elements arrive in uniform random order. · josé soto abner turkieltaub victor verdugo uchile...
TRANSCRIPT
· · ·
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
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rs
Sample phase Select best seen so far
rs+1 rs+2 rs+3
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rt
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r100
r1 r2
r3 r4
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rs
Sample phase Select best seen so far
rs+1 rs+2 rs+3
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rt
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r100
r1 r2 r3
r4
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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
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r100
r1 r2 r3 r4
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rs
Sample phase Select best seen so far
rs+1
rs+2 rs+3
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rt
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r100
r1 r2 r3 r4
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rs
Sample phase Select best seen so far
rs+1 rs+2
rs+3
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rt
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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!