Using Order Constraints in Crowd Data Sourcing

Using Order Constraints in Crowd Data Sourcing

Antoine Amarilli^1,2, Yael Amsterdamer^3,4, Tova Milo⁴, Pierre Senellart^1,2 February 12th, 2018

1Télécom ParisTech

2École normale supérieure

3Bar Ilan University

4Tel Aviv University

1/16

Introduction

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

Taxonomy of items for a store

withcategories. Ask the crowd to classify items

Monotonicity: compatibility increases as we go up. Best categories? Naiveanswer... Cleveranswer...

2/16

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

Taxonomy of items for a store withcategories.

Ask the crowd to classify items

Monotonicity: compatibility increases as we go up. Best categories? Naiveanswer... Cleveranswer...

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

Taxonomy of items for a store withcategories.

Ask the crowd to classify items

Monotonicity: compatibility increases as we go up. Best categories? Naiveanswer... Cleveranswer...

2/16

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

Taxonomy of items for a store withcategories.

Monotonicity: compatibility increases as we go up. Best categories? Naiveanswer... Cleveranswer...

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

Taxonomy of items for a store withcategories.

Ask the crowd to classify items

Monotonicity: compatibility increases as we go up. Best categories? Naiveanswer... Cleveranswer...

2/16

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

0.1

Taxonomy of items for a store withcategories.

Monotonicity: compatibility increases as we go up. Best categories? Naiveanswer... Cleveranswer...

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

0.1

Taxonomy of items for a store withcategories.

Ask the crowd to classify items

Monotonicity: compatibility increases as we go up. Best categories? Naiveanswer... Cleveranswer...

2/16

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

0.1

0.9

Taxonomy of items for a store withcategories.

Monotonicity: compatibility increases as we go up. Best categories? Naiveanswer... Cleveranswer...

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

0.1

0.9

Taxonomy of items for a store withcategories.

Ask the crowd to classify items

Monotonicity: compatibility increases as we go up. Best categories? Naiveanswer... Cleveranswer...

2/16

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

0.1

0.9

0.5

Taxonomy of items for a store withcategories.

Monotonicity: compatibility increases as we go up. Best categories? Naiveanswer... Cleveranswer...

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

0.1

0.9

0.5

Taxonomy of items for a store withcategories.

Ask the crowd to classify items

Monotonicity: compatibility increases as we go up. Best categories? Naiveanswer... Cleveranswer...

2/16

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

0.1

0.9

0.5

0.5

Taxonomy of items for a store withcategories.

Monotonicity: compatibility increases as we go up. Best categories? Naiveanswer... Cleveranswer...

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

0.1

0.9

0.5

0.5

Monotonicity: compatibility increases as we go up.

Best categories? Naiveanswer... Cleveranswer...

2/16

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

0.1

0.9

0.5

0.5

Monotonicity: compatibility increases as we go up.

Naiveanswer... Cleveranswer...

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

0.1

0.9

0.5

0.5

Monotonicity: compatibility increases as we go up.

Best categories? Naiveanswer...

Cleveranswer...

2/16

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

0.1

0.9

0.5

0.5

Monotonicity: compatibility increases as we go up.

Cleveranswer...

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

0.1

0.9

0.5

0.5

Monotonicity: compatibility increases as we go up.

Best categories? Naiveanswer... Cleveranswer...

2/16

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

0.1

0.9

0.5

0.5

= 0.5

Monotonicity: compatibility increases as we go up.

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

0.1

0.9

0.5

0.5

= 0.5

≥0.9

Monotonicity: compatibility increases as we go up.

Best categories? Naiveanswer... Cleveranswer...

2/16

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

0.1

0.9

0.5

0.5

= 0.5

≥0.9

≥0.5

Monotonicity: compatibility increases as we go up.

Example 1: Classifying products

Products

Clothing Electronics

Shoes Cell Phones

TVs Watches Diving Gear

Wearable Devices

Smartphones Diving Watches

Sports

0.1

0.9

0.5

0.5

= 0.5

≥0.9

≥0.5

Monotonicity: compatibility increases as we go up.

Best categories? Naiveanswer... Cleveranswer...

2/16

Example 2: Finding popular activity combinations Taxonomy of activities

activity research social

lunch diving

Taxonomy of activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

Example 2: Finding popular activity combinations Taxonomy of activities

activity research social

lunch diving

Taxonomy of activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

3/16

Example 2: Finding popular activity combinations Taxonomy of activities

activity research social

lunch diving

Taxonomy of activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

Example 2: Finding popular activity combinations Taxonomy of activities

activity research social

lunch diving

Taxonomy of activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

3/16

Example 2: Finding popular activity combinations Taxonomy of activities

activity research social

lunch diving

Taxonomy of activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

Example 2: Finding popular activity combinations Taxonomy of activities

activity research social

lunch diving

Taxonomy of activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

3/16

Example 2: Finding popular activity combinations Taxonomy of activities

activity research social

lunch diving

Taxonomy of activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

Example 2: Finding popular activity combinations Taxonomy of activities

activity research social

lunch diving

Taxonomy of activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

3/16

Example 2: Finding popular activity combinations Taxonomy of activities

activity research social

lunch diving

Taxonomy of activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

Example 2: Finding popular activity combinations Taxonomy of activities

activity research social

lunch diving

Taxonomy of activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

3/16

Example 2: Finding popular activity combinations Taxonomy of activities

activity research social

lunch diving

Taxonomy of activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

Example 2: Finding popular activity combinations Taxonomy of activities

activity research social

lunch diving

Taxonomy of activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

3/16

Example 2: Finding popular activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

•

More specific combinations are less frequent

•

^Askcrowd questions:

•

^{Is{diving}frequent?}

⇒ Yes!

•

^{{lunch,diving}?}

⇒ No!

•

^{lunch}?

⇒ Yes!

•

^{{research,lunch}?}

⇒ Yes!

•

^{{research,diving}?}

⇒ No!

Example 2: Finding popular activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

•

More specific combinations are less frequent

•

^Askcrowd questions:

•

^{Is{diving}frequent?}

⇒ Yes!

•

^{{lunch,diving}?}

⇒ No!

•

^{lunch}?

⇒ Yes!

•

^{{research,lunch}?}

⇒ Yes!

•

^{{research,diving}?}

⇒ No!

4/16

Example 2: Finding popular activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

•

More specific combinations are less frequent

•

^Askcrowd questions:

•

^{Is{diving}frequent?}

⇒ Yes!

•

^{{lunch,diving}?}

⇒ No!

•

^{lunch}?

⇒ Yes!

•

^{{research,lunch}?}

⇒ Yes!

•

^{{research,diving}?}

⇒ No!

Example 2: Finding popular activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

•

More specific combinations are less frequent

•

^Askcrowd questions:

•

^{Is{diving}frequent?}

⇒ Yes!

•

^{{lunch,diving}?}

⇒ No!

•

^{lunch}?

⇒ Yes!

•

^{{research,lunch}?}

⇒ Yes!

•

^{{research,diving}?}

⇒ No!

4/16

Example 2: Finding popular activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

•

More specific combinations are less frequent

•

^Askcrowd questions:

•

^{Is{diving}frequent?}

⇒ Yes!

•

^{{lunch,diving}?}

⇒ No!

•

^{lunch}?

⇒ Yes!

•

^{{research,lunch}?}

⇒ Yes!

•

^{{research,diving}?}

⇒ No!

Example 2: Finding popular activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

•

More specific combinations are less frequent

•

^Askcrowd questions:

•

^{Is{diving}frequent?}

⇒ Yes!

•

^{{lunch,diving}?}

⇒ No!

•

^{lunch}?

⇒ Yes!

•

^{{research,lunch}?}

⇒ Yes!

•

^{{research,diving}?}

⇒ No!

4/16

Example 2: Finding popular activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

•

More specific combinations are less frequent

•

^Askcrowd questions:

•

^{Is{diving}frequent?}

⇒ Yes!

•

^{{lunch,diving}?}

⇒ No!

•

^{lunch}?

⇒ Yes!

•

^{{research,lunch}?}

⇒ Yes!

•

^{{research,diving}?}

⇒ No!

Example 2: Finding popular activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

•

More specific combinations are less frequent

•

^Askcrowd questions:

•

^{Is{diving}frequent?}

⇒ Yes!

•

^{{lunch,diving}?}

⇒ No!

•

^{lunch}?

⇒ Yes!

•

^{{research,lunch}?}

⇒ Yes!

•

^{{research,diving}?}

⇒ No!

4/16

Example 2: Finding popular activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

•

More specific combinations are less frequent

•

^Askcrowd questions:

•

^{Is{diving}frequent?}

⇒ Yes!

•

^{{lunch,diving}?}

⇒ No!

•

^{lunch}?

⇒ Yes!

•

^{{research,lunch}?}

⇒ Yes!

•

^{{research,diving}?}

⇒ No!

Example 2: Finding popular activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

•

More specific combinations are less frequent

•

^Askcrowd questions:

•

^{Is{diving}frequent?}

⇒ Yes!

•

^{{lunch,diving}?}

⇒ No!

•

^{lunch}?

⇒ Yes!

•

^{{research,lunch}?}

⇒ Yes!

•

^{{research,diving}?}

⇒ No!

4/16

Example 2: Finding popular activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

•

More specific combinations are less frequent

•

^Askcrowd questions:

•

^{Is{diving}frequent?}

⇒ Yes!

•

^{{lunch,diving}?}

⇒ No!

•

^{lunch}?

⇒ Yes!

•

^{{research,lunch}?}

⇒ Yes!

•

^{{research,diving}?}

⇒ No!

Example 2: Finding popular activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

•

More specific combinations are less frequent

•

^Askcrowd questions:

•

^{Is{diving}frequent?}

⇒ Yes!

•

^{{lunch,diving}?}

⇒ No!

•

^{lunch}?

⇒ Yes!

•

^{{research,lunch}?}

⇒ Yes!

•

^{{research,diving}?}

⇒ No!

4/16

Example 2: Finding popular activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

•

More specific combinations are less frequent

•

^Askcrowd questions:

•

^{Is{diving}frequent?}

⇒ Yes!

•

^{{lunch,diving}?}

⇒ No!

•

^{lunch}?

⇒ Yes!

•

^{{research,lunch}?}

⇒ Yes!

•

^{{research,diving}?}

⇒ No!

Example 2: Finding popular activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

•

More specific combinations are less frequent

•

^Askcrowd questions:

•

^{Is{diving}frequent?}

⇒ Yes!

•

^{{lunch,diving}?}

⇒ No!

•

^{lunch}?

⇒ Yes!

•

^{{research,lunch}?}

⇒ Yes!

•

^{{research,diving}?}

⇒ No!

4/16

Example 2: Finding popular activity combinations

nil activity research social research

social lunch diving

research diving research

lunch lunch

diving research

lunch diving

•

More specific combinations are less frequent

•

^Askcrowd questions:

•

^{Is{diving}frequent?}

⇒ Yes!

•

^{{lunch,diving}?}

⇒ No!

•

^{lunch}?

⇒ Yes!

•

^{{research,lunch}?}

⇒ Yes!

•

^{{research,diving}?}

⇒ No!

Example 3: Estimating unknown values

small sweet

medium sweet

large sweet

small salty

medium salty

large salty tiny

both

small both

medium both

large both

•

^{How muchfood}do people eat at conference buffets?

•

^{Let’s ask}fellow organizers!

•

^{How toestimatequantities}

for my own conference?

•

^Someorder relationsare implied

→ How tocompletethe missing values?

5/16

Example 3: Estimating unknown values

small sweet

medium sweet

large sweet

small salty

medium salty

large salty tiny

both

small both

medium both

large both

•

^{How muchfood}do people eat at conference buffets?

•

^{Let’s ask}fellow organizers!

•

^{How toestimatequantities}

for my own conference?

•

^Someorder relationsare implied

→ How tocompletethe missing values?

Example 3: Estimating unknown values small

sweet

medium sweet

large sweet

small salty

medium salty

large salty tiny

both

small both

medium both

large both

•

^{How muchfood}do people eat at conference buffets?

•

^{Let’s ask}fellow organizers!

•

^{How toestimatequantities}

for my own conference?

•

^Someorder relationsare implied

→ How tocompletethe missing values?

5/16

Example 3: Estimating unknown values small

sweet

medium sweet

large sweet

small salty

medium salty

large salty tiny

both

small both

medium both

large both

•

^{How muchfood}do people eat at conference buffets?

•

^{Let’s ask}fellow organizers!

•

^{How toestimatequantities}

for my own conference?

•

^Someorder relationsare implied

→ How tocompletethe missing values?

Example 3: Estimating unknown values small

sweet

medium sweet

large sweet

small salty

medium salty

large salty tiny

both

small both

medium both

large both

•

^{How muchfood}do people eat at conference buffets?

•

^{Let’s ask}fellow organizers!

•

^{How toestimatequantities}

for my own conference?

•

^Someorder relationsare implied

→ How tocompletethe missing values?

5/16

Example 3: Estimating unknown values small

sweet

medium sweet

large sweet

small salty

medium salty

large salty tiny

both

small both medium

both

large both

•

^{How muchfood}do people eat at conference buffets?

•

^{Let’s ask}fellow organizers!

•

^{How toestimatequantities}

for my own conference?

•

^Someorder relationsare implied

→ How tocompletethe missing values?

Example 3: Estimating unknown values small

sweet

medium sweet

large sweet

small salty

medium salty

large salty tiny

both

small both medium

both

large both

•

^{How muchfood}do people eat at conference buffets?

•

^{Let’s ask}fellow organizers!

•

^{How toestimatequantities}

for my own conference?

•

^Someorder relationsare implied

→ How tocompletethe missing values?

5/16

Example 3: Estimating unknown values small

sweet

medium sweet

large sweet

small salty

medium salty

large salty tiny

both

small both medium

both

large both

•

^{How muchfood}do people eat at conference buffets?

•

^{Let’s ask}fellow organizers!

•

^{How toestimatequantities}

for my own conference?

•

^Someorder relationsare implied

→ How tocompletethe missing values?

Example 3: Estimating unknown values 10

•

•

•

20

• 15

•

???

100

•

^{How muchfood}do people eat at conference buffets?

•

^{Let’s ask}fellow organizers!

•

^{How toestimatequantities}

for my own conference?

•

^Someorder relationsare implied

→ How tocompletethe missing values?

5/16

Example 3: Estimating unknown values 10

•

•

•

20

• 15

•

???

100

•

^{How muchfood}do people eat at conference buffets?

•

^{Let’s ask}fellow organizers!

•

^{How toestimatequantities}

for my own conference?

•

^Someorder relationsare implied

→ How tocompletethe missing values?

Example 3: Estimating unknown values 10

36

68

10

20

60 15

46

ࠑࠐ

100

•

^{How muchfood}do people eat at conference buffets?

•

^{Let’s ask}fellow organizers!

•

^{How toestimatequantities}

for my own conference?

•

^Someorder relationsare implied

→ How tocompletethe missing values?

5/16

General problem

•

•

•

•

•

•

•

•

•

•

•

^{Severalitemswithvalues}

•

Order relationson the values

•

Some values areknown and the others areunknown

→ Estimatethe unknown values

→ Find the nextcrowd questionto ask to obtain more known values

General problem

•

•

•

•

•

•

•

•

•

•

•

^{Severalitemswithvalues}

•

Order relationson the values

•

Some values areknown and the others areunknown

→ Estimatethe unknown values

→ Find the nextcrowd questionto ask to obtain more known values

6/16

General problem

•

•

•

•

•

•

•

•

•

•

•

^{Severalitemswithvalues}

•

Order relationson the values

•

Some values areknown and the others areunknown

→ Estimatethe unknown values

→ Find the nextcrowd questionto ask to obtain more known values

General problem

•

•

•

•

•

•

•

•

•

•

•

^{Severalitemswithvalues}

•

Order relationson the values

•

Some values areknown and the others areunknown

→ Estimatethe unknown values

→ Find the nextcrowd questionto ask to obtain more known values

6/16

General problem

•

•

•

•

•

•

•

•

•

•

•

^{Severalitemswithvalues}

•

Order relationson the values

•

Some values areknown and the others areunknown

→ Estimatethe unknown values

→ Find the nextcrowd questionto ask to obtain more known values

General problem

???

???

???

???

20

???

15

???

???

100

•

^{Severalitemswithvalues}

•

Order relationson the values

•

Some values areknown and the others areunknown

→ Estimatethe unknown values

→ Find the nextcrowd questionto ask to obtain more known values

6/16

General problem

???

???

???

???

20

???

15

???

???

100

•

^{Severalitemswithvalues}

•

Order relationson the values

•

Some values areknown and the others areunknown

→ Estimatethe unknown values

→ Find the nextcrowd questionto ask to obtain more known values

General problem

???

???

???

???

20

???

15

???

???

100

•

^{Severalitemswithvalues}

•

Order relationson the values

•

Some values areknown and the others areunknown

→ Estimatethe unknown values

→ Find the nextcrowd questionto ask to obtain more known values

6/16

Estimating Missing Values

Estimating missing values (Antoine, Yael, Pierre, Tova, ICDT’17)

???

???

???

???

20

???

15

???

???

100

•

^{Knownandunknownvalues}

withorder relation

•

Estimate theunknown values

without asking more crowd questions

7/16

Easy case: total order 0

???

70

???

???

100

•

If the items aretotally ordered, what can we do?

→ Linear interpolation!

→ Can wegeneralizethis if the order is not total?

Easy case: total order 0

???

70

???

???

100

•

If the items aretotally ordered, what can we do?

→ Linear interpolation!

→ Can wegeneralizethis if the order is not total?

8/16

Easy case: total order 0

35 70 80 90 100

•

If the items aretotally ordered, what can we do?

→ Linear interpolation!

→ Can wegeneralizethis if the order is not total?

Easy case: total order 0

35 70 80 90 100

•

If the items aretotally ordered, what can we do?

→ Linear interpolation!

→ Can wegeneralizethis if the order is not total?

8/16

Polytope interpretation

•

Introduce onevariableper item:

→ x,y,z,w

•

^{Write the}order constraints:

→ x≥y,w≤y,y≤z

•

^{Write the}known values:

→ w= 0.3,z= 0.9

•

^Theselinear constraintsdefine anadmissible polytope x≥y

w≤y y≤z w= 0.3 z= 0.9

x≥y w≤y y≤z w= 0.3 z≤1

→ Estimate unknown values as thecenter of massof the polytope!

Polytope interpretation

•

Introduce onevariableper item:

→ x,y,z,w

•

^{Write the}order constraints:

→ x≥y,w≤y,y≤z

•

^{Write the}known values:

→ w= 0.3,z= 0.9

•

^Theselinear constraintsdefine anadmissible polytope x≥y

w≤y y≤z w= 0.3 z= 0.9

x≥y w≤y y≤z w= 0.3 z≤1

→ Estimate unknown values as thecenter of massof the polytope!

9/16

Polytope interpretation

•

Introduce onevariableper item:

→ x,y,z,w

•

^{Write the}order constraints:

→ x≥y,w≤y,y≤z

•

^{Write the}known values:

→ w= 0.3,z= 0.9

•

^Theselinear constraintsdefine anadmissible polytope x≥y

w≤y y≤z w= 0.3 z= 0.9

x≥y w≤y y≤z w= 0.3 z≤1

→ Estimate unknown values as thecenter of massof the polytope!

Polytope interpretation

•

Introduce onevariableper item:

→ x,y,z,w

•

^{Write the}order constraints:

→ x≥y,w≤y,y≤z

•

^{Write the}known values:

→ w= 0.3,z= 0.9

•

^Theselinear constraintsdefine anadmissible polytope

x≥y w≤y y≤z w= 0.3 z= 0.9

x≥y w≤y y≤z w= 0.3 z≤1

→ Estimate unknown values as thecenter of massof the polytope!

9/16

Polytope interpretation

•

Introduce onevariableper item:

→ x,y,z,w

•

^{Write the}order constraints:

→ x≥y,w≤y,y≤z

•

^{Write the}known values:

→ w= 0.3,z= 0.9

•

^Theselinear constraintsdefine anadmissible polytope x≥y

w≤y y≤z w= 0.3 z= 0.9

x≥y w≤y y≤z w= 0.3 z≤1

→ Estimate unknown values as thecenter of massof the polytope!

Polytope interpretation

•

Introduce onevariableper item:

→ x,y,z,w

•

^{Write the}order constraints:

→ x≥y,w≤y,y≤z

•

^{Write the}known values:

→ w= 0.3,z= 0.9

•

^Theselinear constraintsdefine anadmissible polytope x≥y

w≤y y≤z w= 0.3 z= 0.9

x≥y w≤y y≤z w= 0.3 z≤1

→ Estimate unknown values as thecenter of massof the polytope!

9/16

Polytope interpretation

•

Introduce onevariableper item:

→ x,y,z,w

•

^{Write the}order constraints:

→ x≥y,w≤y,y≤z

•

^{Write the}known values:

→ w= 0.3,z= 0.9

•

^Theselinear constraintsdefine anadmissible polytope x≥y

w≤y y≤z w= 0.3 z= 0.9

x≥y w≤y y≤z w= 0.3 z≤1

Polytope interpretation

•

Introduce onevariableper item:

→ x,y,z,w

•

^{Write the}order constraints:

→ x≥y,w≤y,y≤z

•

^{Write the}known values:

→ w= 0.3,z= 0.9

•

^Theselinear constraintsdefine anadmissible polytope x≥y

w≤y y≤z w= 0.3 z= 0.9

x≥y w≤y y≤z w= 0.3 z≤1

→ Estimate unknown values as thecenter of massof the polytope!_9/16

Results of our ICDT’17 paper

•

^Fortotal orders, the polytope method gives the same result as linear interpolation

•

Brute-forcealgorithm to compute the center of mass

•

Intractability resultsfor this task and for computing the top-kitems

•

Tractable caseswhen the order is atree

Results of our ICDT’17 paper

•

^Fortotal orders, the polytope method gives the same result as linear interpolation

•

Brute-forcealgorithm to compute the center of mass

•

Intractability resultsfor this task and for computing the top-kitems

•

Tractable caseswhen the order is atree

10/16

Results of our ICDT’17 paper

•

^Fortotal orders, the polytope method gives the same result as linear interpolation

•

Brute-forcealgorithm to compute the center of mass

•

Intractability resultsfor this task and for computing the top-kitems

•

Tractable caseswhen the order is atree

Results of our ICDT’17 paper

•

^Fortotal orders, the polytope method gives the same result as linear interpolation

•

Brute-forcealgorithm to compute the center of mass

•

Intractability resultsfor this task and for computing the top-kitems

•

Tractable caseswhen the order is atree

10/16

Open problems

0

???

???

???

???

???

???

???

???

100

•

^{Are there}more generalposets/polytopes where we can interpolate efficiently?

(generalizing trees, e.g., treelike posets)

•

^Fixingone value to its interpolated value canchangeother interpolated values! (unlike linear interpolation)

Open problems

0

???

???

???

???

???

???

???

???

100

•

^{Are there}more generalposets/polytopes where we can interpolate efficiently?

(generalizing trees, e.g., treelike posets)

•

^Fixingone value to its interpolated value canchangeother interpolated values!

(unlike linear interpolation)

11/16

Asking Questions

Asking questions (Antoine, Yael, Tova, ICDT’14)

•

•

•

•

•

•

•

•

•

•

•

Unknown values areBoolean:

either0or1

•

The Boolean function ismonotone with respect to the order

•

^{Ask the}right questionsto determine the Boolean function completely

12/16

Asking questions (Antoine, Yael, Tova, ICDT’14)

•

•

•

•

•

•

•

•

•

•

•

Unknown values areBoolean:

either0or1

•

The Boolean function ismonotone with respect to the order

•

^{Ask the}right questionsto determine the Boolean function completely

Asking questions (Antoine, Yael, Tova, ICDT’14)

•

•

•

•

•

•

•

•

•

•

•

Unknown values areBoolean:

either0or1

•

The Boolean function ismonotone with respect to the order

•

^{Ask the}right questionsto determine the Boolean function completely

12/16

Asking questions (Antoine, Yael, Tova, ICDT’14)

•

•

•

•

•

•

•

•

•

•

•

Unknown values areBoolean:

either0or1

•

The Boolean function ismonotone with respect to the order

•

^{Ask the}right questionsto determine the Boolean function completely

Asking questions (Antoine, Yael, Tova, ICDT’14)

•

•

•

•

•

•

•

•

•

•

•

Unknown values areBoolean:

either0or1

•

The Boolean function ismonotone with respect to the order

•

^{Ask the}right questionsto determine the Boolean function completely

12/16

Easy case: total order

•

•

•

•

•

•

•

•

If the items aretotally ordered, what can we do?

→ Binary search!

→ Can wegeneralizethis if the order is not total?

Easy case: total order

•

•

•

•

•

•

•

•

If the items aretotally ordered, what can we do?

→ Binary search!

→ Can wegeneralizethis if the order is not total?

13/16

Easy case: total order

•

•

•

•

•

•

•

•

If the items aretotally ordered, what can we do?

→ Binary search!

→ Can wegeneralizethis if the order is not total?

Easy case: total order

•

•

•

•

•

•

•

•

If the items aretotally ordered, what can we do?

→ Binary search!

→ Can wegeneralizethis if the order is not total?

13/16

Easy case: total order

•

•

•

•

•

•

•

•

If the items aretotally ordered, what can we do?

→ Binary search!

→ Can wegeneralizethis if the order is not total?

Easy case: total order

•

•

•

•

•

•

•

•

If the items aretotally ordered, what can we do?

→ Binary search!

→ Can wegeneralizethis if the order is not total?

13/16

Easy case: total order

•

•

•

•

•

•

•

•

If the items aretotally ordered, what can we do?

→ Binary search!

→ Can wegeneralizethis if the order is not total?

Easy case: total order

•

•

•

•

•

•

•

•

If the items aretotally ordered, what can we do?

→ Binary search!

→ Can wegeneralizethis if the order is not total?

13/16

Easy case: total order

•

•

•

•

•

•

•

•

If the items aretotally ordered, what can we do?

→ Binary search!

→ Can wegeneralizethis if the order is not total?

Easy case: total order

•

•

•

•

•

•

•

•

If the items aretotally ordered, what can we do?

→ Binary search!

→ Can wegeneralizethis if the order is not total?

13/16

Easy case: total order

•

•

•

•

•

•

•

•

If the items aretotally ordered, what can we do?

→ Binary search!

→ Can wegeneralizethis if the order is not total?

Results of our ICDT’14 paper

•

Lower bound: each crowd query gives one bit so we need logarithmically many crowd queries in the number of possible Boolean functions

•

Matching upper bound:we can always query an item that splits the remaining Boolean functions evenly

•

Output-sensitive complexity:we can determine the function in a number of crowd queries linear in the “frontier”

•

Computational hardness, e.g., of finding the best-split element

14/16

Results of our ICDT’14 paper

•

Lower bound: each crowd query gives one bit so we need logarithmically many crowd queries in the number of possible Boolean functions

•

Matching upper bound:we can always query an item that splits the remaining Boolean functions evenly

•

Output-sensitive complexity:we can determine the function in a number of crowd queries linear in the “frontier”

•

Computational hardness, e.g., of finding the best-split element