· · ·

r1 r2 r3 r4 rn�1 rn

Elements arrive in uniform random order.

Total order � over R.

1

1 2

2 3 1

· · ·

r1 r2 r3 r4 rn�1 rn

Elements arrive in uniform random order.

Total order � over R.

1

1 2

2 3 1

· · ·

r1 r2 r3 r4 rn�1 rn

Elements arrive in uniform random order.

Total order � over R.

1

1 2

2 3 1

Select one element: What is the best stopping rule?

1

Strong algorithms for the OrdinalMatroid Secretary Problem

José Soto Abner Turkieltaub Victor Verdugo

UChile UChile UChile-ENS

Approximation and Networks, Collége de France

Paris, June 7, 2018

r1

r2 r3 r4

· · ·

rs

Sample phase Select best seen so far

rs+1 rs+2 rs+3

· · ·

rt

· · ·

r100

r1 r2

r3 r4

· · ·

rs

Sample phase Select best seen so far

rs+1 rs+2 rs+3

· · ·

rt

· · ·

r100

r1 r2 r3

r4

· · ·

rs

Sample phase Select best seen so far

rs+1 rs+2 rs+3

· · ·

rt

· · ·

r100

r1 r2 r3 r4

· · ·

rs

Sample phase

Select best seen so far

rs+1 rs+2 rs+3

· · ·

rt

· · ·

r100

r1 r2 r3 r4

· · ·

rs

Sample phase Select best seen so far

rs+1

rs+2 rs+3

· · ·

rt

· · ·

r100

r1 r2 r3 r4

· · ·

rs

Sample phase Select best seen so far

rs+1 rs+2

rs+3

· · ·

rt

· · ·

r100

r1 r2 r3 r4

· · ·

rs

Sample phase Select best seen so far

rs+1 rs+2 rs+3

· · ·

rt

· · ·

r100

r1 r2 r3 r4

· · ·

rs

Sample phase Select best seen so far

rs+1 rs+2 rs+3

· · ·

rt

· · ·

r100

r1 r2 r3 r4

· · ·

rs

Sample phase Select best seen so far

rs+1 rs+2 rs+3

· · ·

rt

· · ·

r100

P(select best) ⇡ � s100 ln

� s100

�.

r1 r2 r3 r4

· · ·

rs

Sample phase Select best seen so far

rs+1 rs+2 rs+3

· · ·

rt

· · ·

r100

P(select best) ⇡ � s100 ln

� s100

�.

(?) Conditioning on the history at time t ,

Pt (select best) �t�1Y

j=s+1

Pt (rj is not the second best) =t�1Y

j=s+1

j � 1j

=s

t � 1,

r1 r2 r3 r4

· · ·

rs

Sample phase Select best seen so far

rs+1 rs+2 rs+3

· · ·

rt

· · ·

r100

(?) Conditioning on the history at time t ,

Pt (select best) �t�1Y

j=s+1

Pt (rj is not the second best) =t�1Y

j=s+1

j � 1j

=s

t � 1,

P(select best) �s

100

100X

t=s+1

1t � 1

⇡s

100

Z 100

s

dxx

= �s

100ln

✓s

100

◆

sample size s

probability

10030 40

1e

s = 37 ⇡ 100/e maximizes the probability!

Who solved it?

Lindley 1961, scientific publication.

Scientific American 1960, puzzle.

For every element in OPT, P(element is selected) � 1/↵.

Obs. By picking s ⇠ Binomial(100, 1/e), algorithm is 1/e prob-competitive.

Es

0

@ 1100

100X

j=s+1

t�1Y

j=s+1

j � 1j

1

A � �p ln p ... optimize here

For every element in OPT, P(element is selected) � 1/↵.

Obs. By picking s ⇠ Binomial(100, 1/e), algorithm is 1/e prob-competitive.

Es

0

@ 1100

100X

j=s+1

t�1Y

j=s+1

j � 1j

1

A � �p ln p ... optimize here

· · ·10 5 7 9 20 1

Utility competitive ⇠ E(w(ALG)) �1↵

w(OPT ).

7

7

12

7

12

20

7

12

202

7

12

202

9

7

12

202

9

8

7

12

202

9

8

7

12

202

9

8

w(ALG) = 28 w(OPT) = 41

This solution is 41/28 ⇡ 1.46 utility-competitive.

Offline: Select greedily

1. Greedy in decreasing order returns the optimal OPT(E).

2. For every F ✓ E , OPT(E) \ F ✓ OPT(F ).

· · ·

r3 � r2 � r1 � r4 · · · rm�1 � rm

OPT

neighbours

terminals arrive online!

r1

r2 r3 r5r4

neighbours

terminals arrive online!

r1 r2

r3 r5r4

neighbours

terminals arrive online!

r1 r2

r3 r5r4

neighbours

terminals arrive online!

r1 r2 r3

r5r4

neighbours

terminals arrive online!

r1 r2 r3

r5r4

neighbours

terminals arrive online!

r1 r2 r3 r5r4

neighbours

terminals arrive online!

r1 r2 r3 r5r4

{r1, r2, r3} is a feasible selection

{r2, r3, r4} is not a feasible selection

r1 r2

· · ·

· · ·r3

Sample phase

rs

Sample phase

rtrs+1 rs+2

· · · · · ·

Select if is in best feasible solution so far

If rt 2 OPT, it could only be blocked by one other current best!

r1 r2

· · ·

· · ·r3

Sample phase

rs

Sample phase

rtrs+1 rs+2

· · · · · ·

Select if is in best feasible solution so far

If rt 2 OPT, it could only be blocked by one other current best!

r1 r2

· · ·

· · ·r3

Sample phase

rs

Sample phase

rtrs+1 rs+2

· · · · · ·

Select if is in best feasible solution so far

If rt 2 OPT, it could only be blocked by one other current best!

Pt (select rt ) �t�1Y

j=s+1

Pt (rj 2 OPTj is not matched to green) =t�1Y

j=s+1

j � 1j

=s

t � 1.

More generally ...

Suppose we have an algorithm for a matroid class such that:

(Feasibility) Returns an independent set.

(Sampling) Sample s ⇠ Bin(m, p) elements and reject them.

(k-forbidden) Let r an element in OPT arriving at time t > s.For each s < i < t , a random set Fi of size atmost k , such that:If for each s < i < t , the element arriving /2 Fi then r is selected.

More generally ...

Suppose we have an algorithm for a matroid class such that:

(Feasibility) Returns an independent set.

(Sampling) Sample s ⇠ Bin(m, p) elements and reject them.

(k-forbidden) Let r an element in OPT arriving at time t > s.For each s < i < t , a random set Fi of size atmost k , such that:If for each s < i < t , the element arriving /2 Fi then r is selected.

Thm. By setting the right sampling probability, a k -forbidden algorithm is ↵prob-competitive, where

↵ =

(e if k = 1,kk/(k�1) if k � 2.

More generally ...

Suppose we have an algorithm for a matroid class such that:

(Feasibility) Returns an independent set.

(Sampling) Sample s ⇠ Bin(m, p) elements and reject them.

(k-forbidden) Let r an element in OPT arriving at time t > s.For each s < i < t , a random set Fi of size atmost k , such that:If for each s < i < t , the element arriving /2 Fi then r is selected.

Thm. By setting the right sampling probability, a k -forbidden algorithm is ↵prob-competitive, where

↵ =

(e if k = 1,kk/(k�1) if k � 2.

Proof. Study the quantity

Es

0

@ 1100

100X

j=s+1

t�1Y

j=s+1

j � kj

1

A .

Further applications of the technique

Previous Our results (p)Transversal e (u) eµ-Gammoid µe (u) µµ/(µ�1)

Graphic 2e (u) 4Laminar 9.6 (p) 3

p3 ⇡ 5.19

k -column sparse ke (u) kk/(k�1)

Matching - 4Graph packings - µµ/(µ�1)

k -framed - kk/(k�1)

Hypergraphic - 4Semiplanar - 44/3

Good necessary condition for optimality

e.g. current offline optimum

+

Small forbidden sets

e.g. study some combinatorial witness

I O(1)-forbidden algorithm for general matroids.I

Strongest version: 1-forbidden algorithm for general matroids.I Apply the framework over non-matroidal settings.

I O(1)-forbidden algorithm for general matroids.I

Strongest version: 1-forbidden algorithm for general matroids.I Apply the framework over non-matroidal settings.

Merci!