Course

MATH-504: Integer optimisation

Lectures in this course (31)

Explores transference bounds, ellipsoids, and determining the shortest vector.

Covers the fundamentals of integer optimization, including integer programming, dynamic programming, and approximation algorithms.

Covers the 2D case of polynomial encroaching and vertices of polyhedra.

Covers the theory of linear and integer programming, focusing on the integer hull and rational polyhedra.

Explores convex sets, Minkowski-Weyl theorem, and Separation theorem in convex analysis.

Covers the concept of the integer hull in the general case, focusing on the proof of claim and the properties of cells.

Explores the Closest Vector Problem and Voronoi cells in lattice reduction algorithms.

Covers the concepts of faces, facets, and vertices in polyhedra, exploring minimal faces and inequalities.

Explores Voronoi analysis for efficient Closest Vector Problem solutions.

Explores equitable coloring in totally unimodular matrices, enabling integral vectors and bicoloring results.

Covers the Branch and Bound optimization technique with LP Relaxation and Integral Optimal Solutions.

Explores the computation of Voronoi cells, linear independence, uniqueness, and solving the Closest Vector Problem.

Covers the Branch and Bound algorithm, focusing on formal description and implementation steps to find optimal integer solutions.

Explores the closest vector problem in lattices and the role of Voronoi cells in determining the closest vector.

Covers Caratheodory bounds for integer programming, focusing on linear programming and the existence of optimal solutions.

Explores dynamic programming for the Knapsack problem, discussing strategies, algorithms, NP-hardness, and time complexity analysis.

Explores integer programming, John's Theorem, and the reduction to shortest vector computations in convex bodies.

Explores dynamic programming for the Knapsack Problem, optimizing weight accumulation and profit subsets efficiently.

Introduces the FPTAS for the Knapsack problem, focusing on achieving an approximation of (1-ε) times the optimal solution.

Explores the complexity of convex hulls and the concept of vertices within them.