elements arrive in uniform random order. · josé soto abner turkieltaub victor verdugo uchile...
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![Page 1: Elements arrive in uniform random order. · José Soto Abner Turkieltaub Victor Verdugo UChile UChile UChile-ENS Approximation and Networks, Collége de France Paris, June 7, 2018](https://reader036.vdocuments.site/reader036/viewer/2022062605/5fd41f42b85ccc2133212071/html5/thumbnails/1.jpg)
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· · ·
r1 r2 r3 r4 rn�1 rn
Elements arrive in uniform random order.
Total order � over R.
1
1 2
2 3 1
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· · ·
r1 r2 r3 r4 rn�1 rn
Elements arrive in uniform random order.
Total order � over R.
1
1 2
2 3 1
![Page 4: Elements arrive in uniform random order. · José Soto Abner Turkieltaub Victor Verdugo UChile UChile UChile-ENS Approximation and Networks, Collége de France Paris, June 7, 2018](https://reader036.vdocuments.site/reader036/viewer/2022062605/5fd41f42b85ccc2133212071/html5/thumbnails/4.jpg)
· · ·
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?
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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
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r1
r2 r3 r4
· · ·
rs
Sample phase Select best seen so far
rs+1 rs+2 rs+3
· · ·
rt
· · ·
r100
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r1 r2
r3 r4
· · ·
rs
Sample phase Select best seen so far
rs+1 rs+2 rs+3
· · ·
rt
· · ·
r100
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r1 r2 r3
r4
· · ·
rs
Sample phase Select best seen so far
rs+1 rs+2 rs+3
· · ·
rt
· · ·
r100
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r1 r2 r3 r4
· · ·
rs
Sample phase
Select best seen so far
rs+1 rs+2 rs+3
· · ·
rt
· · ·
r100
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r1 r2 r3 r4
· · ·
rs
Sample phase Select best seen so far
rs+1
rs+2 rs+3
· · ·
rt
· · ·
r100
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r1 r2 r3 r4
· · ·
rs
Sample phase Select best seen so far
rs+1 rs+2
rs+3
· · ·
rt
· · ·
r100
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r1 r2 r3 r4
· · ·
rs
Sample phase Select best seen so far
rs+1 rs+2 rs+3
· · ·
rt
· · ·
r100
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r1 r2 r3 r4
· · ·
rs
Sample phase Select best seen so far
rs+1 rs+2 rs+3
· · ·
rt
· · ·
r100
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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
�.
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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,
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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
◆
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sample size s
probability
10030 40
1e
s = 37 ⇡ 100/e maximizes the probability!
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Who solved it?
Lindley 1961, scientific publication.
Scientific American 1960, puzzle.
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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
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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 ).
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7
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7
12
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7
12
20
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7
12
202
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7
12
202
9
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7
12
202
9
8
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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.
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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
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neighbours
terminals arrive online!
r1
r2 r3 r5r4
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neighbours
terminals arrive online!
r1 r2
r3 r5r4
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neighbours
terminals arrive online!
r1 r2
r3 r5r4
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neighbours
terminals arrive online!
r1 r2 r3
r5r4
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neighbours
terminals arrive online!
r1 r2 r3
r5r4
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neighbours
terminals arrive online!
r1 r2 r3 r5r4
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neighbours
terminals arrive online!
r1 r2 r3 r5r4
{r1, r2, r3} is a feasible selection
{r2, r3, r4} is not a feasible selection
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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!
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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!
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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.
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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.
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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.
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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 .
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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
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Good necessary condition for optimality
e.g. current offline optimum
+
Small forbidden sets
e.g. study some combinatorial witness
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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.
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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!