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In this work, we study diversity-aware clustering problems where the data points are associated with multiple attributes resulting in intersecting groups. A clustering solution needs to ensure that the number of chosen cluster centers from…
Partly on the basis of heuristic arguments from physics it has been suggested that the performance of certain types of algorithms on random $k$-SAT formulas is linked to phase transitions that affect the geometry of the set of satisfying…
The satisfiability threshold for constraint satisfaction problems is that value of the ratio of constraints (or clauses) to variables, above which the probability that a random instance of the problem has a solution is zero in the large…
It is crucial to generate crafted SAT formulas with predefined solutions for the testing and development of SAT solvers since many SAT formulas from real-world applications have solutions. Although some generating algorithms have been…
The solution space of many classical optimization problems breaks up into clusters which are extensively distant from one another in the Hamming metric. Here, we show that an analogous quantum clustering phenomenon takes place in the ground…
We give the first efficient algorithm to approximately count the number of solutions in the random $k$-SAT model when the density of the formula scales exponentially with $k$. The best previous counting algorithm for the permissive version…
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,…
We consider the K-satisfiability problem on a regular d-ary rooted tree. For this model, we demonstrate how we can calculate in closed form, the moments of the total number of solutions as a function of d and K, where the average is over…
One of the most studied models of SAT is random SAT. In this model, instances are composed from clauses chosen uniformly randomly and independently of each other. This model may be unsatisfactory in that it fails to describe various…
Using powerful Multicanonical Ensemble Monte Carlo methods from statistical physics we explore the realization space of random K satisfiability (KSAT) in search for computational hard problems, most likely the 'hardest problems'. We search…
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 report a cluster of results regarding the difficulty of finding approximate ground states to typical instances of the quantum satisfiability problem $k$-QSAT on large random graphs. As an approximation strategy, we optimize the solution…
Discrete combinatorial optimization has a central role in many scientific disciplines, however, for hard problems we lack linear time algorithms that would allow us to solve very large instances. Moreover, it is still unclear what are the…
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…
A novel framework for consensus clustering is presented which has the ability to determine both the number of clusters and a final solution using multiple algorithms. A consensus similarity matrix is formed from an ensemble using multiple…
The computational complexity of solving random 3-Satisfiability (3-SAT) problems is investigated. 3-SAT is a representative example of hard computational tasks; it consists in knowing whether a set of alpha N randomly drawn logical…
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…
We study the behavior of ASAT, a heuristic for solving satisfiability problems by stochastic local search near the SAT/UNSAT transition. The heuristic is focused, i.e. only variables in unsatisfied clauses are updated in each step, and is…
Relation between problem hardness and solution space structure is an important research aspect. Model d-k-CSP generates very hard instances when $r=1$ and $r$ is near 1, where $r$ represents normalized constraint density. We find that when…
A variational approach to finite connectivity spin-glass-like models is developed and applied to describe the structure of optimal solutions in random satisfiability problems. Our variational scheme accurately reproduces the known replica…