world problems. Can we make this more precise? Proposal: We consider structures we call backdoor sets. Idea came out of study of heavy-tailed phenomena in runtime distributions for some SAT solvers.

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9 Backdoor Sets – Initial Motivation Explains why restarting a solver often is an effective strategy Implies a wide range of possible solution times, often including short runs How to explain short runs? Heavy-tailed distributions and Randomization. Certain problems, when solved by randomized backtracking, yield a runtime distribution that is heavy-tailed Pr[solution found in time t] ~ 1/t^c, 0 < c < 2

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Explaining short runs: Backdoors to tractability Informally: A backdoor set to a given problem instance is a subset of its variables such that, once assigned values, the remaining instance simplifies to a tractable class. Formally: We define notion of a “sub-solver” (handles tractable substructure of problem instance) backdoor set and strong backdoor set

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11 Defining a sub-solver Definition is general enough to encompass many polynomial time propagation methods. (Also those for which we do not know a clean characterization of the tractable subclass.) Valid for other encoding languages besides SAT: e.g., Mixed Integer Programming and Constraint Satisfaction Problems

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12 Backdoor set (for satisfiable instances): Strong backdoor set (applies to satisfiable or inconsistent instances): Defining backdoors

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Backdoors can be surprisingly small: Backdoors help explain how a solver can get “lucky” on certain runs: backdoor sets are identified early on in backtracking search. Most recent: Other combinatorial domains. E.g. Graphplan planning, near constant size backdoors (2 or 3 variables) in certain domains. (Hoffman, Gomes, Selman ’03) Backdoors capture critical problem resources (bottlenecks).

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14 Constraint Satisfaction Problem The Constraint Satisfaction Problem (CSP): A finite set of n variables is given and with each variable is associated a non-empty finite domain. A constraint on k variables X 1,…,X k is a relation R(X 1,…,X k ) D 1 x …x D k. A solution to a CSP is an assignment of values to all the variables, satisfying all the constraints. (Satisfaction of a constraint = the relation holds) (Dechter 86, Freuder 82, Mackworth 77, Tsang 93, van Beek and Dechter 97)

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15 Explicit Algorithms for Finding/Exploiting Backdoor Sets We cover three kinds of strategies for dealing with instances with small backdoor sets: A deterministic algorithm A randomized algorithm –Provably better worst-case performance over the deterministic one A heuristic randomized algorithm –Assumes existence of a good heuristic for choosing variables to branch on –We believe this is close to what happens in practice

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16 Deterministic Generalized Iterative Deepening

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17 Assumption: There exists a backdoor whose size is bounded by a function of n (call it B(n)) Idea: Repeatedly choose random subsets of variables that are slightly larger than B(n), searching these subsets for the backdoor Randomized Generalized Iterative Deepening

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18 Randomized Generalized Iterative Deepening

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19 Deterministic Versus Randomized Deterministic algorithm Randomized algorithm Suppose variables have 2 possible values (e.g. SAT) k For B(n) = n/k, algorithm runtime is c n c

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20 Complete Randomized Depth First Search with Heuristic Assume we have the following. DFS, a generic depth first search randomized backtrack search solver with: (polytime) sub-solver A Heuristic H that (randomly) chooses variables to branch on, in polynomial time H has probability 1/h of choosing a backdoor variable (h is a fixed constant) Call this ensemble (DFS, H, A)

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21 Polytime Restart Strategy for (DFS, H, A) Essentially: If there is a small backdoor, then (DFS, H, A) has a restart strategy that runs in polytime.

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Runtime Table for Algorithms DFS,H,A B(n) = upper bound on the size of a backdoor, given n variables When the backdoor is a constant fraction of n, there is an exponential improvement between the randomized and deterministic algorithm

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Summary Introduced notion of a “backdoor set” of variables. 1)More closely captures combinatorics of a problem instance, as dealt with in practice. 2)Provides insight into restart strategies. 3) Backdoors can be surprisingly small in practice. 4) Search heuristics + randomization can be used to find them, provably efficiently.

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