and Matroids. Soheil Ehsani. January 2018. Joint work with M. . Hajiaghayi. , T. . Kesselheim. , S. . Singla. The problem consists of an . initial setting . and a . sequence of events. .. We have to take particular actions . ID: 704998
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Prophet Secretaryfor Combinatorial Auctionsand Matroids
Soheil EhsaniJanuary 2018
Joint work with M.
The problem consists of an initial setting and a
sequence of events.We have to take particular actions
during time.The goal is maximize expected reward or minimize expected cost.
Two important fundamental problems are
Secretary problem and Prophet Inequality.
Online Decision MakingSlide3
There is an
set of numbers
.In an online fashion and at each step we observe the value of .The goal is to select one value , such that is maximized.First introduced by Dynkin in 1960’s and first studied in mechanism design by Hajiaghayi, Kleinberg and Parkes, EC’04 (introduced the value version v.s. the classic rank version and obtained the first bound for the general multi-choice version).
In an online fashion and at each step
we observe a value .The goal is to select one value , upon its arrival, such that is maximized.First introduced by Krengel and Sucheston in 1970’s and first studied in computer science by Hajiaghayi, Kleinberg and Sandholm, AAAI’07 (introduced the first bound for the general multi-choice version).Slide5
Compute a threshold (virtual price)
Select the first item
The above method
).And is tight! A Simple Tight Algorithm! with probability otherwise
Similar to prophet inequality, but the boxes arrive in a random permutation.
.At each time , we observe and randomly draw .The goal is to make an irrevocable and immediate selection such that is maximized.Can we achieve an approximation factor better than for prophet secretary?There is a approximation algorithm using different thresholds [EHLM ESA’15].There is a -approximation algorithm for i.i.d.
and large market instances [AEEHKL STOC’17].
Prophet Inequality has been generalized to many settings:
½-approximation for matroids [KW STOC’12].½-approximation for combinatorial auctions
-approximation for best-order
matroids [YAN SODA’11
Is there a better than ½-approximation for prophet inequality in random order? Combinatorial Auctions: we show a approximation algorithm (incentive-compatible for unit demand).Matroids: we show a approximation algorithm.Single Items: we show a single threshold approximation algorithm (tight). YESSlide8
Generalization: Combinatorial Auctions
Combinatorial Auctions:There is a set
of items and of buyers, each with a valuation function
allocate items to buyers to
maximize social welfare, i.e. Valuation functions are XOS, i.e. for some additive functions .Prophet Inequality Variant:We know distributions for buyers’ valuations.Buyers arrive one by one and we have to allocate a bundle to them immediately.Slide9
Matroids:Set of elements
there exists such that .E.g. graphic matroids, transversal matroids.Prophet Inequality:Given a set of elements, their probability distribution on their values, and a matroid .At each step we draw and should decide weather to collect it (add to set ).The goal is to collect an independent set A of matroid with maximum value, i.e. .Slide10
First we convert the problem into a continuous-time setting.
Assume each buyer arrives at time
(independently at random).
Then we focus on designing dynamic-pricing algorithms: if we serve buyer
Her gain is
Her virtual payment is This approach allows us to split the objective function into revenue and utility:
Lower bounding Social Welfare
The goal is to maximize social welfare, i.e.
We define a
to capture the expected remaining value at time
Residual function is defined for a given instance of prophet secretary and a dynamic-pricing algorithm.Definition: a residual function has following properties:
be a prophet secretary instance
Propose a dynamic pricing algorithm
Show there exists a residual function for , , and any Using a good discount function prove is approximationSlide13
Tuning the Discount Function
Lemma: If an algorithm has a residual function then s
results in a
Second property of residual functions says: .From integration by parts we know: .Therefore,
Now by adding Revenue and Utility we get:
Algorithm: Combinatorial Auctions
Preprocess:Set a base price for each item based on its contribution to the optimum,
and uniformly at
random from .Assignments:Buyer selects each remaining item if . Lemma: The following function is a residual for the algorithm: .Slide15
Algorithm: Single Item
Algorithm:Preprocess the base price
for arrival times.
The item is assigned to the first buyer
such that .The existence of a residual function in conjunction with the above tuning of the discount function results in a approximation algorithm.We need to show is a residual function.
Analysis: Single Items
: note that
of selling item at time (for price ).Third property:
For the conditional utility of buyer
given arrival time
approximation algorithms for most important generalizations of single item.Can we beat this barrier? [ACK arXiv 2017] improve by
for single item.
What about the i.i.d. case? [AEEHKL STOC’17] present 0.74 approximation algorithm for single item.
Hajiaghayi (UMD)Sahil Singla (CMU)
Thanks to my co-authors!
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