The overall targets of the course is to expose some aspects of the deep and important dialectic exchange between the search for algorithmic solutions and the study of the complexity of problems. Algorithms are the backbone and the substance of information technologies, but at the same time their study goes beyond the "mere" computer science and is pervasive to all the disciplines that are problem-bearers. The design of an algorithm starts from the study of the structure of the problem to be solved and it usually represents the highest achievement of this process. The study of algorithms requires and offers methodologies and techniques of problem solving, logical and mathematical skills. The course therefore aims to provide students with fundamental skills and methodologies for the analysis of problems and the design of the algorithms for solving them. Particular emphasis is given to the efficiency of the algorithms themselves, and the theory of computational complexity plays a profound methodological role in the analysis of problems. With reference to the overall didactic aims of the Master program, the course leads students to deepen and expand the three-year training in the field of analysis and evaluation of problems, algorithms, and computational models, providing a wealth of advanced tools to address non-trivial problems in different IT fields. The students will acquire logical-mathematical skills, techniques, experience and methodologies useful in the analysis of algorithmic problems, from detecting their structure and analyzing their computational complexity to designing efficient algorithms, and then planning and conducting their implementation. Besides that, The course provides the foundations of computational complexity theory, focussing on: the theory of NP-completeness; approximation algorithms and basic approaches for the analysis of the approximation guarantee of algorithms for hard problems; and parameterized approaches to hard problems. The student will apply the main algorithmic techniques: recursion, divide and conquer, dynamic programming, some data structures, invariants and monovariants. The student will develop sensitivity about which problems can be solved efficiently and with which techniques, acquiring also dialectical tools to place the complexity of an algorithmic problem and identify promising approaches for the same, looking at the problem to grasp its structure. She will learn to produce, discuss, evaluate, and validate conjectures, and also independently tackle the complete path from the analysis of the problem, to the design of a resolver algorithm, to the coding and experimentation of the same, even in research contexts either in the private sector or at research institutions. Based on the basic notions acquired of computational complexity, the students will be able to employ reductions, standard techniques in complexity theory, to analyze the structural properties of computational problems and identify possible alternative approaches (approximation, parameterization) in the absence of (provably) efficient solutions After attending the course the students will be able to: classify intractable computational problems; understand and verify a formal proof; read and understand a scientific article where a new algorithm is presented together with the analysis of its computational complexity.