# The Design of Approximation Algorithms

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Algorithms by Jeff Erickson

# 1 An introduction to approximation algorithms

Definition 1.1: An $\alpha$-approximation algorithm is …

Definition 1.2: A polynomial-time approximation scheme (PTAS) is …

Theorem 1.3: For any MAX SNP-hard problem, there does not exist a polynomial-time approximation scheme, unless $P = NP$.

An example: the maximum clique problem

# 2 Greedy algorithm and local search

## 2.1 Scheduling jobs with deadlines on a single machine

Assume that all due dates are negative, which implies that the optimal value is always positive. We shall give a 2-approximation algorithm for this special case.

## 2.2 The k-center problem

A greedy 2-approximation algorithm.

There is no $\alpha$-approximation algorithm for the k-center problem for $\alpha < 2$ unless $P = NP$.

## 2.3 Scheduling jobs on identical parallel machines

the local search algorithm and the greedy algorithm are 2-approximation.

the longest processing time rule is a 4/3-approximation algorithm.

## 2.4 The traveling salesman problem

the double-tree algorithm is a 2-approximation algorithm.

the Christofides’ algorithm is a 3/2-approximation algorithm.

## 2.6 Finding minimum-degree spanning trees

The locally optimal tree: no move that can reduce the degree of any node having degree between $\Delta(T) - l$ and $\Delta(T)$.

Theorem 2.19: Let T be a locally optimal tree. Then $\Delta(T) \le 2OPT + l$, where $l = \lceil \log_2{n} \rceil$.

Theorem 2.20: The algorithm finds a locally optimal tree in polynomial time.