Cutset Formulation: MST Problem

In course

Description

This lecture covers the cutset formulation for the Minimum Spanning Tree (MST) Problem, exploring decision variables, constraints, and the quality of the formulation. It delves into the subtour elimination formulation, discussing its strength and complexity. The instructor explains the Gomory Cutting Planes method, detailing the idea, algorithm, and examples. The lecture concludes with LP-based Branch & Bound, illustrating the algorithm and its application through an example.

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https://mediaspace.epfl.ch/media/0_2jx244m8

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Linear programming

Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints.

Integer programming

An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear. Integer programming is NP-complete. In particular, the special case of 0-1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems.

Linear programming relaxation

In mathematics, the relaxation of a (mixed) integer linear program is the problem that arises by removing the integrality constraint of each variable. For example, in a 0–1 integer program, all constraints are of the form The relaxation of the original integer program instead uses a collection of linear constraints The resulting relaxation is a linear program, hence the name.

Branch and cut

Branch and cut is a method of combinatorial optimization for solving integer linear programs (ILPs), that is, linear programming (LP) problems where some or all the unknowns are restricted to integer values. Branch and cut involves running a branch and bound algorithm and using cutting planes to tighten the linear programming relaxations. Note that if cuts are only used to tighten the initial LP relaxation, the algorithm is called cut and branch. This description assumes the ILP is a maximization problem.

Cutting-plane method

In mathematical optimization, the cutting-plane method is any of a variety of optimization methods that iteratively refine a feasible set or objective function by means of linear inequalities, termed cuts. Such procedures are commonly used to find integer solutions to mixed integer linear programming (MILP) problems, as well as to solve general, not necessarily differentiable convex optimization problems. The use of cutting planes to solve MILP was introduced by Ralph E. Gomory.