Algorithm Design

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Chapter 11 Approximation Algorithms
f!:~ Designing the Algorithm: The Pricing Method
Even though (!1.12) gave us an approximation algorithm with a provable
guarantee, we will be able to do better. Our approach forms a nice illustration
of the pricing method for designing approximation algorithms.
The Pricing Method to Minimize Cost The pricing method (also known as
the primal-dual method) is motivated by an economic perspective. For the
case of the Vertex Cover Problem, we will think of the weights on the nodes
as costs, and we will think of each edge as having to pay for its "share" of
the cost of the vertex cover we find. We have actually just seen an analysis of
this sort, in the greedy algorithm for Set Cover from Section 11.3; it too can be
thought of as a pricing algorithm. The greedy algorithm for Set Cover defined
values Cs, the cost the algorithm paid for covering element s. We can think of
c s as the element s’s "share" of the cost. Statement (11.9) shows that it is very
natural to think of the values cs as cost-shares, as the sum of the cost-shares
~s~U cs is the cost of the set cover e returned by the algorithm, ~siee wi. The
key to proving that the algorithm is an H(d*)-approximafion algorithm was a
certain approximate "fairness" property for the cost-shares: (11.10) shows that
the elements in a set Sk are charged by at most an H([SkD factor more than
the cost of covering them by the set S~.
In this section, we’ll develop the pricing technique through another application,
Vertex Cover. Again, we will think of the weight tvi of the vertex i
as the cost for using i in the cover. We will think of each edge e as a separate
"agent" who is willing to "pay" something to the node that covers it. The algorithm
will not only find a vertex cover S, but also determine prices Pe >- 0
for each edge e ~ E, so that if each edge e E pays the price Pe, this will in
total approximately cover the cost of S. These prices Pe are theanalogues of
Cs from the Set Cover Algorithm.
Thinking of the edges as agents suggests some natural fairness rules for
prices, analogous to the property proved by (11.10). First of all, selecting a
vertex i covers all edges incident to i, so it would be "unfair" to charge these
incident edges in total more than the cost of vertex i. We call prices Pe fair if,
for each vertex i, the edges adjacent to i do not have to pay more than the
cost of the vertex: ~e=(i,j) Pe < ~Vi" Note that the property proved by (11.10)
for Set Cover is an approximate fairness condition, while in the Vertex Cover
algorithm we’ll actually use the exact fairness defined here. A useful fact about
fair prices is that they provide a lower bound on the cost of any solution.
(11.13) For any vertex cover S*, and any nonnegative and lair prices
have ~e~ Pe