Algorithm Design

660
Chapter 11 Approximation Mgofithms
The linear programming rounding algorithm for the Weighted Vertex Cover
Problem is due to Hochbaum (1982). The rounding algorithm for Generalized
Load Balancing is due to Lenstra, Shmoys, and Tardos (1990); see the survey
by Hall (1996) for other results in this vein. As discussed in the text, these
two results illustrate a widely used method for .designing approximation algorithms:
One sets up an integer programming formulation for the problem,
transforms it to a related [but not equivalent) linear programming problem,
and then rounds the resulting solution. Vaziranl (2001) discusses many further
applications of this technique.
Local search and randomization are two other powerful techniques for
designing approximation algorithms; we discuss these connections in the next
two chapters.
One topic that we do not cover in this book is inapproximabilitY. Just as
one can prove that a given NP-hard problem can be approximated to within
a certain factor in polynomial time, one can also sometimes establish lower
bounds, showing that if the problem could be approximated to witl.g.’n better
than some factor c in polynomial time, then it could be solved optimally,
thereby proving T = 3q~P. There is a growing body of work that establishes such
limits to approximability for many NP-hard problems. In certain cases, these
positive and negative results have lined up perfectly to produce an approximation
threshold, establishing for certain problems that there is a polynomial-time
approximation algorithm to within some factor c, and it is impossible to do
better unless ~P = %f~P. Some of the early results on inapproximabilitY were not
rove, but more recent work
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and Lund (1996).
Notes oft the Exercises Exercises 4 and 12 are based on results of Dorit
Hochbaum. Exercise 11 is based on results of Sartaj Sahni, Oscar Ibarra, and
Chul Kim, and of Dorit Hochbaum and David Shmoys.
Cha~ter
Local Search
In the previous two chapters, we have considered techniques for dealing with
computationally intractable problems: in Chapter 10, by identifying structured
special cases of NP-hard problems, and in Chapter 11, by designing polynomialtime
approximation algorithms. We now develop a third and final topic related
~
Local search is a very general technique; it describes any algorithm that
"explores" the space of possible solutions in a sequential fashion, moving
in one step from a current solution to a "nearby" one. The generality and
flexibility of this notion has the advantage that it is not difficult to design
a local search approach to almost any computationally hard problem; the
counterbalancing disadvantage is that it is often very difficult to say anything
precise or provable about the quality of the solutions that a local search
algorithm finds, and consequently very hard to tell whether one is using a
good local search algorithm or a poor one.
Our discussion of local search in this chapter will reflect these trade-offs.
Local search algorithms are generally heuristics designed to find good, but
not necessarily optimal, solutions to computational problems, and we begin
by talking about what the search for such solutions looks like at a global
level. A useful intuitive basis for this perspective comes from connections with
energy minimization principles in physics, and we explore this issue first. Our
discussion for this part of the chapter will have a somewhat different flavor
from what we’ve generally seen in the book thus far; here, we’ll introduce
some algorithms, discuss them qualitatively, but admit quite frankly that we
can’t prove very much about them.
There are cases, however, in which it is possible to prove properties
of local search algorithms, and to bound their performance relative to an