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Lecture

Linear Programming Duality

In course

CS-450: Algorithms II

A first graduate course in algorithms, this course assumes minimal background, but moves rapidly. The objective is to learn the main techniques of algorithm analysis and design, while building a repertory of basic algorithmic solutions to problems in many ...

Description

This lecture covers the concept of linear programming duality, focusing on constraints, variables, solutions, and the relationship between primal and dual LP. The instructor explains the duality theorem, fractional vertices, and the integral property of LP. Examples of bipartite matching and the strong duality theorem are also discussed.

Instructors (3)

Mikhail Kapralov

Ola Nils Anders Svensson

Alessandro Chiesa

Alessandro Chiesa is a faculty member in computer science. He conducts research in complexity theory, cryptography, and security, with a focus on the theoretical foundations and practical implementations of cryptographic proofs that are short and easy to verify. He is a co-author of several zkSNARK libraries, and is a co-inventor of the Zerocash protocol. He has co-founded Zcash and StarkWare Industries. He is a recipient of a Sloan Research Fellowship (2021), an Okawa Foundation Research Grant (2020), and Google Faculty Research Awards (2018 and 2017). He was included in MIT Technology Review's "35 Innovators Under 35" list in 2018.

Official source

Ontological neighbourhood

Theoretical computer science

Algorithms and data structures: Combinatorial optimization

Related concepts (32)

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.

Fractional matching

In graph theory, a fractional matching is a generalization of a matching in which, intuitively, each vertex may be broken into fractions that are matched to different neighbor vertices. Given a graph G = (V, E), a fractional matching in G is a function that assigns, to each edge e in E, a fraction f(e) in [0, 1], such that for every vertex v in V, the sum of fractions of edges adjacent to v is at most 1: A matching in the traditional sense is a special case of a fractional matching, in which the fraction of every edge is either 0 or 1: f(e) = 1 if e is in the matching, and f(e) = 0 if it is not.

Matching in hypergraphs

In graph theory, a matching in a hypergraph is a set of hyperedges, in which every two hyperedges are disjoint. It is an extension of the notion of matching in a graph. Recall that a hypergraph H is a pair (V, E), where V is a set of vertices and E is a set of subsets of V called hyperedges. Each hyperedge may contain one or more vertices. A matching in H is a subset M of E, such that every two hyperedges e_1 and e_2 in M have an empty intersection (have no vertex in common).

Maximum cardinality matching

Maximum cardinality matching is a fundamental problem in graph theory. We are given a graph G, and the goal is to find a matching containing as many edges as possible; that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this problem is equivalent to the task of finding a matching that covers as many vertices as possible.

Matching (graph theory)

In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated as a network flow problem. Given a graph G = (V, E), a matching M in G is a set of pairwise non-adjacent edges, none of which are loops; that is, no two edges share common vertices.

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