Bid Optimization in Broad-Match Ad auctions

0.22 MB·English

by  Eyal Even-dar

Bid Optimization for Broad Match Ad Auctions
Eyal Even Dar∗ Yishay Mansour † Vahab S. Mirrokni∗ S. Muthukrishnan∗
Uri Nadav ‡
9
ABSTRACT Keywords
0
0 Ad auctions in sponsored search support “broad match” Sponsored Search, Ad Auctions, Optimal bidding, Bid Op-
2 thatallowsanadvertisertotargetalargenumberofqueries timization

n whilebiddingonlyonalimited number. Whilegivingmore
a expressivenesstoadvertisers,thisfeaturemakesitchalleng- 1. INTRODUCTION
J ing to optimize bidsto maximize theirreturns: choosing to
3 bid on a query as a broad match because it provides high Sponsoredsearchisalargeandthrivingmarketwiththree
distinct players. Users go to search engines such as Yahoo!
2 profit results in one bidding for related queries which may
or Google and pose queries; in the process, they express
yield low or even negative profits.
their intention and preferences. Advertisers seek to place
] Weabstractandstudythecomplexityofthebidoptimiza-
T advertisements and target them to users’ intentions as ex-
tion problemwhichistodetermineanadvertiser’sbidsona
pressed by their queries. Finally, search engines provide a
G subsetofkeywords(possiblyusingbroadmatch)sothather
suitable mechanism for doing this. Currently, the mecha-
s. padrovfierttiissemraisxiamlliozweded. Itnotbhiedqouneraylllaqnugeuraiegseamsobdreolawdhmenattchhe, nism relies on having advertisers bid on the search issued
c by theuser, and thesearch engine torun an auction at the
we present an linear programming (LP)-based polynomial-
[ time the user poses the query to determine the advertise-
time algorithm that gets the optimal profit. In the model
ments that will be shown to the user. As is standard, the
1 in which anadvertisercan only bidon keywords,ie., asub-
advertiseronlypaysiftheuserclicksontheirad(the”pay-
v set of keywords as an exact or broad match, we show that
per-click” model), and the amount they pay is determined
4 thisproblemisnotapproximablewithinanyreasonableap-
5 proximationfactorunlessP=NP.Todealwiththishardness by the auction mechanism, but will be no larger than their
7 result,wepresentaconstant-factorapproximationwhenthe bid.
3 Inthispaper,weassumetheperspectiveoftheadvertiser.
optimalprofitsignificantlyexceedsthecost. Thisalgorithm
. The advertisers need to target their ad campaigns to users’
1 is based on roundinga naturalLP formulation of theprob-
queries. Thus, they need to determine the set S of queries
0 lem. Finally, we study a budgeted variant of the problem,
of their interest. Once that is determined, they need to
9 and show that in the query language model, one can find
strategize in the auction that takes place for each of the
0 two budget constrained ad campaigns in polynomial time
queries in S. A lot of research has focused on the game
: that implement the optimal bidding strategy. Our results
v theory and optimization behind these auctions, both from
are the first to address bid optimization under the broad
i the search engine [2, 20, 7, 1, 13, 5] and advertiser [4, 9,
X match feature which is common in ad auctions.
6, 14] points of view. There has been relatively little prior
r research onhowadvertiserstarget theircampaign, i.e.,how
a
Categories andSubject Descriptors they determine theset S.
The criterion for choosing S is for the advertiser to pick
F.2 [Theory of Computation]: Analysis of Algorithms
a set of keyphrases that searchers may use in their query
and Problem Complexity; J.4 [Computer Applications]:
whenlookingfortheirproducts. Thecentralchallengethen
SocialandBehavioralSciences—Economics;H.4[Information
is to match the advertisers keyphrases with the potential
Systems Applications]: Miscellaneous
queries issued by the users. It is difficult if not impossi-
ble for the advertisers to identify all possible variations of
General Terms keyphrasesthat auserlooking fortheirproductmay usein
their query. As an example, consider a vendorwho chooses
Algorithm, Theory, Economics
the keyphrase tennis shoes. Users searching for them may
∗Google Research, 76 9th Ave, New York, NY 10011, usesingularorplural,synonymsandothervariations(“clay
{evendar,mirrokni,muthu}@google.com court footwear”), may misspell (“tenis shoe”), use exten-
†GoogleResearchandTel-avivUniversity,769thAve,New sions (“white tennis shoes”) or reorder the words (“shoes
York,NY 10011, [email protected] lawn tennis”). In fact, users may even search using words
‡Tel-aviv University,Tel-Aviv,69978, [email protected] not found in the keyphrase (“Wimbledon gear”, ”US Open
Shoes”, “hard court soles”), and may still be of interest to
Copyrightisheldbytheauthor/owner(s).
the advertiser. These artifacts such as plurals, synonyms,
WWW2009,April20-24,2009,Madrid,Spain.
. misspellings, extensions, and reorderings are very common,
and the problems get compounded since typical ad cam- search users develop and prefer over time. Thus, the ad-
paignscompriseseveralkeyphrases,eachwithitsownsetof vertisers bid implicitly on queries on which they can not
artifacts. directlycontrolthetradeoffbetweenthecostandthevalue.
Major search engines help advertisers address this chal- Querydependenceintroducesacomplexoptimizationprob-
lengebyprovidingastructuredbiddinglanguage. Whilethe lem of trading off the benefits of bidding on a keyphrase
specificdetailsdifferfromsearchenginetosearchengine[21, against the impact of bidding on its dependent queries. In
22, 23], at the highest level, the bidding language supports thesponsoredsearchworld,thereisakeenawarenessofthis
two match types: exact and broad. In exact matchtype complexity of bidding, and most search engines and third-
(called “exact” in MSN AdCenter and Google, and “stan- partybiddingagentsprovidedetailedtipsandguidelinesfor
dard” in Yahoo), ad would be eligible to appear when a advertisers [27, 28]. Beyond these guidelines, what is miss-
user searches for the specific keyphrase without any other ing is a clear theoretical understandingof thetradeoffs and
terms in the query, and words in the keyphrase need to thecomplexityofthebiddingproblemthatadvertisersface.
appear in that order. In broad matchtype (called “broad” We initiate principled study of bidding in presence of
in MSN, related to “phrase” and “broad” in Google, and broad matches. Specifically, our contributions are as fol-
“advanced match type” in Yahoo), the system automati- lows.
callymakesadvertiserseligibleonrelevantvariationsoftheir
keyphrases including for the various artifacts listed earlier, 1. We abstract two models — query and keyword lan-
evenifthesearchtermsarenotinthekeyphraselists. Thus, guage models — to study bidding optimization prob-
thesearchenginesautomatetheaspectofdetectingartifacts lems.
and matching the query to keyphrases of interest to adver- In the query language model, the advertiser bids di-
tisers.1 Thus the task of advertisers becomes determining rectly on user queries and wishes to determine which
the keyphrasesand choosing thematch typeon each. query if any to bid on, to maximize expected profit.
The question we address here is, how does an advertiser This models both the theoretical extreme where an
bid in presenceof thesematch types? Say each queryq has advertiser can bid on any of the queries the search
a value v(q) per click for the advertiser that is known to enginewillsee,andthepracticalrealitywherethead-
theadvertiserandisprivate. Further,weletc(q)betheex- vertiser has a select set of queries in mind and wishes
pectedpriceperclickandletn(q)betheexpectednumberof only to optimize within that set. In the keyword lan-
clicks. Thesearestatisticalestimatesprovidedbythesearch guage model, advertisers may bid only on a subset
engines [24, 25, 26]. Then, we consider two optimization of queries, and broad match implicitly derives bids as
problems: (i) in one variant, we assume that the advertiser needed. This directly models the common reality.
wishes tomaximizetheir expected profit, that is, (v(q)−
Pq
c(q))n(q), and (ii) in the other variant, given a budget B 2. We present efficient, polynomial time algorithms for
for the advertiser, we assume that the advertiser wishes to thebidoptimizationproblemunderthesetwomodels.
maximize their expected value, that is, Pqv(q)n(q) subject Inquerybidding, weget apolynomial-time algorithm
to the condition that the expected spend c(q)n(q) does that maximizes the profit, using a reduction to the
Pq
not exceed the budget. well-known Min-Cut problem in graphs. This is in
Thetechnical challenge arises duetoquery dependencies. contrasttothepoorperformanceofnaturalgreedyal-
When one bids on a keyphrase for query q, as a result of gorithmsforthisproblem. Wealsostudythebudgeted
a broad match, it may apply to query q′ as well. The ad- variant of the problem, and propose a novel strategy
vertiserhasdifferentvaluesv(q) andv(q′) onthesebecause usingtwodistinctbudgetedadcampaignthatgetsthe
usersfor q and q′ differon theirintentionsandtherefore on optimal profit. We do so by studying the structure of
their respective values to the advertiser. So, the advertiser the basic feasible solutions of a corresponding linear
may make good profit on q and may wish to bid on that programming formulation of the problem.
query,butisthenforced toimplicitly bidonq′ aswell, and
For keyword bidding, we show that even limited in-
may even make negative profit on q′! Under what circum-
stances are NP-Hard to not only optimize, but even
stances is it now desirable for theadvertiser to bid for q?
to approximate; to deal with this hardness result, we
Note that query dependence is a fundamental aspect of
present a constant-factor approximation when adver-
sponsoredsearchsinceadvertiserscanrealisticallyonlychoose
tisers profit following an optimal bid is considerably
and strategize on a small set of keyphrases because of the
greaterthanhercost. Thisresultisbasedonapplying
effort involved, and have to typically rely on the search en-
a randomized rounding method on the optimal frac-
gine to carefully apply their strategy to variants of their
tional solutions of the linear programming relaxation
keyphrases. But beyond that, even an ad campaign that
of the problem.
is willing to exert a lot of effort and use a large number of
keyphrases or relies on a search engine to provide rich bid- Theserepresentthefirstknowntheoreticalresultsforthe
dinglanguages [11]willstill findit impossible toincludeall problem of bid optimization in presence of broad matches,
search variations of the keyphrases as exact matches, and a problem advertisers face now since this feature is offered
mustnecessarilyrelyonbroadmatchforthevariationsthat by the major search engines. Prior research in bid opti-
mization for advertisers [4, 6, 16] primarily focused on de-
1These match types may be further modified by ensuring termining suitable bids for exact match typesand does not
thattheadbenotshownonoccurrenceofcertainkeywords
studythequerydependenceandimplicitbids; [9,14]stud-
in the query; this feature (called “negative” in MSN and
iedtheproblemofmaximizingthenumberofclicks,andnot
Google or ”excluded” by Yahoo) and other targeting crite-
ria associated with keyphrasecampaigns do not change the the profit which is the more standard metric. At the tech-
discussion and the results here. nical core, our challenge is to tradeoff positive profit from
bidding on a keyphrase that applies to one query q against intersections among broad matches for different keywords,
possibly a negative profit from the implied bids of broad we first consider a bidding language in which an advertiser
match on queries q′. This query dependence is a novel fea- can specify a bid for every query q but only as a broad-
ture in sponsored search auctions, not explicitly studied in match. We refer to this language as thequery language.
prior literature, and our results for this problem may have To allow the most accurate description of an advertisers
applications beyond,in thegeneral auction theory area. value per query, the ultimate way is to let the advertiser
Finally, we report experimental results on a small family specify all possible queries with exact or broad match, and
of instancesofthebidoptimizations problem,andcompute amonetarybidforeachofthem. Ifanadvertiserisallowed
the optimal bidding using the integer linear programming to bid on each type of query as an exact match as well as
formulation. Our main observation in these experiments is broad match, she can decide for each query independent of
that by considering only the broad match, we do not lose theotherqueries,andthecomplexityofthebiddingproblem
muchin themaximumprofit of thesolution. This supports is not captured in such biddinglanguage.
our hope that under reasonable circumstances (similar to To capture the complexity of the optimal bidding prob-
theonesinourexperiments),consideringonlybroadmatch lem and the fact that advertisers may only bid on a subset
is effective, and in turn, that would enable advertisers to of queries, we study the keyword language that allows ad-
focus on campaigns with small lists of keyphrases. vertisers to place a bid only on (single) keywords or short
phrases. More precisely, in the keyword language, we as-
2. MODEL sume that advertisers are allowed to bid only on a subset
S ⊂Q of queries.
Weconsidertheoptimizationproblemsthatanadvertiser
A further improvement of this language would allow the
faces while bidding in an auction for queries with a broad
advertiser to specify, besides a value bid for s∈S, whether
match feature.
s is to bematched exactly or broadly.
Abidb∈R|Q|insomebiddinglanguageisassociatedwith
+
The Advertiser. We consider a single advertiser who is
asetof‘winningqueries’denotedbyϕ(b)={q∈Q|b(q)≥
interested in showing her ad to users after they search for
c(q)}. A subset T of queries which is a winning set of some
queries from a set Q. The advertiser has some utility from bid b is referred to as a feasible winning set. The utility
having a user click on her ad. In reality, clicks associated associated with a winning set T is
with different queries may bring have utility to the adver-
tiser; The advertiser has a value of v(q) units of monetary u(T)= (v(q)−c(q))n(q),
X
value associated with a ‘click’ that follows a query q∈Q. q∈T
We assume a posted price model where prices are posted
where v(·) and n(·) are advertiser specific.
and the search volume of every query as well as its click
A feasible winning set with optimal utility is referred to
throughrate(i.e.,theprobabilitythatuserswouldclickher
as an optimal winning set.
ad) are known to the advertiser. Namely, every query q is
associated with a pair of parameters, known to the adver-
The Auction. For every query, the auctioneer should de-
tiser, (c(q),n(q)), where c(q) is the per click cost of q, and
cide the bid of every advertiser. This decision is easy for
n(q)istheexpectednumberofclicksthatwouldresultfrom
queries on which the advertiser bids explicitly (as an exact
winningq(theexpectednumberofclickscanbedetermined
match). However, for the queries that the advertiser has
fromthesearchvolumeofqandtheadvertiser’sspecificclick
not bid directly, but only through a broad match frame-
through rate for q).
work,theauctioneershouldcomputeanappropriatebidfor
Thus, when an advertiser wins a query q, her overall
theadvertiser to participate in the auction.
profit 2 from winning, denoted w(q) is
Anaturalwayforsettingsuchavalueistoaggregate the
w(q)=(v(q)−c(q))n(q) . bid values of all the phrases matched by the query. While
thereareseveralchoicesfortheaggregation method,inthis
Notethatalthougheachqueryhasapositivevalue,winning
paper,weconsiderthemaxaggregation operator—whena
it may result an overall negative profit.
queryq matchesphrases w ,...,w from theadvertiserlist
1 k
of phrases, its bid is interpreted as b(q)=maxib(wi).
Biddinglanguages. Abiddinglanguageisawayforanad- Wecannowstateformally thebidoptimization problem.
vertisertospecifyhervalueorwillingnesstopayforqueries. Givenadvertiser’sspecificdata(AsetQ,valueforqueriesv,
Eventually,theauctioneerneedstohaveabidforeverypos- search volume and click through rates n(·) ) and a bidding
sible query 3. The choice of a bidding language is critical language L, an optimal bid b∗, is a feasible bid in the lan-
for the auction mechanism. At the one extreme, it may guageLthatmaximizestheadvertisers’utilityfromwinning
be infeasible to allow an advertiser to specify explicitly her a set ϕ(b) of queries. Formally,
value for every possible query. On the other hand, a lan-
guage that is too restrictive would not allow an advertiser
b∗ ∈argmax {u(ϕ(b))}. (2.1)
to communicate her preferences properly. b∈L
In order to study the complexity of the optimal bidding
inthebroadmatchframeworkwhiletakingintoaccountthe Querydependencies. Wesaythataqueryqdependsona
2In this paper, we use terms utility and profit inter- queryq′ ifwinningqueryq′ implieswinningqueryq. Inthe
broadmatchauctioninwhichthebidinterpretationstrategy
changably.
3A bid of 0 for a query may be regarded as the default in isdoneusingthemaxoperator,thishappensifqmatchesq′
a case where the advertiser is not explicitly interested in a broadly,and its cost c(q) is less than that of c(q′). In other
query q and norin queries that q match broadly. words, if a bid b wins q′, it must be that b(q′) ≥ c(q′), but
the interpreted bid for q is then at least b(q′) ≥ c(q) since winning the set of n−1 queries made of pairs of keywords
c(q′)≥c(q), hence the bid b must be winning q as well. As in which this keyword appears. The value of a keyword
aresult,thecoststructureincursasetofpairs(q′,q)where is set to $2; The value of a each pair is set to 1−1.5/n.
the first entry of each pair q′ ∈ S is a valid phrase in the So,everykeywordattainsapositiveutilityof$1,andevery
biddinglanguageandthesecondentryisavalidqueryinthe pair causes a loss of $1.5/n. Initially, Max-Margin Greedy’s
set ofqueriesQsuchthatwinningqueryq′ implieswinning bid is empty. At this point Max-Margin Greedy is stuck
query q. This set of pairs is denoted byC and formally: — every single query it adds to the winning set results in
a negative overall utility. Thus, this instance, Max-Margin
Greedywouldyield0utility. Anoptimalsolutionwinsallthe
C ={(q′,q)|q′ ∈S,q∈Q,q matches q′ broadly ,c(q′)≥c(q)}. queriesandhasautilityofn×(2−1)− n (1−1.5−1)= n−3.
`2´ n 4
Moreover, we define D(q) ={q′|(q′,q) ∈C}, and N(q) = One can explore other variants of greedy algorithms for
{q′|(q,q′)∈C}. this problem. For example, a natural greedy algorithm is
Max-Rate Greedy algorithm: Initially setthewinningsetto
Budget-constrained Ad Campaigns. A variant of the the empty set, and then iteratively, add a bid on a query
optimal bidding problem in the broad match framework is with the highest ratio of marginal profit over the marginal
to find a set of queries to bid on that maximizes the total cost, or the query with the highest ratio of marginal value
valueofthequerieswon bytheadvertisersubjecttoabud- over marginal cost. We note that all these iterative greedy
getconstraint,i.e,ourgoalistobidonasubsetT ofqueries algorithms pefrom poorly for the above example. Even a
tomaximize v(q)n(q)subjecttothebudgetconstraint significant look-ahead will not resolve thisbad example.
c(q)n(qP)q≤∈TB. Tohandlesuchabudgetconstraint,we WeturntothenextalgorithmOptBid1 forcomputingan
aPssqu∈mTe that one can run a budget-constrained ad campaign optimal winning set. OptBid1 is a solution to thefollowing
by bidding on a subset T of keywords and setting a budget integer linear program:
B. Assuming B′ = c(q)n(q), there are two possibili-
Pq∈T
ties in this budget-constrained ad campaign: (i) If B′ ≤B, ILP : max X Xqiw(qi)
the auction is run in a normal way and the value from this
qi∈Q
adcampaignfortheadvertiseris v(q)n(q),(ii)Onthe
otherhand,ifB′ >B,weassumePthq∈aTt thequeriesarriveat For every pair (qj,qi)∈C : Xqi −Xqj ≥0
the same rate and as a result, for each query, we get B′ ∀qi∈Q : Xqi ∈{0,1} (3.1)
B
fraction of the value of an ad campaign without the bud- Foreveryqueryq,anintegralvariableXq isa0-1variable
getconstraint. Inotherwords,thevaluethattheadvertiser which is equal to 1 if and only if q belongs to the winning
gets is v(q)n(q)B′. We can also interpret the above set of queries. In order to solve the above ILP, we relax it
Pq∈T B
assumption by a throttling method in which, in order to to a linear program where instead of integer 0-1 variables,
cope with the budget constraint, at each step, we let the wehavefractionalXq variableswithvaluesbetween0and1
advertiser participate in the auction with probability B′. (0≤Xq ≤1). Here, we observe that the integrality gap of
B
thislinearprogrammingrelaxationis1,i.e.,foranyinstance
3. BIDDINGINTHE QUERYLANGUAGE of this linear program, there exists an optimal solution X∗
inwhichallthevaluesareintegerX∗ ∈{0,1}forallq∈Q.
q
In this section, we study the query language that allows
placingabidoneveryquery. Weobservethatinthequery Lemma 3.2. The integrality gap of the linear program-
language, the task of computing an optimal bid is equiva- ming relaxation of the ILP 3.1 is one.
lenttothatofcomputinganoptimalwinningset: Givenan
optimal feasible set T set a bid b(q) =c(q) for every query Proof. The lemma follows from the fact the the con-
q∈T with positive weight and b(q)=0 otherwise. straintmatrixoftheLPrelaxationofILP3.1istotallyuni-
modular4. A sufficient condition for a matrix to be totally
Lemma 3.1. Abidb derivedfrom anoptimalwinningset uni-modular is that every row has either two non-zero en-
T, as described above, is an optimal bid. tries, one is 1 and the other −1, or a single non zero entry
Proof. By construction, the bid b wins all the queries with value 1 or −1. An integer program whose constraint
matrix is totally uni-modular and whose right hand side is
with positive weight from T, and every other query must
integer can be solved by linear programming since all its
belong to T (otherwise T would not be feasible).
basic feasible solutions are integer (see [15] pp. 316).
We therefore consider algorithms for computing an opti-
mal feasible winning set. First, we consider a greedy al- The above lemma implies the following polynomial-time
gorithm, denoted by Max-Margin Greedy. Initially, Max- algorithm OptBid1 for optimal bidding in the query lan-
Margin Greedy sets the winning set to be empty. Then, it- guage: computeabasicfeasiblesolutionX∗oftheLPrelax-
eratively,itaddsabidonaquerywiththehighestmarginal ation of ILP 3.1, and find a biddingstrategy corresponding
benefittothewinningsetutility. Unfortunately,Max-Margin to thewinning set of X∗, i.e., {q∈Q|Xq∗ =1}.
Greedy fails to compute an optimal winning set due to the TherunningtimeofAlgorithmOptBid1 isthatofsolving
following example. a linear program with Ω(|Q|2) constraints, which although
polynomial in |Q|, might be inefficient. Next, we present a
faster algorithm, OptBid2.
Example. Consider a set Q of queries which contains n
keywordsandanother`n2´queries,eachofwhichisapairof 4AmatrixAistotallyuni-modularifeverysquaresubmatrix
keywords. The cost of each query is set to $1. Hence, the ofitisuni-modular,i.e.,everysubmatrixhasadeterminant
query dependencies is such that winning a keyword implies of 0, -1or +1.
“a b ” ∞ set Zq := w(q)−Pq′|(q,q′)∈CYq,q′. It is straightforward to
“a” verify that this is a feasible solution of DUAL with value
8 ∞ Pq∈Q+w(q)−c.
Now,observethatT isafeasiblesetinthequerylanguage.
“a b c” 11 For every pair (q,q′) ∈ C, we have that if q ∈ T then also
9 ∞ “b” q′ ∈ T. Otherwise, the edge (q,q′), with weight ∞, would
s 8 t be part of thecut. Thus, thevalue of themin cut is
5
“e f g ” c= |w(q)|+ w(q).
∞ 7 X X
4 “e f ” q∈Q−∩T q∈Q+\T
∞
“e f h ” and therefore,
u(T) = Xw(q)= X wq− X w(q)+ X w(q)
Q - Q+ q∈T q∈Q+ q∈Q+\T q∈Q−∩T
= X wq−c.
Figure 1: An example of running algorithm Opt- q∈Q+
Bid2on the setofqueries (withthe followingprofit:
We already found a feasible solution for the dual of ILP,
{(a,11),(b,8),(ab,−8),(abc,−9),(ef,7),(efg,−5),(efh,−4)}),
withthesamevalue. Wethereforeconclude,usingtheweak
and dependency graph as illustrated. ObtBid2 will
duality theorem, that T is an optimal solution of ILP.
choose the winning set {a,b,ab,abc}. The optimal
bid is {(a,11),(b,8)}.
4. BIDDINGINTHEKEYWORDLANGUAGE
Inthissection,westudyoptimalbiddingforthekeyword
Forthepurposeof presentingAlgorithm OptBid2,wede- language, wheretheadvertiserisrestricted tobidonasub-
fine a weighted flow graph G = (V,E), derived from the set of (possibly short) queries S ⊂Q.
input. ThevertexsetofGisV ={s,t}∪Q+∪Q−,wheres Notethatinthecasethatallquerieshavepositiveutility,
is a source node, t is a target node and Q+ and Q− are the optimal bid is trivial by simply placing a high bid for
the sets of queries with positive/non-positive weights re- everyqueryinS. Inaddition,findingtheoptimalbidwhen
spectively, i.e., Q+ ≡ {q | w(q) > 0}. The source vertex allqueriesareassociated with anegativeutilityistrivial(a
sisconnectedtoeachvertexq∈Q− withanedgeofweight bidof$0foreveryphraseinS isoptimal). Moreover,inthe
|w(q)|=|(v(q)−c(q))n(q)|. Thetargetvertextisconnected case of uniform value from every query, the optimal bid is
with each vertexp∈Q+ with an edge of weight w(p). Two easy—auniformbidequaltotheuniformvalueguarantees
vertices q ∈ Q−,p ∈ Q+ are connected with an edge of winning every query with positive weight and losing every
weight ∞ if and only if (p,q)∈C. query with negative weight, which is of course optimal. In
realisticsettings,somequerieshavepositiveutilityandsome
Algorithm OptBid2 havenegativeutility. Inthiscasetheproblemoffindingthe
1. Computeamin-cut ofG. LetS,T bethetwosides of optimalbidbecomesintractable. Moreprecisely,asweshow
thecut. now, evenwhenthesetof queriesQismadeupfrom single
keywordsandpairsofkeywords,thisproblembecomeshard
2. Assume,withoutlossofgenerality,thatt∈T. Return to approximate within a factor of |S|1−ǫ, for every ǫ>0:
T\{t},thatis,thesetofqueriesthatareonthesame
side of thecut as t is an optimal winning set. Theorem 4.1. Inthekeywordlanguagebroadmatchframe-
work, it is NP-hard to approximate the optimal value of the
The running time of Algorithm OptBid2 is that of min-
optimal bidding problem within a factor of |S|1−ǫ, for any
cut, i.e., O(|Q|3) [18].
ǫ>0.
Theorem 3.3. Algorithm OptBid2findsanoptimalwin- Proof. We give a factor-preserving reduction to the in-
ning set.
dependentsetproblem. GivenagraphGwithnnodes,and
Proof. We show T is an optimal winning set using the medges,weconstructthefollowinginstanceofourproblem:
dual program of ILP. put a singleton keyword for each node v of G with weight
wv =−(deg(v)−1),and putaqueryconsisting of apair of
DUAL : min X Zq keywords corresponding to each edge e of G with weight 1.
q∈Q+ The maximum value we can get from picking a keyword is
∀q∈Q+ : X Yq,q′+Zq ≥w(q) 1in,atnhde owuetpguett.thItiscvaanlubeeifseaellnotfhiatstntheieghobptoirmsudmonsootluatpiopneairs
q′:(q,q′)∈C
anindependentsetofnodes(sinceotherwise, wegetzeroor
∀q′ ∈Q− : X Yq,q′ ≤−wq′ (Noticethat wq′ ≤0) negativefromapickednode),andasaresult,themaximum
q:(q,q′)∈C value is thesame as thesize of theindependent set.
∀(q,q′)∈C : Yq,q′ ≥0 4.1 A Constant-FactorApproximation
∀q∈Q+ : Zq ≥0 Inthissection,inlightoftheabovehardnessresult,wede-
Letf =(fe)e∈E beamaximumflowinG,withvaluec. For signaconstant-factorapproximationalgorithmforaspecial
every (q,q′) ∈ C, set Yq,q′ := fq,q′ and for every q ∈ Q+, caseoftheoptimalbiddingprobleminthekeywordlanguage
in whichthecost part oftheoptimal solution islessthan 1 s ∈ S, such that s ∈ D(p) (or equivalently (s,q) ∈ C). It
c
of thevalue part of theoptimal solution, for some constant is not hard to see that the above rounded integral solution
c > 1. Recall that D(q) = {q′|q′ ∈ S,(q′,q) ∈ C}. Our correspond to a feasible bidding strategy. In particular, we
algorithm is constant-factor approximation if for any query can implement this bidding strategy by putting an exact
q ∈ Q\S, |D(q)| is less than a constant c′. We present match bid of b(s)=c(s) for any s∈S if R′ =1 (i.e., with
s
our result for the case that each query q ∈ Q can be in probability Rs) and then by putting a broad match bid of
the broad-match set of at most two queries q ,q ∈ S, i.e., b(s) = p for query s ∈ S if W′p = 1 (i.e., with probability
1 2 s
|D(q)| ≤ 2. However, our result can be extended to more Wp(1−ǫ)).
s
general settings in which query q ∈ Q can be the broad A query s ∈ S is selected if bid b(s) for this query is
match for a constant number of queries q1,...,qc′ ∈S (for at least c(s). As a result, a query s ∈ S is selected with
a constant c′). probability(1−ǫ)Zsc(s)+Rs. Moreover,foraqueryq∈Q\S
Basedontheabovediscussion,weassumethat|D(q)|≤2 for which (s,q)∈C and (r,q)∈C, query q is selected if the
for any query q ∈ Q. Note that the hardness result of the bid for either of the queries s or r is at least c(q), i.e., with
keywordlanguageholdsevenforinstancesinwhich|D(q)|≤ probability
2 for all queries q ∈ Q. Also, let E = {c(q)|q ∈ Q}. Our
algorithm is based on a linear programming relaxation of
theoptimalbiddingproblemforthekeywordlanguage. The Pr[ query q is selected] =
integer linear program is as follows: 1−(1−(1−ǫ)Zc(q))(1−(1−ǫ)Zc(q)) =
s r
ILP-Approx max Ps∈S(Zsc(s)+Rs)ws (1−ǫ)(Zsc(q)+Zrc(q))−(1−ǫ)2Zrc(q)Zsc(q).
+Pq∈Q\SYqwq Therefore,theexpectedutilityofthesolutionafterimple-
∀q∈Q\S,(s,q)∈C,(r,q)∈C Yq ≤Zsc(q)+Zrc(q) mentingtheintegralsolution(V′,W′,Z′,Y′)(orbiddingas
desribed above) is:
∀q∈Q\S,(s,q)∈C Yq ≥Zsc(q)
∀s∀∈sS∈,pS,,pp′∈∈EE ZspZ=spP+tR∈Es,≤t≤p1Wst Xs∈S((1−ǫ)Zc(s)+Rs)ws+
∀q∈Q\S Yq ∈{0,1} X “(1−ǫ)(Zsc(q)+Zrc(q))−(1−ǫ)2Zsc(q)Zrc(q)”wq
q∈Q\S;r,s∈D(q)
∀s∈S Rs∈{0,1}
∀s∈S,p∈E Wp,Zp ∈{0,1} Next, we derive a lower bound and an upper bound on
s s
theprobability thatthebidgenerated asabove,wins query
(4.1)
q. Wewill show that
Where thevariables correspond to thefollowing: 1
Yq(1−ǫ)(1− (1−ǫ))≤Pr[q is selected]≤(1−ǫ)2Yq.
• Wp foranys∈S istheindicatorvariablecorrespond- 2
s (4.2)
ing to thebid of p on querys (as a broad match),
Considerthefollowingsetofinequalitiesthatholdforevery
• Zp for any s∈S is theindicator variable correspond- query q that dependson queries s,r∈D(q):
s
ing to the bid of at most p on query s (as a broad
match),
Zsc(q)Zrc(q) ≤qZsc(q)Zrc(q) ≤ (4.3)
• Rinsgftoortahneyesxa∈ctSmisattchhebiniddiocnatqoruevrayrisa,ble correspond- Zsc(q)+2 Zrc(q) ≤Yq ≤Zsc(q)+Zrc(q),
• Yspqofnodrinagnytoqw∈innQin\gSquiesrtyhqe(iansdaicaretosurltvaorfiabbidledicnogrroen- Thefirstinequalityfollowstheconstraints0≤Zsc(q),Zrc(q) ≤
1andthesecondinequalityisthearithmeticgeometricmean
queries in S).
inequality. The third inequality follows the summation of
We relax the integer 0-1 variables in this integer linear the inequalities Zrc(q) ≤ Yq and Zsc(q) ≤ Yq and the last
program to fractional variables between zero and one, and inequality appears as a constraint in theLP.
then compute an optimal fractional solution for this LP re- The left hand side in Inequality4.2 follows since
laxation. Thenweroundthisfractionalsolutiontoconstruct
a feasible (integral) biddingstrategy.
(1−ǫ)(Zc(q)+Zc(q))−(1−ǫ)Zc(q)(1−ǫ)Zc(q) ≤
Rounding to an Integral Solution. Given a fractional s r s r
solution (V,Z,W,Y) to the LP, we round it to an integral (1−ǫ)2Yq−(1−ǫ)2Zsc(q)Zrc(q) ≤(1−ǫ)2Yq
solution (V′,W′,Z′,Y′) as follows: for every query s ∈ S,
we set V′ =1 with probability 1, and V′ =0 otherwise. If And theright hand side follows
s s
Vs′ =1,wesetWs′p=0forallp∈E. Otherwise,foreachs∈ (1−ǫ)(Zc(q)+Zc(q))−(1−ǫ)2Zc(q)Zc(q) ≥
S,forallp∗ ∈E,wechoosep∗withprobabilityproportional s r s r
btoeWdespt∗er(m1−inǫe)d(lfaotrera)naanpdprsoeptrWiat′pe∗sm=a1l,laconndstfoarntanǫythpa6=twpi∗ll, (1−ǫ)(Zsc(q)+Zrc(q))−(1−ǫ)221(Zsc(q)+Zrc(q))≥
s 1
pwe∈sEetaWnds′ps=∈0S., wAeftseertsZet′ptin=g all W′ vaWria′tb.leFs,infaolrlye,afcohr (1−ǫ)(Zsc(q)+Zrc(q))(1− 2(1−ǫ))≥
s Pt∈E,t≤p s 1
any q ∈ Q\S, Yq′ = 1 if and only if Zs′c(q) = 1 for some Yq(1−ǫ)(1− 2(1−ǫ)).
In the summation that describes the overall utility from (c,M) (c`,1) (1,0) (1,0) (1,0)
the queries, the probability for selecting query q, is multi-
T
1
plied by both the value and the cost of q. Using Inequality
4.3,weget alowerboundontheexpectedvaluefrom q and
T
an upperbound on the expected cost of q. 2
Letusdenotetheoptimalutilityanybiddingstrategycan
achieve by U∗ = V∗−C∗, where V∗ and C∗ are the value T
3
and cost part of the objective utility function, respectively.
LetU∗ =UE∗+UB∗ whereUE∗ istheutilityresultingfromthe T4
exact match bidding, and U∗ is the utility from the broad
B
match bidding. Similarly we define U∗ = V∗ −C∗, and
E E E
U∗ =V∗−C∗ (whereV andC correspondtothevalueand L R n
B B B
thecostofeachpartofthesolution). Knowingthatforeach
queryw(q)=v(q)n(q)−c(q)n(q),theexpectedutilityofthe Figure 2: An illustration of a bidding problem with
abovealgorithmbasedonrandomizedroundingoftheLPis a large integrality gap (k=3).
at least:
1
(1−ǫ)(1− 2(1−ǫ))VB∗−(2−2ǫ)CB∗ +VE∗−CE∗ ≥ more important difference is the extra budget constraint.
Because of this extra linear constraint, the following linear
1
(1−ǫ)(1− (1−ǫ))V∗−max(1,2−2ǫ)C∗. programmingrelaxationofthisIPisnottotallyunimodular
2
anymore.
Lemma 4.2. By setting ǫ = 0 or ǫ = 1/2 in the above
algorithm,wegetthatUALG ≥ 1V∗−2C∗ orUALG≥ 3V∗−
C∗, respectively. 2 8 Budgeted−LP : max X Xqiv(qi)n(qi)
qi∈Q
Given the abovelemma, we conclude thefollowing: For every pair (qj,qi)∈C : Xqi −Xqj ≥0
Theorem 4.3. Forinstancesoftheoptimalbiddingprob- X Xqic(qi)n(qi)≤B
lem in which C∗ ≤ V∗ and each query depend on at most 2 qi∈Q
4
other queries (i.e.,|D(q)|≤2 foreach q∈Q\S), the above ∀qi ∈Q : 0≤Xqi ≤1 (5.2)
randomized algorithm is a 1-approximation algorithm.
6 In fact, we can show that the integrality gap of this LP
canbeverylarge, andthusonecannotroundthefractional
In order to extend the above result for the more general
caseinwhich|D(q)|≤c′foraconstantc′,weshouldaddthe solution of this linear programming relaxation to a good
integral solution (as we did for IP 3.1).
tinheeqaubaolivteyrYeqsu≥ltPtost∈hDe(qf)oZlloscw(qi)n,ga:nFdotrhaennyocnoencsatanngtecn′,ertahleizree Lemma 5.1. The integrality gap of linear programming
existtwoconstantscandc′suchthatforinstancesoftheop- relaxation 5.2 can be arbitrarily large.
timalbiddingprobleminwhichC∗ ≤ Vc∗ and|D(q)|≤c′for Proof. Consider the following instance of the bidding
eachq∈Q\S,thereexistsaconstant-factorapproximation problemwithabudgetconstraintinthequerylanguage: Let
algorithm. thesetofqueriesQ=R∪L∪T whereR={r ,r ,...,r },
1 2 k+1
L = {l ,l ,...,l }, and T = T ∪T ∪···∪T , where
1 2 k+1 1 2 k+1
5. BUDGETCONSTRAINTS Ti ={ti1,ti2,...,tin}. Foranyqueryq∈L, wehavec(q)=
In this section, we study the problem with an additional c and v(q)=M, and for q∈R, c(q)=c′ and v(q)=1. We
budget constraint, i.e, we have a budget limit B and the assume that c>>c′ >>n>>1. For any q ∈T, v(q)=0,
total cost of our bidding strategy should not exceed this and c(q)=1. Also, let thequery dependencystructurebe:
limit(B). Our goal is to maximize the total value subject
C ={(ri,lj)|1≤i,j ≤k+1,i6=j}∪
to this budget constraint. More formally, the problem is as
follows: {(ri,ti1|1≤i≤k+1}∪
{(tij,ti(j+1)|1≤i≤k+1,1≤j ≤n−1}.
Budgeted−IPmax max X Xqiv(qi)n(qi) deIfitniitsioenass.yWtoeaclhseocksetthtahtebCumdgaettchBes=tche+ckocs′t+ann.dInqutehriys
qi∈Q
instanceofthebiddingproblem,theoptimalintegersolution
∀(qj,qi)∈C : Xqi −Xqj ≥0 canpickatmostonenodefromthesetL(giventhebudget
X Xqic(qi)n(qi)≤B constraint of B and thefact that c>>c′ >>n>>1), and
qi∈Q thustheoptimal integer solution is M+k.
∀qi∈Q : Xqi ∈{0,1} (5.1) x=On the(c+otkhc′e+rnh)and,.aTshaevfraalcuteioonfatlhsisolfuraticotnio,nwalescoaluntisoent
Similar to IP 3.1, for every query q, an integral variable ((k+1)c+(k+1)c′+n)
Xq indicates whether q belongs to the set of queries won is[(k+1)M+k+1]((k+1()cc++k(ck′++1n))c′+n) whichisapproximately
by an theoptimal solution or not. There are two difference (k+1)M. As a result, the ratio between the optimal frac-
betweentheaboveIPandtheIPforquerylanguagewithout tion solution and the integral solution can be as large as k.
the budget constraint: One difference is in the objective This proves that the integrality gap of the LPis arbitrarily
function in which instead of w(qi), we have v(qi), and the large.
In fact, not only the LP for the budgeted problem is not has |V(Gi)|−1 edges. Now, we observe that for any good
integral, but also the optimal bidding problem with a bud- component Gi, thereexists exactly one equation in P1. For
getconstraintisanNP-hardproblemeveninthequerylan- anygoodcomponentGi,thereare|V(Gi)|−1equationscor-
guage. The NP-hardness follows from the fact that this responding to edges of Gi, and at least one equation in P1.
problem is harder than the knapsack problem. In fact, the If there two such equations in P1 for nodes of Gi, then the
knapsackproblem isaspecial caseof thisproblem in which union of these two equations and equations corresponding
thesetof queriesareonlythekeywordsandC =∅. Despite to edges of Gi form |V(Gi)|+1 equations in P all defined
the large integrality gap of the above LP and NP-hardness ononlyvariableonV(Gi). Asaresult,theseequationscan-
of the problem, in the following we show how one can use not beindependent,which contradicts with thefact that P
a certain set of optimal solutions of this LP to implement is a set of independent equations. As a result, each good
twobudgetconstrainedadcampaignsforanadauctionwith component Gi corresponds to exactly |V(Gi)| equations in
broad-matchthatachievestheoptimalfractionalsolutionof P.
theaboveLP.Weshowthisfactbyprovingtheexistenceof LetY bethesetofvariablesX∗ ingood connected com-
qi
optimalsolutionsfortheLPwithcertainstructuralproper- ponents of G(X∗,P ). Thus, Y corresponds to a set of all
2
ties. The key structural lemma is thefollowing: integral variables X∗. The above discussion implies that
qi
there are exactly |Y| equations in P characterizing all vari-
Lemma 5.2. TheLinearProgram 5.2hasatleastoneop- ablesinY. Asaresult,thereare|P|−|Y|equationsuniquely
timalsolutionX∗ forwhichthereexistsavalueX suchthat: identifyingallthe(fractional)variablesinQ\Y. Notingthat
foreachqueryq,Xq∗ ∈{0,1,X}. Moreover, thisoptimalso- |P|=|Q|,weconcludethat |Q\Y|equationsuniquelyiden-
lution can be found in polynomial time. tify all the (fractional) variables in Q\Y, and none of these
Proof. We prove this fact by showing that an optimal equations are of the form Xq∗i = 0 or Xq∗i = 1. At most
one of these equations correspond to a tight budget con-
basic feasible solution of the LP satisfies the desired prop-
erties. From standard linear programming theory,we know straint (Pqi∈QXq∗ic(qi)n(qi) = B), and thus we have at
least|Q\Y|−1equationsoftheformX∗ =X∗ onvariables
that such a basic solution exists, and can be found in poly- qi qj
nomial time. Consider an optimal basic feasible solution Q\Y inbadcomponents. SincethereisnocycleingraphG
X∗ of the LP 5.2. Since the LP has |Q| variables, a basic and verticesin Q\Y haveat least |Q\Y|−1edges amongst
feasible solution can be uniquely characterized by |Q| tight them,theyshouldallbelongtothesameconnectedcompo-
inequalities. In other words, there is a set P of |Q| inde- nent(i.e,theonlyonebadconnectedcomponent),andthus
pendent linear equations among the linear constraints that have the same value Xq∗i =X. This completes the proof of
characterize X∗. Let P ⊂ P be a set of these |Q| linear thelemma.
1
equations of the form X∗ = 0 or X∗ = 1. Each linear
equation in P correspondqis to an integqrial variable X∗. Let UsingLemma5.2,wecanshowthatinthequerylanguage
1 qi model, one can implement the optimal fractional solution
S1 be the set of queries qi corresponding to these integral
variables X∗. Also let P ⊂ P be a set of these |Q| linear using two budget-constrained ad campaigns. To formally
equations oqfithe form X∗2 = X∗ . We construct a graph show this, we assume that queries arrive at the same rate,
qi qj andbyputtingabudgetconstraintonanadcampaign(that
G(X∗,P )whosevertexsetV(G)isthesetofvariablesX∗
2 qi includes a set of queries), the budget on all queries is con-
for all queries qi ∈Q as follows: we put an edge between a sumed at the same rate untilit gets used completely.
vertexX∗ toavertexX∗ ifandonlyifX∗ =X∗ isalin-
qi qj qi qj
earequationinP2. Thus,ifthereispathbetweentwonodes Theorem 5.4. In the query language model, there ex-
X∗ and X∗ in G, we have X∗ = X∗ . Let the connected
qi qj qi qj ists a polynomial-time algorithm that computes two budget-
componentsofGbeG1,G2,...,Gt. Asaresult,foranytwo constrained ad campaigns that implement an optimal bid-
variablesXq∗i andXq∗j inthesameconnectedcomponentGp dingstrategyachievingthemaximumvaluefortheadvertiser
for 1 ≤ p ≤ t, we have X∗ = X∗ (since there is a path
qi qj given a budget constraint.
between any two nodesin thesame connected component.)
WesaythataconnectedcomponentGp isabadcomponent Proof. The polynomial-time algorithm is as follows:
if none of the nodes in Gp are in S1. Otherwise, we say 1. SolveLP5.2andcomputeanoptimalsolutionX∗such
tthheatnGumpbisearogfoobdadcocmompopnoennet.ntIsnisthaetfmololoswtionnge,,waendshtohwistwhailtl that Xq∗i ∈{0,1,X} for all queries qi.
prove the lemma. In order to prove this claim, we need to 2. Let S and S be the sets of queries with the cor-
0 1
provethe following lemma: responding integral variables X∗ = 0 and X∗ = 1,
qi qi
respectively.
Lemma 5.3. Graph Gasdefined above does nothaveany
cycle. 3. Let B1 =Pqi∈S1Xq∗ic(qi)n(qi).
Proof. Forcontradiction,assumethatthereexistsacy- 4. Runthe following two ad campaigns:
cle X∗, X∗, ···, X∗ in graph G. Thus, all linear equations
1 2 v
X∗ = X∗ for 1 ≤ i ≤ v −1, and X∗ = X∗ are in P. (a) A campaign with budget B1 on queries in S1.
i i+1 v 1
But these v equations are not independent, and this cycle (b) A campaign with budget B −B on queries in
1
contradicts the fact that P is a set of independent linear Q\(S ∪S ).
0 1
equations.
Lemma5.2showsthatthefirststepofthisalgorithmcan
Lemma5.3provesthatgraphGisaforest, andthuscon- be done in polynomial time. To show the correctness of
nected components Gi for 1≤i≤p are all trees, and each the algorithm, note that assuming that queries arrive at
the same rate, and based on the definition of the budget- and broad match types. Our simulation is composed of 30
constrained adcampaign, thevaluefrom thesetwoadcam- keywords, where we consider all pairs of keywords as the
paigns is: set of possible queries. Therefore, while there are “only”
30keywordsanadvertiserwhoisinterestedinmanagingall
B−B possiblequerieswillhave435keywordstoconsider,whichis
X v(q)n(q)+ B′ 1 X v(q)n(q), tedious for small advertisers. Most advertisers will prefer a
q∈S1 q∈Q\(S0∪S1) campaignwithasmallsetofkeywordswhichtheycaneasily
where B′ = c(q)n(q). Note that X∗ = 1 for trackandevaluate,whichinoursimulationisrepresentedby
qi ∈S1,wehPaveq∈BQ1\(=S0P∪Sq1i)∈S1Xq∗ic(qi)n(qi)=Pqqii∈S1c(qi)n(qi), thhaevecotrhee3s0amkeeycwoosrtd.sT.hTehneestevtuaplueisovfearyqsuiemryplies.dAeltlerqmueinrieeds
and thusB + Xc(q)n(q)=B, therefore,
1 Pq∈Q\(S0∪S1) as follows. The value of a keyword is drawn from a stan-
B−B B−B dardnormaldistribution. Thenetvalueofaqueryiseither
X = 1 = 1. (1) the average net value of its keywords, or (2) the max
c(q)n(q) B′
Pq∈Q\(S0∪S1) value among its keywords, or (3) the min value among its
Thus, thevalue of the optimal solution is: keywords;theprecisenetvalueofaqueryisdecidedaccord-
ing to 1−3 uniformly at random. This setting is loosely
motivated by the intuition that some queries just average
v(q)n(q)+ Xv(q)n(q)=
X X the keywords in the query (like “Canon or Nikon”), some
q∈S1 q∈Q\(S0∪S1) are valued as the best among the keywords (like “Canon
v(q)n(q)+ B−B1 v(q)n(q). DSLR”),andsomevaluedastheworstamongthekeywords
X B′ X (like “Canon calculator”).
q∈S1 q∈Q\(S0∪S1) Running the simulation 15 times, we obtain that the av-
Hence,thetotalvaluefromthesetwocampaignsisthesame erage value obtained by solving the integer linear program
as theoptimal fractional solution, as desired. whileallowingbothexactmatchandbroadmatchwas120.9
while allowing only broad match was 119.2. Furthermore
Finally,weobservethattheoptimalbiddingproblemwith
weobtainthatthemaximumratiobetweenthetwowasless
abudget constraint isAPX-hardforthekeywordlanguage.
than 4 percent. This simple simulation supports our initial
Theorem 5.5. The optimal bidding problem with a bud- hopethatnotusingexactmatchmaybearealisticassump-
tion for some advertisers, in particular, small to medium
getconstraint isAPX-hardforthekeywordlanguage. More-
advertisers. We must remark that a more detailed experi-
over, this problem isnotapproximable withinafactor better
than a multiplicative factor 1− 1. mental study is needed to be more conclusive. Our hope is
e that our LP-based algorithms can indeed be run with rea-
Proof. We give a simple reduction from the maximum sonably sized problems for this purpose.
coverage problem. Inaninstanceofthemaximumcoverage
problem, we are given a family of subset S1,...,Sk ⊂ V, 7. CONCLUDINGREMARKS
and a value w(e) for each element e ∈ V. The goal is
Ourworkinitiatesthestudyofthebidoptimizationprob-
to find a family of k subsets Sa1,...,Sak that maximizes lemforadvertisersinpresenceofacommonfeatureinspon-
. Given an instance of the maximum coverage
Pe∈∪i≤kSai sored search, ie., the broad match type. The central tech-
problem, we define an instance of optimal bidding problem
nical issueisthat choosingtobid onakeyphrasemay yield
as follows: for each subset Si (1 ≤ i≤ k) in the maximum positive profit from some queries, but may commit one to
coverageproblem,weputakeywordsSi inthesetS ofkey- implicitly bid on queries in which the profit may be small
wordsintheoptimalbiddingproblem. Thecostc(s)ofeach
orevennegative. WeproposeLP-basedpolynomial-timeal-
keywordinS is1anditsvalueiszero. Wealsoputaquery
gorithmsfor thisproblem which isoptimalunderthequery
qe ∈ Q corresponding to each element e ∈ V of the maxi- language model, and is an approximation in the keyword
mumcoverageproblem. Thevalueofqueryqe ∈Q\S isone languagemodelforcertaincaseswhileitisNP-Hardtoeven
and the cost of each query is zero. Moreover, we say that
approximate theoptimal solution to any factor, in general.
a query qe corresponding to an element e can be broadly Our work leaves open several research problems. A tech-
matched with any query sSi ∈S corresponding to any sub- nical problem is to extendtheresults hereto themulti-slot
set Si of the maximum coverage problem that includes the case. More precisely, given a “landscape” that is a func-
element e,i.e., e∈Si. Finally,weset thebudgetconstraint tion of bids and gives estimated clicks and cost, obtain the
Btok. Itisnothardtoseethatthemaximumvaluebidding
profit-maximizing bidding strategy. A conceptual problem
strategy on keywordsin this instance with thetotal cost at
istodetermineasuitableapproachforbroadmatchauctions
most B corresponds to the maximum value set-coverage of
where search engines are able to provide faithful estimates
at most k sets in the original problem. This reduction im-
for clicks and cost associated with the broad matches, so
plies that our problem is not approximable within a factor
advertisers can bidaccurately. Thisinvolvesaveraging over
betterthan1−1 unlessP=NP,sincethemaximumcoverage
e manyrelatedqueries. Aprincipledapproachtoformulating
problem is NP-hard to approximate within a factor better
this notion will beof great interest.
than 1− 1 [8].
e Finally,aspecifictechnicalquestionthatremainsopenin
this paper is theapproximability of the budget-constrained
6. ANEXPERIMENTAL STUDY
version of theoptimal biddingproblem in thekeyword lan-
In this section, we report results from an experimental guage. Aninterestingaspectofthisproblemisthefollowing
study to address how much an ad campaign loses by us- relation tosubmodularoptimization. GivenasubsetT ⊂S
ing solely broad match rather than a combination of exact ofkeywordsonwhichwecanbid,letusdenotethetotalcost
and value of the queries that we win as a result of bidding [14] S. Muthukrishnan,M. P´al, and Z. Svitkina.
on keywords in T by C(T) and V(T). One can check that Stochastic models for budget optimization in
both the value and cost functions V,C : 2S → R are set- search-based advertising. In Proc. Workshop on
cover-type set functions, and thus they are monotone and Internet and Network Economics (WINE), 2007.
submodular. 5 The optimal bidding problem with a bud- [15] Christos H. Papadimitriou and KennethSteiglitz.
get constraint in the query language is, therefore, to find Combinatorial Optimization. Prentice Hall
a subset T ⊂ S of keywords that maximizes the submod- [16] P. Rusmevichientongand D.Williamson. An
ular function V(T) subject to the submodular constraint adaptive algorithm for selecting profitable keywords
C(T) ≤ B. Constant-factor approximation algorithms are for search-based advertising services. ACM
knownformaximizingageneralmonotonesubmodularfunc- Conference on Electronic Commerce 2006: 260-269.
tion subject to a knapsack (modular) constraint [19], but [17] A. Schrijver. Combinatorial Optimization. Springer,
maximizing submodular functions subject to a submodular
2003.
constraint (as in our case) is an open question.
[18] M. Stoerand F. Wagner. A simple min-cut
algorithm. Journal of theACM, 44(4):585-591, July
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