Algorithms for Matching and Clustering Using only Ordinal - - PowerPoint PPT Presentation

▶
algorithms for matching and clustering
SMART_READER_LITE
LIVE PREVIEW

Algorithms for Matching and Clustering Using only Ordinal - - PowerPoint PPT Presentation

Sep 14, 2022 594 likes •956 views

Blind, Greedy, and Random: Algorithms for Matching and Clustering Using only Ordinal Information Elliot Anshelevich (together with Shreyas Sekar) Rensselaer Polytechnic Institute (RPI), Troy, NY Maximum Utility Matching Edges have

slide-1
SLIDE 1

Blind, Greedy, and Random: Algorithms for Matching and Clustering Using only Ordinal Information

Elliot Anshelevich (together with Shreyas Sekar) Rensselaer Polytechnic Institute (RPI), Troy, NY

slide-2
SLIDE 2

Maximum Utility Matching

  • Edges have weight, want to form matching with maximum weight
  • For example, weight can represent compatibility, utility from matching this pair

100 90 90 50 75 A B C D

slide-3
SLIDE 3

Maximum Utility Matching

  • Edges have weight, want to form matching with maximum weight
  • For example, weight can represent compatibility, utility from matching this pair

Goal: maximize social welfare = total utility

100 90 90 50 75 A B C D

slide-4
SLIDE 4

Maximum Utility Matching

  • Edges have weight, want to form matching with maximum weight
  • For example, weight can represent compatibility, utility from matching this pair

What if we only know ordinal preference information? 100 90 90 50 75 ? ? ? ? ? A B C D A B C D A > B > D B > C > A A > D > C B > C > D Truth What we know

slide-5
SLIDE 5

Ordinal Approximations

What if we only know ordinal preference information? Goal: Compute max-utility matching using only ordinal information. 100 90 90 50 75 ? ? ? ? ? A B C D A B C D A > B > D B > C > A A > D > C B > C > D

slide-6
SLIDE 6

Ordinal Approximations

What if we only know ordinal preference information? Goal: Compute max-utility matching using only ordinal information. 100 3 3 1 2 ? ? ? ? ? A B C D A B C D A > B > D B > C > A A > D > C B > C > D

slide-7
SLIDE 7

Ordinal Approximations

What if we only know ordinal preference information? Goal: Compute max-utility matching using only ordinal information. Approximate max-utility matching using only ordinal information. 100 90 90 50 75 ? ? ? ? ? A B C D A B C D A > B > D B > C > A A > D > C B > C > D

slide-8
SLIDE 8

Ordinal Approximations

What if we only know ordinal preference information? Goal: Compute max-utility matching using only ordinal information. Approximate max-utility matching using only ordinal information. 100 90 90 50 75 ? ? ? ? ? A B C D A B C D A > B > D B > C > A A > D > C B > C > D Truth What we know

slide-9
SLIDE 9

Greedy Algorithm

  • Pick edge (X,Y) of maximum weight.
  • Remove X and Y, and repeat.

Classic algorithm; produces 2-approximation. 100 90 90 50 75 ? ? ? ? ? A B C D A B C D A > B > D B > C > A A > D > C B > C > D Truth What we know

slide-10
SLIDE 10

Greedy Algorithm

  • Pick edge (X,Y) such that X is Y’s first choice, and Y is X’s first choice.
  • Remove X and Y, and repeat.

Classic algorithm; produces 2-approximation no matter what the true weights are! 100 90 90 50 75 ? ? ? ? ? A B C D A B C D A > B > D B > C > A A > D > C B > C > D Truth What we know

slide-11
SLIDE 11

Ordinal Approximation for Metric

Can we do better? Will look at metric weights, i.e., weights that obey triangle inequality. Will provide a

  • 1.6-ordinal approximation

(nothing better than 1.25 is possible)

  • Framework for ordinal approximations:

useful for clustering problems, traveling salesman, etc.

y x z A B C z ≤ x + y

slide-12
SLIDE 12

Maximum Weight Metric Matching

  • Diverse Team Formation
  • Want partners with complementary skills
  • Matching is teams of two
slide-13
SLIDE 13

Maximum Weight Metric Matching

  • Diverse Team Formation
  • Want partners with complementary skills
  • Matching is teams of two
  • Homophily

y x z A B C z ≥ 1/3 (x + y)

slide-14
SLIDE 14

Ordinal Approximation for Metric

Can we do better? Will look at metric weights, i.e., weights that obey triangle inequality. Will provide a

  • 1.6-ordinal approximation

(nothing better than 1.25 is possible)

  • Framework for ordinal approximations:

useful for clustering problems, traveling salesman, etc.

y x z A B C z ≤ x + y

slide-15
SLIDE 15

Blind, Greedy, and Random: Algorithms for Matching and Clustering Using only Ordinal Information

100 90 90 50 75 ? ? ? ? ? A B C D A B C D A > B > D B > C > A A > D > C B > C > D

slide-16
SLIDE 16

Blind, Greedy, and Random: Algorithms for Matching and Clustering Using only Ordinal Information

Random: Pick a random matching For metric weights: produces 2-approximation to maximum-weight matching!

  • Can we take better of two algorithms? Don’t even know what

“better” is!

  • Can we mix over two solutions? Yes, but can do even better.
slide-17
SLIDE 17

1.6-approximation to Max Weight Matching

  • Run Greedy until match 2/3 of the nodes
slide-18
SLIDE 18

1.6-approximation to Max Weight Matching

  • Run Greedy until match 2/3 of the nodes
slide-19
SLIDE 19

1.6-approximation to Max Weight Matching

  • Run Greedy until match 2/3 of the nodes
slide-20
SLIDE 20

1.6-approximation to Max Weight Matching

  • Run Greedy until match 2/3 of the nodes

Claim: Top half of edges in Greedy Matching are already 2-approx to Max-Weight Matching.

slide-21
SLIDE 21

1.6-approximation to Max Weight Matching

  • Run Greedy until match 2/3 of the nodes

Claim: Top half of edges in Greedy Matching are already 2-approx to Max-Weight Matching. Claim: Running Greedy until 2/3 of nodes are matched is a 2-approx.

slide-22
SLIDE 22

1.6-approximation to Max Weight Matching

  • Run Greedy until match 2/3 of the nodes
  • Solution 1: Form random matching on rest of nodes
slide-23
SLIDE 23

1.6-approximation to Max Weight Matching

  • Run Greedy until match 2/3 of the nodes
  • Solution 1: Form random matching on rest of nodes
  • Solution 2: Form random bipartite matching to rest of nodes
slide-24
SLIDE 24

1.6-approximation to Max Weight Matching

  • Run Greedy until match 2/3 of the nodes
  • Solution 1: Form random matching on rest of nodes
  • Solution 2: Form random bipartite matching to rest of nodes
  • Take each solution with probability 1/2
slide-25
SLIDE 25

Lower Bound Example

1+ε 1 1 ? ? ? ? A B C D A B C D A > B > D B > A > C A > D > C B > C > D 2 1 1 A B C D 1 1 1 1 1-ε OPT/E[any alg] is no better than 1.25

slide-26
SLIDE 26

Ordinal Approximations Using this as a Black Box

Full Information Ordinal Approximation Maximum Weight Matching 1 1.6 Max k-sum clustering 2 2 Densest k-subgraph 2 4 Max Metric Traveling Salesman (TSP) 1.14 1.88

slide-27
SLIDE 27

Ordinal Approximations Using this as a Black Box

Full Information Black Box Reduction Ordinal Approximation Maximum Weight Matching 1  1.6 Max k-sum clustering 2 2 2 Densest k-subgraph 2 2( for k-matching) 4 Max Metric Traveling Salesman (TSP) 1.14 4/3 1.88

slide-28
SLIDE 28

Ordinal Approximations Using this as a Black Box

Full Information Black Box Reduction Ordinal Approximation Maximum Weight Matching 1 1.6 1.6 Max k-sum clustering 2 3.2 2 Densest k-subgraph 2 4 4 Max Metric Traveling Salesman (TSP) 1.14 2.14 1.88

slide-29
SLIDE 29

Truthful Matching

  • Running Greedy to form perfect matching is truthful
  • Running Greedy to form k-matching is not truthful

A B C D A > B > D B > A > C A > D > C B > C > D

slide-30
SLIDE 30

Truthful Matching

  • Running Greedy to form perfect matching is truthful
  • Running Greedy to form k-matching is not truthful

A B C D A > B > D D > A > B B > A > C C > B > A A > D > C B > C > D

slide-31
SLIDE 31

Truthful Matching

  • Running Greedy to form perfect matching is truthful
  • Running Greedy to form k-matching is not truthful

A B C D A > B > D D > A > B B > A > C C > B > A A > D > C B > C > D

slide-32
SLIDE 32

Truthful Matching

  • Running Greedy to form perfect matching is truthful
  • Running Greedy to form k-matching is not truthful
  • Instead can use Random Serial Dictatorship: 2-approximation

A B C D A > B > D B > A > C A > D > C B > C > D

slide-33
SLIDE 33

Truthful Matching

  • Running Greedy to form perfect matching is truthful
  • Running Greedy to form k-matching is not truthful
  • Instead can use Random Serial Dictatorship: 2-approximation

A B C D A > B > D B > A > C A > D > C B > C > D Take top preference of random node Remove these nodes from graph Repeat

slide-34
SLIDE 34

Ordinal Approximations Using this as a Black Box

Full Information Truthful Ordinal Approximation Improved (non black-box) Maximum Weight Matching 1 1.76 1.6 Max k-sum clustering 2 2 2 Densest k-subgraph 2 6 4 Max Metric Traveling Salesman (TSP) 1.14 2 1.88

slide-35
SLIDE 35

Other Ordinal Problems

  • Ordinal problems in social choice
  • Facility location
  • Min-cost matching, Minimum Spanning Trees
  • Non-metric shortest path vs longest tour

B A C B > A > C