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... Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with @math variables and @math clauses, there is a value of @math beyond which the CSP will be unsatisfiable with high probability. Strong refutation is the problem of certifying that no variable assignment satisfies more than a constant fraction of clauses; this is the natural algorithmic problem in the unsatisfiable regime (when @math ). Intuitively, strong refutation should become easier as the clause density @math grows, because the contradictions introduced by the random clauses become more locally apparent. For CSPs such as @math -SAT and @math -XOR, there is a long-standing gap between the clause density at which efficient strong refutation algorithms are known, @math , and the clause density at which instances become unsatisfiable with high probability, @math . In this paper, we give spectral and sum-of-squares algorithms for strongly refuting random @math -XOR instances with clause density @math in time @math or in @math rounds of the sum-of-squares hierarchy, for any @math and any integer @math . Our algorithms provide a smooth transition between the clause density at which polynomial-time algorithms are known at @math , and brute-force refutation at the satisfiability threshold when @math . We also leverage our @math -XOR results to obtain strong refutation algorithms for SAT (or any other Boolean CSP) at similar clause densities. Our algorithms match the known sum-of-squares lower bounds due to Grigoriev and Schonebeck, up to logarithmic factors. Additionally, we extend our techniques to give new results for certifying upper bounds on the injective tensor norm of random tensors. Finding a satisfying assignment for a k-CNF formula (k ≥ 3), assuming such exists, is a notoriously hard problem. In this work we consider the uniform distribution over satisfiable k-CNF formulas with a linear number of clauses (clause-variable ratio greater than some constant). We rigorously analyze the structure of the space of satisfying assignments of a random formula in that distribution, showing that basically all satisfying assignments are clustered in one cluster, and agree on all but a small, though constant proportion, number of variables. This observation enables us to describe a polynomial time algorithm that finds whp a satisfying assignment for such formulas, thus asserting that most satisfiable k-CNF formulas are easy (whenever the clause-variable ratio is greater than some constant). This should be contrasted with the setting of very sparse k-CNF formulas (which are satisfiable whp), where experimental results show some regime of clause density to be difficult for many SAT heuristics. One explanation for this phenomena, backed up by partially non-rigorous analytical tools from statistical physics, is the complicated clustering of the solution space at that regime, unlike the more “regular” structure that denser formulas possess. Thus in some sense, our result rigorously supports this explanation. key words: computational and structural complexity, algorithms and data structures, message passing algorithms, SAT. Let @math be a nontrivial @math -ary predicate. Consider a random instance of the constraint satisfaction problem @math on @math variables with @math constraints, each being @math applied to @math randomly chosen literals. Provided the constraint density satisfies @math , such an instance is unsatisfiable with high probability. The problem is to efficiently find a proof of unsatisfiability. We show that whenever the predicate @math supports a @math - probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree @math (which runs in time @math ) refute a random instance of @math . In particular, the polynomial-time SOS algorithm requires @math constraints to refute random instances of CSP @math when @math supports a @math -wise uniform distribution on its satisfying assignments. Together with recent work of [LRS15], our result also implies that polynomial-size semidefinite programming relaxation for refutation requires at least @math constraints. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random CSPs. For every constraint predicate @math , they give a three-way hardness tradeoff between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent algorithmic results of [AOW15] and [RRS16], this full three-way tradeoff is , up to lower-order factors. Let I be a random 3CNF formula generated by choosing a truth assignment φ for variables x 1 , ..., x n uniformly at random and including every clause with i literals set true by φ with probability p i , independently. We show that for any 0 ≤ η 2 , η 3 ≤ 1 there is a constant d min so that for all d ≥ d min , a spectral algorithm similar to the graph coloring algorithm of [1] will find a satisfying assignment with high probability for p 1 = d n2, p 2 = η 2 d n2, and p3 = η 3 d n2. Appropriately setting η 2 and η 3 yields natural distributions on satisfiable 3CNFs, not-all-equal-sat 3CNFs, and exactly-one-sat 3CNFs. We present a simple probabilistic algorithm for solving k-SAT and more generally, for solving constraint satisfaction problems (CSP). The algorithm follows a simple local search paradigm (S. , 1992): randomly guess an initial assignment and then, guided by those clauses (constraints) that are not satisfied, by successively choosing a random literal from such a clause and flipping the corresponding bit, try to find a satisfying assignment. If no satisfying assignment is found after O(n) steps, start over again. Our analysis shows that for any satisfiable k-CNF-formula with n variables this process has to be repeated only t times, on the average, to find a satisfying assignment, where t is within a polynomial factor of (2(1-1 k)) sup n . This is the fastest (and also the simplest) algorithm for 3-SAT known up to date. We consider also the more general case of a CSP with n variables, each variable taking at most d values, and constraints of order l, and analyze the complexity of the corresponding (generalized) algorith m. It turns out that any CSP can be solved with complexity at most (d spl middot (1-1 l)+ spl epsiv ) sup n . Model checking and counter-example guided abstraction refinement are examples of applications of SAT solving requiring the production of models for satisfiable formulas. Better than giving a truth value to every variable, one can provide an implicant, i.e. a partial assignment of the variables such that every full extension is a model for the formula. An implicant is prime if every assignment is necessary. Since prime implicants contain no literal irrelevant for the satisfiability of the formula, they are considered as highly refined information. We here propose a novel algorithm that uses data structures found in modern CDCL SAT solvers to efficiently compute prime implicants starting from an existing model. The original aspects are (1) the algorithm is based on watched literals and a form of propagation of required literals, adapted to CDCL solvers (2) the algorithm works not only on clauses, but also on generalized constraints (3) for clauses and, more generally for cardinality constraints, the algorithm complexity is linear in the size of the constraints found. We implemented and evaluated the algorithm with the Sat4j library. 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 Abstract In this paper, we present improved exact and parameterized algorithms for the maximum satisfiability problem. In particular, we give an algorithm that computes a truth assignment for a boolean formula F satisfying the maximum number of clauses in time O(1.3247 m | F |), where m is the number of clauses in F , and | F | is the sum of the number of literals appearing in each clause in F . Moreover, given a parameter k , we give an O(1.3695 k +| F |) parameterized algorithm that decides whether a truth assignment for F satisfying at least k clauses exists. Both algorithms improve the previous best algorithms by Bansal and Raman for the problem. This article offers a simplified approach to the distribution theory of randomly weighted averages or @math -means @math , for a sequence of i.i.d.random variables @math , and independent random weights @math with @math and @math . The collection of distributions of @math , indexed by distributions of @math , is shown to encode Kingman's partition structure derived from @math . For instance, if @math has Bernoulli @math distribution on @math , the @math th moment of @math is a polynomial function of @math which equals the probability generating function of the number @math of distinct values in a sample of size @math from @math : @math . This elementary identity illustrates a general moment formula for @math -means in terms of the partition structure associated with random samples from @math , first developed by Diaconis and Kemperman (1996) and Kerov (1998) in terms of random permutations. As shown by Tsilevich (1997) if the partition probabilities factorize in a way characteristic of the generalized Ewens sampling formula with two parameters @math , found by Pitman (1992), then the moment formula yields the Cauchy-Stieltjes transform of an @math mean. The analysis of these random means includes the characterization of @math -means, known as Dirichlet means, due to Von Neumann (1941), Watson (1956) and Cifarelli and Regazzini (1990) and generalizations of L 'evy's arcsine law for the time spent positive by a Brownian motion, due to Darling (1949) Lamperti (1958) and Barlow, Pitman and Yor (1989). We investigate geometrical properties of the random K-satisfiability problem using the notion of x-satisfiability: a formula is x-satisfiable is there exist two SAT-assignments differing in Nx variables. We show the existence of a sharp threshold for this property as a function of the clause density. For large enough K, we prove that there exists a region of clause density, below the satisfiability threshold, where the landscape of Hamming distances between SAT-assignments experiences a gap: pairs of SAT-assignments exist at small x, and around x=12, but they do not exist at intermediate values of x. This result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis (the cavity method), and its algorithmic counterpart (the survey propagation algorithm). Our method uses elementary probabilistic arguments (first and second moment methods), and might be useful in other problems of computational and physical interest where similar phenomena appear. Abstract Many AI problems, when formalized, reduce to evaluating the probability that a propositional expression is true. In this paper we show that this problem is computationally intractable even in surprisingly restricted cases and even if we settle for an approximation to this probability. We consider various methods used in approximate reasoning such as computing degree of belief and Bayesian belief networks, as well as reasoning techniques such as constraint satisfaction and knowledge compilation, that use approximation to avoid computational difficulties, and reduce them to model-counting problems over a propositional domain. We prove that counting satisfying assignments of propositional languages is intractable even for Horn and monotone formulae, and even when the size of clauses and number of occurrences of the variables are extremely limited. This should be contrasted with the case of deductive reasoning, where Horn theories and theories with binary clauses are distinguished by the existence of linear time satisfiability algorithms. What is even more surprising is that, as we show, even approximating the number of satisfying assignments (i.e., “approximating” approximate reasoning), is intractable for most of these restricted theories. We also identify some restricted classes of propositional formulae for which efficient algorithms for counting satisfying assignments can be given. It is known that random k-CNF formulas have a so-called satisfiability threshold at a density (namely, clause-variable ratio) of roughly 2kln2: at densities slightly below this threshold almost all k-CNF formulas are satisfiable, whereas slightly above this threshold almost no k-CNF formula is satisfiable. In the current work we consider satisfiable random formulas and inspect another parameter—the diameter of the solution space (that is, the maximal Hamming distance between a pair of satisfying assignments). It was previously shown that for all densities up to a density slightly below the satisfiability threshold the diameter is almost surely at least roughly n 2 (and n at much lower densities). At densities very much higher than the satisfiability threshold, the diameter is almost surely zero (a very dense satisfiable formula is expected to have only one satisfying assignment). In this paper we show that for all densities above a density that is slightly above the satisfiability threshold (more precisel... Consider a system of diagonal equations satisfying the property that the (fixed) integral coefficient matrix @math contains no singular @math submatrix. A recent paper of the authors [ 3 ] establishes that whenever @math and @math , then the expected asymptotic formula holds for the number @math of integral solutions @math of ( @math ) with @math @math . Let @math be a @math -ary predicate over a finite alphabet. Consider a random CSP @math instance @math over @math variables with @math constraints. When @math the instance @math will be unsatisfiable with high probability, and we want to find a refutation - i.e., a certificate of unsatisfiability. When @math is the @math -ary OR predicate, this is the well studied problem of refuting random @math -SAT formulas, and an efficient algorithm is known only when @math . Understanding the density required for refutation of other predicates is important in cryptography, proof complexity, and learning theory. Previously, it was known that for a @math -ary predicate, having @math constraints suffices for refutation. We give a criterion for predicates that often yields efficient refutation algorithms at much lower densities. Specifically, if @math fails to support a @math -wise uniform distribution, then there is an efficient algorithm that refutes random CSP @math instances @math whp when @math . Indeed, our algorithm will "somewhat strongly" refute @math , certifying @math , if @math then we get the strongest possible refutation, certifying @math . This last result is new even in the context of random @math -SAT. Regarding the optimality of our @math requirement, prior work on SDP hierarchies has given some evidence that efficient refutation of random CSP @math may be impossible when @math . Thus there is an indication our algorithm's dependence on @math is optimal for every @math , at least in the context of SDP hierarchies. Along these lines, we show that our refutation algorithm can be carried out by the @math -round SOS SDP hierarchy. Finally, as an application of our result, we falsify assumptions used to show hardness-of-learning results in recent work of Daniely, Linial, and Shalev-Shwartz. 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. For input @math , let @math denote the set of outputs that are the "legal" answers for a computational problem @math . Suppose @math and members of @math are so large that there is not time to read them in their entirety. We propose a model of local computation algorithms which for a given input @math , support queries by a user to values of specified locations @math in a legal output @math . When more than one legal output @math exists for a given @math , the local computation algorithm should output in a way that is consistent with at least one such @math . Local computation algorithms are intended to distill the common features of several concepts that have appeared in various algorithmic subfields, including local distributed computation, local algorithms, locally decodable codes, and local reconstruction. We develop a technique, based on known constructions of small sample spaces of @math -wise independent random variables and Beck's analysis in his algorithmic approach to the Lov ' a sz Local Lemma, which under certain conditions can be applied to construct local computation algorithms that run in polylogarithmic time and space. We apply this technique to maximal independent set computations, scheduling radio network broadcasts, hypergraph coloring and satisfying @math -SAT formulas.