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 = ...

Abstract. The paper extends the idea of negative representations of information for enhancing privacy. Simply put, a set DB of data elements can be represented in terms of its complement set. That is, all the elements not in DB are depicted and DB itself is not explicitly stored. We review the negative database (NDB) representation scheme for storing a negative image compactly and propose a design for depicting a multiple record DB using a collection of NDBs—in contrast to the single NDB approach of previous work. Finally, we present a method for creating negative databases that are hard to reverse in practice, i.e., from which it is hard to obtain DB, by adapting a technique for generating 3-SAT formulas. 1

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 = 1 2, 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.

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In this paper we study the model of ε-smoothed k-CNF formulas. Starting from an arbitrary instance F with n variables and m = dn clauses, apply the ε-smoothing operation of flipping the polarity of every literal in every clause independently at random with probability ε. Keeping ε and k fixed, and letting the density d = m/n grow, it is rather easy to see that for d ≥ ε −k ln 2, F becomes whp unsatisfiable after smoothing. We show that a lower density that behaves roughly like ε −k+1 suffices for this purpose. We also show that our bound on d is nearly best possible in the sense that there are k-CNF formulas F of slightly lower density that whp remain satisfiable after smoothing. One consequence of our proof is a new lower bound of Ω(2 k /k 2) on the density up to which Walksat solves random k-CNFs in polynomial time whp. We are not aware of any previous rigorous analysis showing that Walksat is successful at densities that are increasing as a function of k. 1

We study a model of random uniform hypergraphs, where a random instance is obtained by adding random edges to a large hypergraph of a given density. The research on this model for graphs has been started by Bohman et al. (Random Struct Algorithms 22 (2003) 33–42), and

The XOR-satisfiability (XORSAT) problem requires finding an assignment of n Boolean variables that satisfymexclusiveOR(XOR)clauses, wherebyeachclause constrains a subset of the variables. We consider random XORSAT instances, drawn uniformly at random from the ensemble of formulae containing n variables and m clauses of size k. This model presents several structural similarities to other ensembles of constraint satisfaction problems, such as k-satisfiability (k-SAT). For many of these ensembles, as the number of constraints per variable grows, the set of solutions shatters into an exponential number of well-separated components. This phenomenon appears to be related to the difficulty of solving random instances of such problems. We prove a complete characterization of this clustering phase transition for random k-XORSAT. In particular we prove that the clustering threshold is sharp and determine its exact location. We prove that the set of solutions has large conductance below this threshold and that each of the clusters has large conductance above the same threshold. Our proof constructs a very sparse basis for the set of solutions (or the subset within a cluster). This construction is achieved through a low complexity iterative algorithm. 1

A truth assignment is a mapping f that assigns 0 (interpreted as “false”) or 1 (interpreted as “true”) to each variable in its domain; we shall enumerate all the variables in this domain as x1,..., xn. The complement xi of each such variable xi is defined by f(xi) = 1 − f(xi) for all truth assignments f; both xi and xi are called literals; if u = xi then u = xi. A clause is a set of (distinct) literals and a formula (in a conjunctive normal form) is a family of (not necessarily distinct) clauses. A truth assignment satisfies a clause if it maps at least one of its literals to 1; the assignment satisfies a formula if and only if it satisfies each of its clauses. A formula is called satisfiable if it is satisfied by at least one truth assignment; otherwise it is called unsatisfiable. The problem of recognizing satisfiable formulas is known as the satisfiability problem, or SAT for short. Solving SAT by implicit enumeration Given a formula F and a literal v in F, we let F|v denote the “residual formula ” arising from F when f(v) is set at 1: explicitly, this formula is

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These notes are under construction. They constitute a combination of what I have said in the lectures, what I will say in future lectures, and what I will not say due to time constraints. Some sections are very brief, and this is generally because they are not yet written. Some of the “problems and exercises ” describe things that I am actually going to write down in detail in the text. This is because I have used the problems & exercises section in this way to take short notes of things I should not forget to mention.