We consider the robustness of computational hardness of problems whose input is obtained by applying independent random deletions to worst-case instances. For some classical @math -hard problems on graphs, such as Coloring, Vertex-Cover, and Hamiltonicity, we examine the complexity of these problems when edges (or vertices) of an arbitrary graph are deleted independently with probability @math . We prove that for @math -vertex graphs, these problems remain as hard as in the worst-case, as long as @math for arbitrary @math , unless @math . We also prove hardness results for Constraint Satisfaction Problems, where random deletions are applied to clauses or variables, as well as the Subset-Sum problem, where items of a given instance are deleted at random.
Similar reasoning applies to @math formulas. It is well known that there exists @math such that a random @math formula in which each possible clause is added independently with probability @math can be solved with high probability using the pure literal heuristic @cite_15 . As observed in @cite_15 , if this heuristic fails in finding a satisfying assignment for a formula @math it will still fail to find a satisfying assignment if clauses are added to @math . This implies that for any @math -variable @math formula @math , if @math , then the clause-percolated formula is satisfiable and furthermore a satisfying assignment can be found in polynomial time using the pure literal heuristic.
@cite_15
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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 . Let @math be a uniformly distributed random @math -SAT formula with @math variables and @math clauses. We present a polynomial time algorithm that finds a satisfying assignment of @math with high probability for constraint densities @math , where @math . Previously no efficient algorithm was known to find satisfying assignments with a nonvanishing probability beyond @math [A. Frieze and S. Suen, J. Algorithms, 20 (1996), pp. 312-355]. We analyze the pure literal rule heuristic for computing a satisfying assignment to a random 3-CNF formula with n variables. We show that the pure literal rule by itself finds satisfying assignments for almost all 3-CNF formulas with up to 1.63n clauses, but it fails for more than 1.7n clauses. As an aside we show that the value of maximum satisfiability for random 3-CNF formulas is tightly concentrated around its mean. Given a Boolean formula and a satisfying assignment, a flip is an operation that changes the value of a variable in the assignment so that the resulting assignment remains satisfying. We study the problem of computing the shortest sequence of flips (if one exists) that transforms a given satisfying assignment @math to another satisfying assignment @math of an input Boolean formula. Earlier work characterized the complexity of deciding the existence of a sequence of flips between two given satisfying assignments using Schaefer's framework for classification of Boolean formulas. We build on it to provide a trichotomy for the complexity of finding the shortest sequence of flips and show that it is either in P, NP-complete, or PSPACE-complete. Our result adds to the growing set of complexity results known for shortest reconfiguration sequence problems by providing an example where the shortest sequence can be found in polynomial time even though the sequence flips variables that have the same value in both @math an...