  Ad auctions in sponsored search support ``broad match'' that allows an
advertiser to target a large number of queries while bidding only on a limited
number. While giving more expressiveness to advertisers, this feature makes it
challenging to optimize bids to maximize their returns: choosing to bid on a
query as a broad match because it provides high profit results in one bidding
for related queries which may yield low or even negative profits.
  We abstract and study the complexity of the {\em bid optimization problem}
which is to determine an advertiser's bids on a subset of keywords (possibly
using broad match) so that her profit is maximized. In the query language model
when the advertiser is allowed to bid on all queries as broad match, we present
an linear programming (LP)-based polynomial-time algorithm that gets the
optimal profit. In the model in which an advertiser can only bid on keywords,
ie., a subset of keywords as an exact or broad match, we show that this problem
is not approximable within any reasonable approximation factor unless P=NP. To
deal with this hardness result, we present a constant-factor approximation when
the optimal profit significantly exceeds the cost. This algorithm is based on
rounding a natural LP formulation of the problem. Finally, we study a budgeted
variant of the problem, and show that in the query language model, one can find
two budget constrained ad campaigns in polynomial time that implement the
optimal bidding strategy. Our results are the first to address bid optimization
under the broad match feature which is common in ad auctions.
