Using the simplifying assumptions of Example 8.7, suppose that there are three advertisers, A, B, and C. There are three queries, x, y, and z. Each advertiser has a budget of 2. Advertiser A bids only on x; B bids on x and y, while C bids on x, y, and z. Note that on the query sequence xxyyzz, the optimum off-line algorithm would yield a revenue of 6, since all queries can be assigned.
(a) Show that the greedy algorithm will assign at least 4 of these 6 queries.
(b) Find another sequence of queries such that the greedy algorithm can assign as few as half the queries that the optimum off-line algorithm assigns on that sequence.
Example 8.7
Suppose there are two advertisers A and B, and only two possible queries, x and y. Advertiser A bids only on x, while B bids on both x and y. The budget for each advertiser is 2. Notice the similarity to the situation in Example 8.1; the only differences are the fact that the bids by each advertiser are the same and the budgets are smaller here.
Example 8.1
Let us take a very simple example of why knowing the future could help. A manufacturer A of replica antique furniture has bid 10 cents on the search term "chesterfield".2 A more conventional manufacturer B has bid 20 cents on both the terms "chesterfield" and "sofa." Both have monthly budgets of $100, and there are no other bidders on either of these terms. It is the beginning of the month, and a search query "chesterfield" has just arrived. We are allowed to display only one ad with the query.