46 research outputs found
Issues and Challenges in Advertising on the Web
One of the big surprises of the 21st century has been the ability of all sorts of interesting Web applications to support themselves through advertising, rather than subscription. While radio and television have managed to use advertising as their primary revenue source, most media – newspapers and magazines, for example – have had to use a hybrid approach, combining revenue from advertising and subscriptions. A venue for on-line advertising has been search, and much of the effectiveness of search advertising came from the “adwords” model of matching search queries to advertisements. This paper presents the algorithms for optimizing the way of matching search queries to advertisements is done. The algorithms discussed are of unusual type; they are greedy and they are on-line which are used to tackle the adwords problem.DOI:http://dx.doi.org/10.11591/ijece.v4i5.662
Matroid Online Bipartite Matching and Vertex Cover
The Adwords and Online Bipartite Matching problems have enjoyed a renewed
attention over the past decade due to their connection to Internet advertising.
Our community has contributed, among other things, new models (notably
stochastic) and extensions to the classical formulations to address the issues
that arise from practical needs. In this paper, we propose a new generalization
based on matroids and show that many of the previous results extend to this
more general setting. Because of the rich structures and expressive power of
matroids, our new setting is potentially of interest both in theory and in
practice.
In the classical version of the problem, the offline side of a bipartite
graph is known initially while vertices from the online side arrive one at a
time along with their incident edges. The objective is to maintain a decent
approximate matching from which no edge can be removed. Our generalization,
called Matroid Online Bipartite Matching, additionally requires that the set of
matched offline vertices be independent in a given matroid. In particular, the
case of partition matroids corresponds to the natural scenario where each
advertiser manages multiple ads with a fixed total budget.
Our algorithms attain the same performance as the classical version of the
problems considered, which are often provably the best possible. We present
-competitive algorithms for Matroid Online Bipartite Matching under the
small bid assumption, as well as a -competitive algorithm for Matroid
Online Bipartite Matching in the random arrival model. A key technical
ingredient of our results is a carefully designed primal-dual waterfilling
procedure that accommodates for matroid constraints. This is inspired by the
extension of our recent charging scheme for Online Bipartite Vertex Cover.Comment: 19 pages, to appear in EC'1
First-Come-First-Served for Online Slot Allocation and Huffman Coding
Can one choose a good Huffman code on the fly, without knowing the underlying
distribution? Online Slot Allocation (OSA) models this and similar problems:
There are n slots, each with a known cost. There are n items. Requests for
items are drawn i.i.d. from a fixed but hidden probability distribution p.
After each request, if the item, i, was not previously requested, then the
algorithm (knowing the slot costs and the requests so far, but not p) must
place the item in some vacant slot j(i). The goal is to minimize the sum, over
the items, of the probability of the item times the cost of its assigned slot.
The optimal offline algorithm is trivial: put the most probable item in the
cheapest slot, the second most probable item in the second cheapest slot, etc.
The optimal online algorithm is First Come First Served (FCFS): put the first
requested item in the cheapest slot, the second (distinct) requested item in
the second cheapest slot, etc. The optimal competitive ratios for any online
algorithm are 1+H(n-1) ~ ln n for general costs and 2 for concave costs. For
logarithmic costs, the ratio is, asymptotically, 1: FCFS gives cost opt + O(log
opt).
For Huffman coding, FCFS yields an online algorithm (one that allocates
codewords on demand, without knowing the underlying probability distribution)
that guarantees asymptotically optimal cost: at most opt + 2 log(1+opt) + 2.Comment: ACM-SIAM Symposium on Discrete Algorithms (SODA) 201
Prophet Secretary for Combinatorial Auctions and Matroids
The secretary and the prophet inequality problems are central to the field of
Stopping Theory. Recently, there has been a lot of work in generalizing these
models to multiple items because of their applications in mechanism design. The
most important of these generalizations are to matroids and to combinatorial
auctions (extends bipartite matching). Kleinberg-Weinberg \cite{KW-STOC12} and
Feldman et al. \cite{feldman2015combinatorial} show that for adversarial
arrival order of random variables the optimal prophet inequalities give a
-approximation. For many settings, however, it's conceivable that the
arrival order is chosen uniformly at random, akin to the secretary problem. For
such a random arrival model, we improve upon the -approximation and obtain
-approximation prophet inequalities for both matroids and
combinatorial auctions. This also gives improvements to the results of Yan
\cite{yan2011mechanism} and Esfandiari et al. \cite{esfandiari2015prophet} who
worked in the special cases where we can fully control the arrival order or
when there is only a single item.
Our techniques are threshold based. We convert our discrete problem into a
continuous setting and then give a generic template on how to dynamically
adjust these thresholds to lower bound the expected total welfare.Comment: Preliminary version appeared in SODA 2018. This version improves the
writeup on Fixed-Threshold algorithm