Combinatorial optimization and metaheuristics

Today, combinatorial optimization is one of the youngest and most active areas of discrete mathematics. It is a branch of optimization in applied mathematics and computer science, related to operational research, algorithm theory and computational complexity theory. It sits at the intersection of several fields, including artificial intelligence, mathematics and software engineering. Its increasing interest arises for the fact that a large number of scientific and industrial problems can be formulated as abstract combinatorial optimization problems, through graphs and/or (integer) linear programs. Some of these problems have polynomial-time (“efficient”) algorithms, while most of them are NP-hard, i.e. it is not proved that they can be solved in polynomial-time. Mainly, it means that it is not possible to guarantee that an exact solution to the problem can be found and one has to settle for an approximate solution with known performance guarantees. Indeed, the goal of approximate methods is to find “quickly” (reasonable run-times), with “high” probability, provable “good” solutions (low error from the real optimal solution). In the last 20 years, a new kind of algorithm commonly called metaheuristics have emerged in this class, which basically try to combine heuristics in high level frameworks aimed at efficiently and effectively exploring the search space. This report briefly outlines the components, concepts, advantages and disadvantages of different metaheuristic approaches from a conceptual point of view, in order to analyze their similarities and differences. The two very significant forces of intensification and diversification, that mainly determine the behavior of a metaheuristic, will be pointed out. The report concludes by exploring the importance of hybridization and integration methods

Factorization and Criticality in the Anisotropic XY Chain via Correlations

In this review, we discuss the zero and finite temperature behavior of various bipartite quantum and total correlation measures, the skew information-based quantum coherence, and the local quantum uncertainty in the thermal ground state of the one-dimensional anisotropic XY model in transverse magnetic field. We compare the ability of considered measures to correctly detect or estimate the quantum critical point and the non-trivial factorization point possessed by the spin chain.Comment: 29 pages, 8 figures. A review paper accepted for publication in the special issue Entanglement Entropy in the journal Entrop