ICLA 2019 Average Complexity of SAT On Average Case Complexity of SAT Johann A. Makowsky Faculty of Computer Science Technion–Israel Institute of Technology Haifa, Israel janos@cs.technion.ac.il www.cs.technion.ac.il/ ∼ janos File:icla-title.tex 1
ICLA 2019 Average Complexity of SAT This talk is based on an old, but barely referenced paper: On Average Case Complexity of SAT for Symmetric Distributions Makowsky J. A. and Sharell A., Journal of Logic and Computation, 5(1), 71-92 (1995) I want to put these results into an actual perspective. The slides were essentially prepared by Yoni Mircae File:icla-title.tex 2
ICLA 2019 Average Complexity of SAT Outline • Efficient on the average • Flat distributions Distributions for SAT • • Symmetric distributions • Fixed density distributions • Resolution of clauses • More recent work File:icla-title.tex 3
ICLA 2019 Average Complexity of SAT Efficient on the average Back to Outline File:icla-main.tex 4
ICLA 2019 Average Complexity of SAT How to Define “Efficient on the Average”? • A possible definition would be: an algorithm A is efficient-on-average if it runs in expected polynomial time. • Problem : suppose A runs in time n 2 on all inputs of length n except on one input that takes 2 n . Then, the expected running time of A is: E [ T A ] = 2 n − 1 · n 2 + 1 2 n · 2 n = O ( n 2 ) 2 n • However, if B is a simulation of A that takes T 2 A , the expected running time of B is: E [ T B ] = 2 n − 1 · n 4 + 1 2 n · 2 2 n = O (2 n ) 2 n File:icla-main.tex 5
ICLA 2019 Average Complexity of SAT Average Case Complexity : Basic Definitions • A function µ : S → [0 , 1] is a probability density function (pdf) on a countable or finite set S if Σ x ∈ S µ ( x ) = 1 . • A size function for a set S if a function | · | : S → N + such that the set S n = { x ∈ S : | x | = n } is finite. • An input set S is a pair < S, | · | > . • Let S be an input set and µ a pdf on S . The pair < S, µ > is called a global randomization of S . • Let < S, µ > be a global randomization and µ n defined by µ n ( x ) = Pr µ { x | x ∈ S n } . The sequence < S n , µ n > is called a local randomiza- tion of S . Note that each µ n is a pdf on S n . File:icla-main.tex 6
ICLA 2019 Average Complexity of SAT Distributional Problem • Let < S, µ > be a global randomization, ≤ be a linear ordering on S which is polynomial time computable and D ⊆ S . – Let µ ∗ be defined by µ ∗ ( x ) = Σ y ≤ x µ ( x ) µ is effectively computable if µ ∗ is polynomial time computable. – A pair < D, µ > with µ effectively computable is called a distributional problem . We think of D as the set of positive instances of some problem. File:icla-main.tex 7
ICLA 2019 Average Complexity of SAT Weight Function • Given a global randomization < S, µ > we define its weight function w by w ( n ) = Pr µ { S n } . Note that w is a pdf on N + . • If for some constant c > 0 and for every n ∈ N + , w ( n ) ≥ n − c then we say that w is a regular weight function and that the global randomization is regular. • If for every ε > 0: Σ w ( n ) n ε = ∞ then we say that w is a strongly regular weight function and that the global randomization is strongly regular. Example : w ( n ) = n − 1 ( logn ) − 2 Note that a local randomization with a weight function defines a unique global randomization. File:icla-main.tex 8
ICLA 2019 Average Complexity of SAT Local Probabilistic Bounds • Let < S n , µ n > be a local randomization on S and let f : R + → R + . Let T : S → R + , E µ n ( T ) = Σ x ∈ S n T ( x ) µ n ( x ) is the expectation of T on inputs of size n with respect to µ n . We say that: – f is an upper bound on the expectation of T if E µ n ( T ( x )) ≤ f ( n ) . – f is a (local) upper bound in probability on T if lim n →∞ Pr µ n { T ( x ) ≤ f ( n ) } = 1 . – f is a (local) lower bound in probability on T if lim n →∞ Pr µ n { T ( x ) > f ( n ) } = 1 . Results in probabilistic analysis of algorithms are usually expressed with these types of local bounds on T . File:icla-main.tex 9
ICLA 2019 Average Complexity of SAT Probabilistic Bounds • For Average Case Complexity Theory we present here a definition of at most f on the average that was developed in ∗ : Let < S, µ > be a global randomization on S and T : S → R + . For a strictly increasing function f : R + → R + we say that T is at most f on the average w.r.t the global randomization < S, µ > if E µ ( f − 1 ( T ( x )) ) < ∞ | x | and denote this by T ∈ AV B ( < S, µ >, f ) or simply T ∈ AV B ( f ) if the randomization is evident from context. ∗ Shai Ben-David, Benny Chor, Oded Goldreich, and Michael Luby. On the theory of average case complete complexity. Journal of Computer and System Sciences, 44(2):193-219, April 1992. File:icla-main.tex 10
ICLA 2019 Average Complexity of SAT Probabilistic Bounds (cont.) We can now define a (average) complexity class: Let < D, µ > be a distributional problem. We say that < D, µ > is poly- nomial on the average and write < D, µ > ∈ AverP if there is a deterministic algorithm A for D with run-time T A and there is a polynomial p such that T A ∈ AV B ( p ). Theorem 1 (Transfer Theorem for Upper Bounds): Let < S, µ > be a global randomization, < S n , µ n > the implied local randomization and T : S → R + . For any function f : R + → R + : • If f is a convex function then: E µ n ( T ( x )) ≤ f ( n ) ⇒ T ∈ AV B ( f ) • If f is a concave function then: T ∈ AV B ( f ) ⇒ E µ n ( T ( x )) ≤ f ( n ) Proof: Σ a i ) ≤ a i φ ( x i ) Follows from Jensen’s inequality: φ ( Σ a i x i for a real convex function φ Σ a i and positive weights a i . The inequality is reversed if φ is concave. � File:icla-main.tex 11
ICLA 2019 Average Complexity of SAT Probabilistic Bounds (cont.) ∗ ): Theorem 2 (Transfer Theorem for Lower Bounds Let < S, µ > be a global randomization with weight function w and let f, g : R + → R + be two strictly increasing functions. If f is a lower bound in probability on T w.r.t < S n , µ n > and g is sufficiently small for w ( n ) Σ ∞ g − 1 ( f ( n )) = ∞ n =1 n to hold then T / ∈ AV B ( g ) . ∗ Abraham Sharell. On the average case complexity of SAT for flat distributions. Master’s thesis, Technion-Israel Institute of Technology, 1992. File:icla-main.tex 12
ICLA 2019 Average Complexity of SAT Proof: Since g is strictly increasing it suffices to show that E µ ( g − 1 ( T ( x )) ) = ∞ . | x | Reminder : Markov’s inequality: If X is a nonnegative random variable and a > 0, then P ( x ≥ a ) ≤ E ( X ) . a Applying Markov’s inequality to the (strictly positive) random variable g − 1 ( T ( x )) we derive for every n ∈ N + E µ n ( g − 1 ( T ( x ))) > g − 1 ( f ( n )) Pr µ n { g − 1 ( T ( x )) > g − 1 ( f ( n )) } = g − 1 ( f ( n )) Pr µ n { T ( x ) > f ( n ) } Observing that E µ ( g − 1 ( T ( x )) w ( n ) w ( n ) ) = Σ ∞ E µ n ( g − 1 ( T ( x ))) > Σ ∞ g − 1 ( f ( n )) Pr µ n { T ( x ) > f ( n ) } n =1 n =1 | x | n n together with the assumptions in the hypothesis gives the desired result. � File:icla-main.tex 13
ICLA 2019 Average Complexity of SAT Corollary 3 Let < S, µ > be a regular global randomization with weight function w . Let f : R + → R + be a strictly increasing function, and assume that f ( n ) is a lower bound in probability on T w.r.t < S n , µ n > . ∈ AV B ( f ( n ε )) . (i) If w is regular then there exists 0 < ε < 1 so that T / ∈ AV B ( f ( n ε )) . (ii) If w is strongly regular then for every 0 < ε < 1 we have T / File:icla-main.tex 14
ICLA 2019 Average Complexity of SAT Proof: For regular w let c > 1 be a constant so that for all n ∈ N + : w ( n ) ≥ n − c . c and g ( n ) = f ( n ε ). Then g − 1 ( f ( n )) = n c and Set ε = 1 w ( n ) 1 Σ ∞ g − 1 ( f ( n )) ≥ Σ ∞ n = ∞ . n =1 n =1 n By the Transfer Theorem for Lower Bounds we conclude that T / ∈ AV B ( g ). For strongly regular weight functions let 0 < ε < 1 and g ( n ) = f ( n ε ). Then the ε − 1) and by the definition general term in the above sum evaluates to w ( n ) n ( 1 of strongly regular the sum diverges. � File:icla-main.tex 15
ICLA 2019 Average Complexity of SAT Probabilistic Bounds (cont.) Proposition 4 Let f, g : R + → R + be two strictly increasing functions so that: g − 1 ( f ( n )) lim n →∞ = ∞ n If f is a lower bound with probability 1 on T w.r.t < S n , µ n > then there exists a global randomization < S, µ > which is compatible with < S n , µ n > such that T / ∈ AV B ( < S, µ >, g ) . File:icla-main.tex 16
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