7 - On the efficient approximability of constraint satisfaction problems

Summary

Abstract

We discuss some results about the approximability of constraint satisfaction problems. In particular we focus on the question of when an efficient algorithm can perform significantly better than the algorithm that picks a solution uniformly at random.

Introduction

The most famous problem in theoretical computer science is the question of whether the two complexity classes P and NP are equal.

Here P is the set of problems that can be solved in time which is polynomial in the size of the input. This is the mathematical definition aimed to correspond to problems which can be solved efficiently in practice on fairly large instances. One might object that there are very large polynomials but this has rarely been a problem and most problems known to be in P are efficiently solvable in the everyday meaning of the concept.

The class NP is the set of decision problems such that for instances with a positive answer there is a short proof of this state of affairs that can be verified efficiently. One of the most famous problems in NP is the traveling salesman problem, TSP, in which we are given n cites and distances d(c_i, c_j) between the cities. The task is, given an upper bound K, to find a tour that visits all the cities and returns to its origin and is of total length at most K. If there is such a tour then, given the tour, it is easy to verify that indeed it is of the desired quality. Formally “easy” should here be interpreted as computable in time which is polynomial in the input length.