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Generalized Matroid Secretary Problem Sourav Chakraborty (Indian Statistical Institute) Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem The Art of Selling a Car Want to sell my car. Sourav Chakraborty

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Generalized Matroid Secretary Problem

Sourav Chakraborty (Indian Statistical Institute)

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Art of Selling a Car

Want to sell my car.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

SLIDE 3

The Art of Selling a Car

Want to sell my car. 2 L

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

SLIDE 4

The Art of Selling a Car

Want to sell my car. 2 L

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Art of Selling a Car

Want to sell my car. 2 L 1 L

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Art of Selling a Car

Want to sell my car. 2 L 1 L

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Art of Selling a Car

Want to sell my car. 2 L 1 L 1.5 L

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Art of Selling a Car

Want to sell my car. 2 L 1 L 1.5 L

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Art of Selling a Car

Want to sell my car. 2 L 1 L 1.5 L 1.25 L

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Art of Selling a Car

Want to sell my car. 2 L 1 L 1.5 L 1.25 L

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Art of Selling a Car

Want to sell my car. 2 L 1 L 1.5 L 1.25 L 1.75 L

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Art of Selling a Car

Want to sell my car. 2 L 1 L 1.5 L 1.25 L 1.75 L

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Art of Selling a Car

Want to sell my car. 2 L 1 L 1.5 L 1.25 L 1.75 L 2.25 L

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Art of Selling a Car

Want to sell my car. 2 L 1 L 1.5 L 1.25 L 1.75 L 2.25 L What should be my strategy?

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Different Assumptions

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Different Assumptions

Are the bids chosen by an adversary or drawn from some distribution?

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Different Assumptions

Are the bids chosen by an adversary or drawn from some distribution? How much knowledge in advance do the seller have about the bids? (For example: do the seller know the distribution from which the bids are drawn.)

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Different Assumptions

Are the bids chosen by an adversary or drawn from some distribution? How much knowledge in advance do the seller have about the bids? (For example: do the seller know the distribution from which the bids are drawn.) Do the buyers come in a particular order or in some random

rder?

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Different kind of assumptions

(Pessimistic Assumption) All the bids and the ordering of the buyers are chosen by an adversary.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Different kind of assumptions

(Pessimistic Assumption) All the bids and the ordering of the buyers are chosen by an adversary. For any strategy of the seller there is an adversarial strategy such that the expected return for the seller is 1/N of the best bid. (N is the number of bidders.)

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Different kind of assumptions

(Pessimistic Assumption) All the bids and the ordering of the buyers are chosen by an adversary. For any strategy of the seller there is an adversarial strategy such that the expected return for the seller is 1/N of the best bid. (N is the number of bidders.) (Optimistic Approach) All bids are drawn from a distribution (that is known to the seller) and the bidders come in a random

rder.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Different kind of assumptions

(Pessimistic Assumption) All the bids and the ordering of the buyers are chosen by an adversary. For any strategy of the seller there is an adversarial strategy such that the expected return for the seller is 1/N of the best bid. (N is the number of bidders.) (Optimistic Approach) All bids are drawn from a distribution (that is known to the seller) and the bidders come in a random

rder.

There is a simple strategy the guarantees that the expected return is at least half of the best bid.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Different kind of assumptions

(Pessimistic Assumption) All the bids and the ordering of the buyers are chosen by an adversary. For any strategy of the seller there is an adversarial strategy such that the expected return for the seller is 1/N of the best bid. (N is the number of bidders.) (Optimistic Approach) All bids are drawn from a distribution (that is known to the seller) and the bidders come in a random

rder.

There is a simple strategy the guarantees that the expected return is at least half of the best bid. (Middle Path) The bids are chosen by an adversary but the bidders come in a random order.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Different kind of assumptions

(Pessimistic Assumption) All the bids and the ordering of the buyers are chosen by an adversary. For any strategy of the seller there is an adversarial strategy such that the expected return for the seller is 1/N of the best bid. (N is the number of bidders.) (Optimistic Approach) All bids are drawn from a distribution (that is known to the seller) and the bidders come in a random

rder.

There is a simple strategy the guarantees that the expected return is at least half of the best bid. (Middle Path) The bids are chosen by an adversary but the bidders come in a random order. Secretary Problem

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Secretary Problem

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Secretary Problem

Bids: B1, B2, . . . , BN

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Secretary Problem

Bids: B1, B2, . . . , BN Bids are adversarially chosen but come in a random order. Bids come is a online fashion.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Secretary Problem

Bids: B1, B2, . . . , BN Bids are adversarially chosen but come in a random order. Bids come is a online fashion. Once bid Bi is seen, the algorithm has to either REJECTS or SELECTS.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Secretary Problem

Bids: B1, B2, . . . , BN Bids are adversarially chosen but come in a random order. Bids come is a online fashion. Once bid Bi is seen, the algorithm has to either REJECTS or SELECTS. If the algorithm REJECTS it then the bid Bi is lost forever and cannot be selected later.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Secretary Problem

Bids: B1, B2, . . . , BN Bids are adversarially chosen but come in a random order. Bids come is a online fashion. Once bid Bi is seen, the algorithm has to either REJECTS or SELECTS. If the algorithm REJECTS it then the bid Bi is lost forever and cannot be selected later. If the algorithm SELECTS it then Bi is the return the algorithm get.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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The Secretary Problem

Bids: B1, B2, . . . , BN Bids are adversarially chosen but come in a random order. Bids come is a online fashion. Once bid Bi is seen, the algorithm has to either REJECTS or SELECTS. If the algorithm REJECTS it then the bid Bi is lost forever and cannot be selected later. If the algorithm SELECTS it then Bi is the return the algorithm get. GOAL: To maximize Expected Return

maxi Bi

.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Algorithm for Secretary Problem

Simple Algorithm REJECT the bids B1, B2, . . . , BN/2 Let C = max{B1, . . . , BN/2}. For any i > N/2 if Bi is at least C then ACCEPT Bi. With probability 1/4 the highest bid is the second half and the second highest bid is in the first half. So, Expected Return maxi Bi > 1 4. [Lindley, Dynkin (1963)] showed that the competitive ratio is 1/e.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Lets Sell Flight Tickets

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Lets Sell Flight Tickets

Available seats

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Lets Sell Flight Tickets

Available seats Chennai-London (20)

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Lets Sell Flight Tickets

Available seats Chennai-London (20) London-Paris (5) Paris-NY (2) Mumbai-Paris (10) Chennai-Mumbai (40) London-NY (15) London-Mumbai (4)

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Lets Sell Flight Tickets

Available seats Chennai-London (20) London-Paris (5) Paris-NY (2) Mumbai-Paris (10) Chennai-Mumbai (40) London-NY (15) London-Mumbai (4) Bidders B1 Chennai-London-Mumbai ($300)

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Lets Sell Flight Tickets

Available seats Chennai-London (20) London-Paris (5) Paris-NY (2) Mumbai-Paris (10) Chennai-Mumbai (40) London-NY (15) London-Mumbai (4) Bidders B1 Chennai-London-Mumbai ($300) NO

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Lets Sell Flight Tickets

Available seats Chennai-London (20) London-Paris (5) Paris-NY (2) Mumbai-Paris (10) Chennai-Mumbai (40) London-NY (15) London-Mumbai (4) Bidders B1 Chennai-London-Mumbai ($300) NO B2 London-Paris ($100)

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Lets Sell Flight Tickets

Available seats Chennai-London (20) London-Paris (5) Paris-NY (2) Mumbai-Paris (10) Chennai-Mumbai (40) London-NY (15) London-Mumbai (4) Bidders B1 Chennai-London-Mumbai ($300) NO B2 London-Paris ($100) YES

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Lets Sell Flight Tickets

Available seats Chennai-London (20) London-Paris (5) Paris-NY (2) Mumbai-Paris (10) Chennai-Mumbai (40) London-NY (15) London-Mumbai (4) Bidders B1 Chennai-London-Mumbai ($300) NO B2 London-Paris ($100) YES B3 Mumbai-Paris-NY ($700)

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Lets Sell Flight Tickets

Available seats Chennai-London (20) London-Paris (5) Paris-NY (2) Mumbai-Paris (10) Chennai-Mumbai (40) London-NY (15) London-Mumbai (4) Bidders B1 Chennai-London-Mumbai ($300) NO B2 London-Paris ($100) YES B3 Mumbai-Paris-NY ($700) YES

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Lets Sell Flight Tickets

Available seats Chennai-London (20) London-Paris (5) Paris-NY (2) Mumbai-Paris (10) Chennai-Mumbai (40) London-NY (15) London-Mumbai (4) Bidders B1 Chennai-London-Mumbai ($300) NO B2 London-Paris ($100) YES B3 Mumbai-Paris-NY ($700) YES B4 Mumbai-Paris-NY ($800)

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Lets Sell Flight Tickets

Available seats Chennai-London (20) London-Paris (5) Paris-NY (2) Mumbai-Paris (10) Chennai-Mumbai (40) London-NY (15) London-Mumbai (4) Bidders B1 Chennai-London-Mumbai ($300) NO B2 London-Paris ($100) YES B3 Mumbai-Paris-NY ($700) YES B4 Mumbai-Paris-NY ($800) YES

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Lets Sell Flight Tickets

Available seats Chennai-London (20) London-Paris (5) Paris-NY (2) Mumbai-Paris (10) Chennai-Mumbai (40) London-NY (15) London-Mumbai (4) Bidders B1 Chennai-London-Mumbai ($300) NO B2 London-Paris ($100) YES B3 Mumbai-Paris-NY ($700) YES B4 Mumbai-Paris-NY ($800) YES B5 Chennai-London-NY ($950)

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Lets Sell Flight Tickets

Available seats Chennai-London (20) London-Paris (5) Paris-NY (2) Mumbai-Paris (10) Chennai-Mumbai (40) London-NY (15) London-Mumbai (4) Bidders B1 Chennai-London-Mumbai ($300) NO B2 London-Paris ($100) YES B3 Mumbai-Paris-NY ($700) YES B4 Mumbai-Paris-NY ($800) YES B5 Chennai-London-NY ($950) YES

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Lets Sell Flight Tickets

Available seats Chennai-London (20) London-Paris (5) Paris-NY (2) Mumbai-Paris (10) Chennai-Mumbai (40) London-NY (15) London-Mumbai (4) Bidders B1 Chennai-London-Mumbai ($300) NO B2 London-Paris ($100) YES B3 Mumbai-Paris-NY ($700) YES B4 Mumbai-Paris-NY ($800) YES B5 Chennai-London-NY ($950) YES B6 London-Paris-NY ($1200) .. ....

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Lets Sell Flight Tickets

Available seats Chennai-London (20) London-Paris (5) Paris-NY (2) Mumbai-Paris (10) Chennai-Mumbai (40) London-NY (15) London-Mumbai (4) Bidders B1 Chennai-London-Mumbai ($300) NO B2 London-Paris ($100) YES B3 Mumbai-Paris-NY ($700) YES B4 Mumbai-Paris-NY ($800) YES B5 Chennai-London-NY ($950) YES B6 London-Paris-NY ($1200) .. .... What should be my strategy?

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Generalized Matroid Problem

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Generalized Matroid Problem

(U, I) is a matroid. U is a universe (all possible itineraries). I is the set of independent sets (allowed combinations of the elements of the universe). Bids: (U1, B1), (U2, B2), . . . , (UN, BN) ∈ (U, R+)

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Generalized Matroid Problem

(U, I) is a matroid. U is a universe (all possible itineraries). I is the set of independent sets (allowed combinations of the elements of the universe). Bids: (U1, B1), (U2, B2), . . . , (UN, BN) ∈ (U, R+) Bids are adversarially chosen but come in a random order. Bids come is a online fashion. Once bid (Ui, Bi) is seen, the algorithm has to either REJECTS

r SELECTS.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Generalized Matroid Problem

(U, I) is a matroid. U is a universe (all possible itineraries). I is the set of independent sets (allowed combinations of the elements of the universe). Bids: (U1, B1), (U2, B2), . . . , (UN, BN) ∈ (U, R+) Bids are adversarially chosen but come in a random order. Bids come is a online fashion. Once bid (Ui, Bi) is seen, the algorithm has to either REJECTS

r SELECTS.

If the algorithm REJECTS it then the bid (Ui, Bi) is lost forever.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Generalized Matroid Problem

(U, I) is a matroid. U is a universe (all possible itineraries). I is the set of independent sets (allowed combinations of the elements of the universe). Bids: (U1, B1), (U2, B2), . . . , (UN, BN) ∈ (U, R+) Bids are adversarially chosen but come in a random order. Bids come is a online fashion. Once bid (Ui, Bi) is seen, the algorithm has to either REJECTS

r SELECTS.

If the algorithm REJECTS it then the bid (Ui, Bi) is lost forever. If the algorithm SELECTS (Ui, Bi) then it adds Ui to its set S of selected items. S must be in I.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Generalized Matroid Problem

(U, I) is a matroid. U is a universe (all possible itineraries). I is the set of independent sets (allowed combinations of the elements of the universe). Bids: (U1, B1), (U2, B2), . . . , (UN, BN) ∈ (U, R+) Bids are adversarially chosen but come in a random order. Bids come is a online fashion. Once bid (Ui, Bi) is seen, the algorithm has to either REJECTS

r SELECTS.

If the algorithm REJECTS it then the bid (Ui, Bi) is lost forever. If the algorithm SELECTS (Ui, Bi) then it adds Ui to its set S of selected items. S must be in I. The final return is the weight of S. Where, weight of a set S is

i:Ui∈S Bi.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

SLIDE 55

Generalized Matroid Problem

(U, I) is a matroid. U is a universe (all possible itineraries). I is the set of independent sets (allowed combinations of the elements of the universe). Bids: (U1, B1), (U2, B2), . . . , (UN, BN) ∈ (U, R+) Bids are adversarially chosen but come in a random order. Bids come is a online fashion. Once bid (Ui, Bi) is seen, the algorithm has to either REJECTS

r SELECTS.

If the algorithm REJECTS it then the bid (Ui, Bi) is lost forever. If the algorithm SELECTS (Ui, Bi) then it adds Ui to its set S of selected items. S must be in I. The final return is the weight of S. Where, weight of a set S is

i:Ui∈S Bi.

GOAL: To maximize Expected Return

maxS∈I Weight of S.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Current Status

Introduced by Babaioff et al (2007).

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Current Status

Introduced by Babaioff et al (2007). They also gave a strategy that has expected return at least 1/ log d of the OPT, where d is the dimension of the matroid (the size of the biggest set in I).

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Current Status

Introduced by Babaioff et al (2007). They also gave a strategy that has expected return at least 1/ log d of the OPT, where d is the dimension of the matroid (the size of the biggest set in I). Under various restrictions on the matroid structure or the bids better strategies have been designed. [Survey: Babaioff-Immorlica-Kempe-Klienberg Online auctions and generalized secretary problems (2008)]

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Current Status

Introduced by Babaioff et al (2007). They also gave a strategy that has expected return at least 1/ log d of the OPT, where d is the dimension of the matroid (the size of the biggest set in I). Under various restrictions on the matroid structure or the bids better strategies have been designed. [Survey: Babaioff-Immorlica-Kempe-Klienberg Online auctions and generalized secretary problems (2008)] Even for general matroid the conjecture is constant competitive ratio.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Current Status

Introduced by Babaioff et al (2007). They also gave a strategy that has expected return at least 1/ log d of the OPT, where d is the dimension of the matroid (the size of the biggest set in I). Under various restrictions on the matroid structure or the bids better strategies have been designed. [Survey: Babaioff-Immorlica-Kempe-Klienberg Online auctions and generalized secretary problems (2008)] Even for general matroid the conjecture is constant competitive ratio. Theorem (Chakraborty-Lachish 2012) For the general matroid secretary problem there is a strategy such that the expected return is at least 1/√log d of the OPT.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Current Status

Improvement of the Competitive Ratio O(log d) by Babaioff-Immorlica-Kempe-Klienberg 2008 O(√log d) by Chakraborty-Lachish 2012 O(log log d) by Lachish 2014 O(log log d) by Moran-Svensson-Zenklusen 2015 Better (even constant competitive ratio) algrotihms are known for special matroids.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Formal statement of the Problem - Matroid be a vector space

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Formal statement of the Problem - Matroid be a vector space

V is a vector space of dimension d. Bids: (v1, w(v1)), (v2, w(v2)), . . . , (vN, w(vN)) ∈ (U, R+)

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Formal statement of the Problem - Matroid be a vector space

V is a vector space of dimension d. Bids: (v1, w(v1)), (v2, w(v2)), . . . , (vN, w(vN)) ∈ (U, R+) Bids are adversarially chosen but come in a random order. Once bid (vi, w(vi)) is seen, the algorithm has to either REJECTS or SELECTS. If the algorithm REJECTS it then the bid (vi, w(vi)) is lost. If the algorithm SELECTS (vi, w(vi)) then it adds vi to its set S

f selected items. S must be an independent set.

The final return is the weight of S. Where, weight of a set S is

i:vi∈S w(vi)

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

SLIDE 65

Formal statement of the Problem - Matroid be a vector space

V is a vector space of dimension d. Bids: (v1, w(v1)), (v2, w(v2)), . . . , (vN, w(vN)) ∈ (U, R+) Bids are adversarially chosen but come in a random order. Once bid (vi, w(vi)) is seen, the algorithm has to either REJECTS or SELECTS. If the algorithm REJECTS it then the bid (vi, w(vi)) is lost. If the algorithm SELECTS (vi, w(vi)) then it adds vi to its set S

f selected items. S must be an independent set.

The final return is the weight of S. Where, weight of a set S is

i:vi∈S w(vi)

GOAL: To maximize Expected Return

maxS∈I Weight of S.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

SLIDE 66

Sampling and Selection Phase

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Sampling and Selection Phase

V is a vector space of dimension d. Bids: (v1, w(v1)), (v2, w(v2)), . . . , (vN, w(vN)) ∈ (U, R+) Sampling Phase :Reject (v1, w(v1)), (v2, w(v2)), . . . , (vN/2, w(vN/2)), but record all the data. Selection Phase :Based on the data from the Sampling Phase decide which of the (vN/2+1, w(vN/2+1)), . . . , (vN, w(vN)) to select.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Simplifying the Problem

By loosing a constant in the competitive ratio we can assume the following:

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Simplifying the Problem

By loosing a constant in the competitive ratio we can assume the following: A1 Let M be the maximum weight of all the vectors. M is known to us.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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Simplifying the Problem

By loosing a constant in the competitive ratio we can assume the following: A1 Let M be the maximum weight of all the vectors. M is known to us. A2 For all vectors vi the weight w(vi) is one of {OPT, OP T

, OP T

, . . . , OP T

2log d }.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

SLIDE 71

Simplifying the Problem

By loosing a constant in the competitive ratio we can assume the following: A1 Let M be the maximum weight of all the vectors. M is known to us. A2 For all vectors vi the weight w(vi) is one of {OPT, OP T

, OP T

, . . . , OP T

2log d }.

A3 Let Li be the set of vectors in (v1, w(v1)), . . . , (vN/2, w(vN/2)) that have weight OPT/2i.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

SLIDE 72

Simplifying the Problem

By loosing a constant in the competitive ratio we can assume the following: A1 Let M be the maximum weight of all the vectors. M is known to us. A2 For all vectors vi the weight w(vi) is one of {OPT, OP T

, OP T

, . . . , OP T

2log d }.

A3 Let Li be the set of vectors in (v1, w(v1)), . . . , (vN/2, w(vN/2)) that have weight OPT/2i. A3’ Let Ri be the set of vectors in (vN/2+1, w(vN/2+1)), . . . , (vN, w(vN)) that have weight OPT/2i.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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What is the OPT?

A1 Let M < OPT/√log d be the maximum weight of all the vectors. M is known to us. A2 For all vectors vi the weight w(vi) is one of {OPT, OP T

, OP T

, . . . , OP T

2log d }.

A3 Let Li be the set of vectors in (v1, w(v1)), . . . , (vN/2, w(vN/2)) that have weight OPT/2i. A3’ Let Ri be the set of vectors in (vN/2+1, w(vN/2+1)), . . . , (vN, w(vN)) that have weight OPT/2i.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

SLIDE 74

What is the OPT?

A1 Let M < OPT/√log d be the maximum weight of all the vectors. M is known to us. A2 For all vectors vi the weight w(vi) is one of {OPT, OP T

, OP T

, . . . , OP T

2log d }.

A3 Let Li be the set of vectors in (v1, w(v1)), . . . , (vN/2, w(vN/2)) that have weight OPT/2i. A3’ Let Ri be the set of vectors in (vN/2+1, w(vN/2+1)), . . . , (vN, w(vN)) that have weight OPT/2i.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

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What is the OPT?

A1 Let M < OPT/√log d be the maximum weight of all the vectors. M is known to us. A2 For all vectors vi the weight w(vi) is one of {OPT, OP T

, OP T

, . . . , OP T

2log d }.

A3 Let Li be the set of vectors in (v1, w(v1)), . . . , (vN/2, w(vN/2)) that have weight OPT/2i. A3’ Let Ri be the set of vectors in (vN/2+1, w(vN/2+1)), . . . , (vN, w(vN)) that have weight OPT/2i.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

SLIDE 76

What is the OPT?

A1 Let M < OPT/√log d be the maximum weight of all the vectors. M is known to us. A2 For all vectors vi the weight w(vi) is one of {OPT, OP T

, OP T

, . . . , OP T

2log d }.

A3 Let Li be the set of vectors in (v1, w(v1)), . . . , (vN/2, w(vN/2)) that have weight OPT/2i. A3’ Let Ri be the set of vectors in (vN/2+1, w(vN/2+1)), . . . , (vN, w(vN)) that have weight OPT/2i. For all i the best offline algo on the first half will choose di dim(L1 ∪ L2 ∪ · · · ∪ Li) − dim(L1 ∪ L2 ∪ · · · ∪ Li−1) number of vectors from layer Li.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

SLIDE 77

Main Hurdle: Measure on the interaction between layers

Let us only plan to SELECT elements from Li and Lj.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

SLIDE 78

Main Hurdle: Measure on the interaction between layers

Let us only plan to SELECT elements from Li and Lj. [Simple Case]: Let the span(Li) is disjoint from span(Lj). Then if we greedily select vectors from Li and Lj then our return from Li and Lj is dim(Li)OPT 2i + dim(Lj)OPT 2j .

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

SLIDE 79

Main Hurdle: Measure on the interaction between layers

Let us only plan to SELECT elements from Li and Lj. [Simple Case]: Let the span(Li) is disjoint from span(Lj). Then if we greedily select vectors from Li and Lj then our return from Li and Lj is dim(Li)OPT 2i + dim(Lj)OPT 2j . [Hard Case]: If span(Li) is not disjoint from span(Lj). Thus selecting vectors from Li can obstruct selecting vectors from Lj.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem

SLIDE 80

Main Hurdle: Measure on the interaction between layers

Let us only plan to SELECT elements from Li and Lj. [Simple Case]: Let the span(Li) is disjoint from span(Lj). Then if we greedily select vectors from Li and Lj then our return from Li and Lj is dim(Li)OPT 2i + dim(Lj)OPT 2j . [Hard Case]: If span(Li) is not disjoint from span(Lj). Thus selecting vectors from Li can obstruct selecting vectors from Lj. So we have to understand how the layers are disrupting each other.

Sourav Chakraborty (Indian Statistical Institute) Generalized Matroid Secretary Problem