May 9, 2022 · An approximation algorithm is a way of dealing with NP-completeness for an optimization problem. This technique does not guarantee the best solution. The goal of the approximation algorithm is to come as close as possible to the optimal solution in polynomial time. Such algorithms are called approximation algorithms or heuristic algorithms.

In this section we'll discuss three applications of linear programming to the design and analysis of approximation algorithms. In an undirected graph G = (V; E), if S V is a set of vertices and e is an edge, we say that S covers e if at least one endpoint of e belongs to S. We say that S is a vertex cover if it covers every edge.

Three standard approaches include: Approximation algorithms: procedures which are proven to give solutions within a factor of optimum. Of these approaches, approximation algorithms are arguably the most mathematically satisfying, and will be the subject of discussion for this section.

Approximation algorithms compute near-optimal solutions. Known for thousands of years. For instance, approximations of value of ⇡; some engineers still use 4 these days :-) Consider optimisation problem. Each potential solution has positive cost, we want near-optimal solution.

Approximation Algorithm : Definition Given an optimization problem P, an algorithm Ais said to be an approximation algorithm for P, if for any given instance I, it returns an approximate solution, that is a feasible solution.

Approximation Algorithms • Let P be an optimization problem in NP. • A is called an ρ-approximation algorithm for P if for all inputs I, A produces an output O ∈O I such that [Minimization problem] f(O)6ρ×OPT I, [Maximization problem] f(O)>ρ×OPT I. • ρis called the approximation ratio or the approximation factor.

Optimization question: Can you choose k elements from S such that every element of A is in at least one of these k? (This is a called a cover.) Decision question: Exist a cover of size k or less? Why use approximation? Return an optimal answer in some cases (fail in others?) For a given resource level, achieve a lower performance value?

Jan 4, 2022 · Welcome to a course on approximation algorithms. These are “efficient” algorithms which return a solution “close” to the desired solution, where close is deliberately left vague at this point. Why should one care? For many reasons. Most importantly, there are many problems for which finding the desired solution may be too hard.

To learn techniques for design and analysis of approximation algorithms, via some fundamen-tal problems. To build a toolkit of broadly applicable algorithms/heuristics that can be used to solve a variety of problems. To understand reductions between optimization problems, and to develop the ability to relate new problems to known ones.

An algorithm has approximation ratio r if it outputs solutions with cost such that c/c* ≤ r and c*/c ≤ r where c* is the optimal cost. We focus on ratio (as opposed to difference) because that appears to be more natural for problems of interest