Classifying Approximation Algorithms: Understanding the APX Complexity Class

10/29/2021
by   Arthur Lee, et al.
0

We are interested in the intersection of approximation algorithms and complexity theory, in particular focusing on the complexity class APX. Informally, APX ⊆ NPO is the complexity class comprising optimization problems where the ratio OPT(I)/ALG(I)≤ c for all instances I. We will do a deep dive into studying APX as a complexity class, in particular, investigating how researchers have defined PTAS and L reductions, as well as the notion of APX-completeness, thereby clarifying where APX lies on the polynomial hierarchy. We will discuss the relationship of this class with FPTAS, PTAS, APX, log-APX and poly-APX). We will sketch the proof that Max 3-SAT is APX-hard, and compare this complexity class in relation to BPP, ZPP to elucidate whether randomization is powerful enough to achieve certain approximation guarantees and introduce techniques that complement the design of approximation algorithms such as through primal-dual analysis, local search and semi-definite programming. Through the PCP theorem, we will explore the fundamental relationship between hardness of approximation and randomness, and will recast the way we look at the complexity class NP. We will finish by looking at the "real world" applications of this material in Economics. Finally, we will touch upon recent breakthroughs in the Metric Travelling Salesman and asymmetric travelling salesman problem, as well original directions for future research, such as quantifying the amount of additional compute power that access to an APX oracle provides, elucidating fundamental combinatorial properties of log-APX problems and unique ways to attack the problem of whether the minimum set-cover problem is self-improvable.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
04/28/2021

Verified Approximation Algorithms

We present the first formal verification of approximation algorithms for...
research
07/04/2021

Sublinear-Space Approximation Algorithms for Max r-SAT

In the Max r-SAT problem, the input is a CNF formula with n variables wh...
research
12/19/2018

Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems

Semidefinite programming is a powerful tool in the design and analysis o...
research
07/17/2020

Improved Approximation Algorithms for Tverberg Partitions

Tverberg's theorem states that a set of n points in ^d can be partiti...
research
05/12/2021

Application of the Level-2 Quantum Lasserre Hierarchy in Quantum Approximation Algorithms

The Lasserre Hierarchy is a set of semidefinite programs which yield inc...
research
02/01/2021

The Complexity of Learning Linear Temporal Formulas from Examples

In this paper we initiate the study of the computational complexity of l...

Please sign up or login with your details

Forgot password? Click here to reset