Partly on the basis of heuristic arguments from physics, it has been suggested that the performance of certain types of algorithms on random @math -SAT formulas is linked to phase transitions that affect the geometry of the set of satisfying assignments. But, beyond intuition, there has been scant rigorous evidence that “practical” algorithms are affected by these phase transitions. In this paper we prove that Walksat, a popular randomized satisfiability algorithm, fails on random @math -SAT formulas not very far above clause variable density, where the set of satisfying assignments shatters into tiny, well-separated clusters. Specifically, we prove that Walksat is ineffective with high probability (w.h.p.) if @math , where @math is the number of clauses, @math is the number of variables, and @math is an absolute constant. By comparison, Walksat is known to find satisfying assignments in linear time w.h.p. if @math [A. Coja-Oghlan and A. Frieze, SIAM J. Comput., 43 (2...
The first and the last author obtained negaitve results for message passing algorithms for random @math -SAT that do not require bounds on the number of iterations @cite_5 @cite_11 . Specifically, @cite_5 shows that a basic version of Belief Propagation Guided Decimation fails to find satisfying assignments for densities @math for a certain constant @math . Moreover, @cite_11 shows that a basic version of the conceptually more powerful Survey Propagation Guided Decimation algorithm fails if @math .
@cite_5 @cite_11
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Let Φ be a uniformly distributed random k-SAT formula with n variables and m clauses. Non-constructive arguments show that Φ is satisfiable for clause variable ratios m n ≤ rk 2k ln 2 with high probability (Achlioptas, Moore: SICOMP 2006; Achlioptas, Peres: J. AMS 2004). Yet no efficient algorithm is know to find a satisfying assignment for densities as low as m n rk · ln(k) k with a non-vanishing probability. In fact, the density m n rk · ln(k) k seems to form a barrier for a broad class of local search algorithms (Achlioptas, Coja-Oghlan: FOCS 2008). On the basis of deep but non-rigorous statistical mechanics considerations, a message passing algorithm called belief propagation guided decimation for solving random k-SAT has been forward (Mezard, Parisi, Zecchina: Science 2002; Braunstein, Mezard, Zecchina: RSA 2005). Experiments suggest that the algorithm might succeed for densities very close to rk for k = 3, 4, 5 (Kroc, Sabharwal, Selman: SAC 2009). Furnishing the first rigorous analysis of belief propagation guided decimation on random k-SAT, the present paper shows that the algorithm fails to find a satisfying assignment already for m n ≥ ρ · rk k, for a constant ρ > 0 independent of k. Let @math be a uniformly distributed random @math -SAT formula with @math variables and @math clauses. For clauses variables ratio @math the formula @math is satisfiable with high probability. However, no efficient algorithm is known to provably find a satisfying assignment beyond @math with a non-vanishing probability. Non-rigorous statistical mechanics work on @math -CNF led to the development of a new efficient "message passing algorithm" called [M ', Science 2002]. Experiments conducted for @math suggest that the algorithm finds satisfying assignments close to @math . However, in the present paper we prove that the basic version of Survey Propagation Guided Decimation fails to solve random @math -SAT formulas efficiently already for @math with @math almost a factor @math below @math . Experimental results show that certain message passing algorithms, namely, survey propagation, are very effective in finding satisfying assignments in random satisfiable 3CNF formulas. In this paper we make a modest step towards providing rigorous analysis that proves the effectiveness of message passing algorithms for random 3SAT. We analyze the performance of Warning Propagation, a popular message passing algorithm that is simpler than survey propagation. We show that for 3CNF formulas generated under the planted assignment distribution, running warning propagation in the standard way works when the clause-to-variable ratio is a sufficiently large constant. We are not aware of previous rigorous analysis of message passing algorithms for satisfiability instances, though such analysis was performed for decoding of Low Density Parity Check (LDPC) Codes. We discuss some of the differences between results for the LDPC setting and our results. We study the satisfiability of randomly generated formulas formed by M clauses of exactly K literals over N Boolean variables. For a given value of N the problem is known to be most difficult when α = M N is close to the experimental threshold αc separating the region where almost all formulas are SAT from the region where all formulas are UNSAT. Recent results from a statistical physics analysis suggest that the difficulty is related to the existence of a clustering phenomenon of the solutions when α is close to (but smaller than) αc. We introduce a new type of message passing algorithm which allows to find efficiently a satisfying assignment of the variables in this difficult region. This algorithm is iterative and composed of two main parts. The first is a message-passing procedure which generalizes the usual methods like Sum-Product or Belief Propagation: It passes messages that may be thought of as surveys over clusters of the ordinary messages. The second part uses the detailed probabilistic information obtained from the surveys in order to fix variables and simplify the problem. Eventually, the simplified problem that remains is solved by a conventional heuristic. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005