Introduction to Approximation Algorithms

To date, thousands of natural optimization problems have been shown to be NP-hard [8, 18]. To deal with these problems, two approaches are commonly adopted: (a) approximation algorithms, (b) randomized algorithms. Roughly speaking, approximation algorithms aim to find solutions whose costs are as close to be optimal as possible in polynomial time. Randomized algorithms can be looked at from different angles. We can design algorithms giving optimal solutions in expected polynomial time, or giving expectedly good solutions. Many different techniques are used to devise these kinds of algorithms, which can be broadly classified into: (a) combinatorial algorithms, (b) linear programming based algorithms, (c) semi-definite programming based algorithms, (d) randomized (and derandomized) algorithms, etc. Within each class, standard algorithmic designs techniques such as divide and conquer, greedy method, and dynamic programming are often adopted with a great deal of ingenuity. Designing algorithms, after all, is as much an art as it is science. There is no point approximating anything unless the approximation algorithm runs efficiently, i.e. in polynomial time. Hence, when we say approximation algorithms we implicitly imply polynomial time algorithms. To clearly understand the notion of an approximation algorithm, we first define the so-called approximation ratio. Informally, for a minimization problem such as the VERTEX COVER problem, a polynomial time algorithm A is said to be an approximation algorithm with approximation ratio δ if and only if for every instance of the problem,A gives a solution which is at most δ times the optimal value for that instance. This way, δ is always at least 1. As we do not expect to have an approximation algorithm with δ = 1, we would like to get δ as close to 1 as possible. Conversely, for maximization problems the algorithm A must produce, for each input instance, a solution which is at least δ times the optimal value for that instance. (In this case δ ≤ 1.) The algorithm A in both cases are said to be δ-approximation algorithm. Sometimes, for maximization problems people use the term approximation ratio to refer to 1/δ. This is to ensure that the ratio is at least 1 in both the min and the max case. The terms approximation ratio, approximation factor, performance guarantee, worst case ratio, absolute worst case ratio, are more or less equivalent, except for the 1/δ confusion we mentioned above. Let us now be a little bit more formal. Consider an optimization problem Π, in which we try to minimize a certain objective function. For example, when Π is VERTEX COVER the objective function is the size of a vertex cover. For each instance I ∈ Π, let OPT(I) be the optimal value of the objective function for I . In VERTEX COVER, I is a graph G and OPT(G) depends on the structure of G. Given a polynomial time algorithm A which returns some feasible solution for Π, let A(I) denote the objective value returned by A on in put I . Define