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In the last 30 years it was found that many combinatorial systems undergo phase transitions. One of the most important examples of these can be found among the random k-satisfiability problems (often referred to as k-SAT), asking whether…
An active topic in the study of random constraint satisfaction problems (CSPs) is the geometry of the space of satisfying or almost satisfying assignments as the function of the density, for which a precise landscape of predictions has been…
We initiate the study of constraint satisfaction problems (CSPs) in the presence of counting quantifiers, which may be seen as variants of CSPs in the mould of quantified CSPs (QCSPs). We show that a single counting quantifier strictly…
We study the structure of the solution space and behavior of local search methods on random 3-SAT problems close to the SAT/UNSAT transition. Using the overlap measure of similarity between different solutions found on the same problem…
Constraint satisfaction problem (CSP) is a well-studied combinatorial search problem, in which we are asked to find an assignment of values to given variables so as to satisfy all of given constraints. We study a reconfiguration variant of…
The XOR-satisfiability (XORSAT) problem requires finding an assignment of $n$ Boolean variables that satisfy $m$ exclusive OR (XOR) clauses, whereby each clause constrains a subset of the variables. We consider random XORSAT instances,…
The solution space of a K-satisfiability (K-SAT) formula is a collection of solution clusters, each of which contains all the solutions that are mutually reachable through a sequence of single-spin flips. Knowledge of the statistical…
Constraint satisfaction problems (CSPs) are a class of problems that are ubiquitous in science and engineering. It features a collection of constraints specified over subsets of variables. A CSP can be solved either directly or by reducing…
The random K-satisfiability (K-SAT) problem is an important problem for studying typical-case complexity of NP-complete combinatorial satisfaction; it is also a representative model of finite-connectivity spin-glasses. In this paper we…
Constrained sampling and counting are two fundamental problems in artificial intelligence with a diverse range of applications, spanning probabilistic reasoning and planning to constrained-random verification. While the theory of these…
Using elementary rigorous methods we prove the existence of a clustered phase in the random $K$-SAT problem, for $K\geq 8$. In this phase the solutions are grouped into clusters which are far away from each other. The results are in…
This thesis is divided in two parts. The first presents an overview of known results in statistical mechanics of disordered systems and its approach to random combinatorial optimization problems. The second part is a discussion of two…
The set of solutions of random constraint satisfaction problems (zero energy groundstates of mean-field diluted spin glasses) undergoes several structural phase transitions as the amount of constraints is increased. This set first breaks…
In this short paper we present a survey of some results concerning the random SAT problems. To elaborate, the Boolean Satisfiability (SAT) Problem refers to the problem of determining whether a given set of $m$ Boolean constraints over $n$…
We study a model of constraint satisfaction problems geared towards instances with few variables but with domain of unbounded size (udCSP). Our model is inspired by recent work on FPT algorithms for MinCSP where frequently both upper and…
The fixed template Promise Constraint Satisfaction Problem (PCSP) is a recently proposed significant generalization of the fixed template CSP, which includes approximation variants of satisfiability and graph coloring problems. All the…
Many NP-complete constraint satisfaction problems appear to undergo a "phase transition'' from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds…
For a large number of random constraint satisfaction problems, such as random k-SAT and random graph and hypergraph coloring, there are very good estimates of the largest constraint density for which solutions exist. Yet, all known…
Planting a solution into the random RB model, which is a prototype of random constraint satisfaction problem (CSP) with growing domains, can generate very hard satisfiable CSP benchmarks. We study the solution space structure of the planted…
Given $k$ collections of 2SAT clauses on the same set of variables $V$, can we find one assignment that satisfies a large fraction of clauses from each collection? We consider such simultaneous constraint satisfaction problems, and design…