Set Cover in Sub-linear Time Piotr Indyk Sepideh Mahabadi Ronitt - - PowerPoint PPT Presentation

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Set Cover in Sub-linear Time Piotr Indyk Sepideh Mahabadi Ronitt - - PowerPoint PPT Presentation

May 16, 2023 330 likes β€’1.23k views

Set Cover in Sub-linear Time Piotr Indyk Sepideh Mahabadi Ronitt Rubinfeld MIT Columbia University MIT/TAU Ali Vakilian Anak Yodpinyanee MIT MIT Set Cover Problem Input: Collection of sets 1 , , , each a subset of

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SLIDE 1

Set Cover in Sub-linear Time

Piotr Indyk

MIT

Sepideh Mahabadi

Columbia University

Ronitt Rubinfeld

MIT/TAU

Ali Vakilian

MIT

Anak Yodpinyanee

MIT

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SLIDE 2

Set Cover Problem

Input: Collection β„± of sets 𝑇1, … , 𝑇𝑛, each a subset of 𝒱 = {1, … , π‘œ}

1 4 5 2 3

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SLIDE 3

Set Cover Problem

Input: Collection β„± of sets 𝑇1, … , 𝑇𝑛, each a subset of 𝒱 = {1, … , π‘œ} Output: a subset π’Ÿ of β„± such that:

  • π’Ÿ covers 𝒱
  • |π’Ÿ| is minimized

1 4 5 2 3

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SLIDE 4

Set Cover Problem

Input: Collection β„± of sets 𝑇1, … , 𝑇𝑛, each a subset of 𝒱 = {1, … , π‘œ} Output: a subset π’Ÿ of β„± such that:

  • π’Ÿ covers 𝒱
  • |π’Ÿ| is minimized

Complexity:

  • NP-hard

1 4 5 2 3

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SLIDE 5

Set Cover Problem

Input: Collection β„± of sets 𝑇1, … , 𝑇𝑛, each a subset of 𝒱 = {1, … , π‘œ} Output: a subset π’Ÿ of β„± such that:

  • π’Ÿ covers 𝒱
  • |π’Ÿ| is minimized

Complexity:

  • NP-hard
  • Greedy (ln π‘œ)-approximation algorithm

1 4 5 2 3

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SLIDE 6

Set Cover Problem

Input: Collection β„± of sets 𝑇1, … , 𝑇𝑛, each a subset of 𝒱 = {1, … , π‘œ} Output: a subset π’Ÿ of β„± such that:

  • π’Ÿ covers 𝒱
  • |π’Ÿ| is minimized

Complexity:

  • NP-hard
  • Greedy (ln π‘œ)-approximation algorithm
  • Can’t do better unless P=NP [LY91][RS97][Fei98][AMS06][DS14]

1 4 5 2 3

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SLIDE 7

Set Cover Problem

Input: Collection β„± of sets 𝑇1, … , 𝑇𝑛, each a subset of 𝒱 = {1, … , π‘œ} Output: a subset π’Ÿ of β„± such that:

  • π’Ÿ covers 𝒱
  • |π’Ÿ| is minimized

Complexity:

  • NP-hard
  • Greedy (ln π‘œ)-approximation algorithm
  • Can’t do better unless P=NP [LY91][RS97][Fei98][AMS06][DS14]

1 4 5 2 3

β€œIs it possible to solve minimum set cover in sub-linear time?”

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SLIDE 8

Sub-linear Time Set Cover

Data Access Model ?

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SLIDE 9

Sub-linear Time Set Cover

Data Access Model [NO’08,YYI’12]

EltOf(𝑻, 𝒋): 𝑗th element in 𝑻 SetOf(𝒇, π’Œ): π‘˜th set containing 𝒇

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SLIDE 10

Sub-linear Time Set Cover

Data Access Model [NO’08,YYI’12]

  • No assumption on the order
  • Incidence list in (sub-linear) algorithms for graphs

EltOf(𝑻, 𝒋): 𝑗th element in 𝑻 SetOf(𝒇, π’Œ): π‘˜th set containing 𝒇

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SLIDE 11

Sub-linear Time Set Cover

Data Access Model [NO’08,YYI’12]

  • No assumption on the order
  • Incidence list in (sub-linear) algorithms for graphs
  • Sublinear in 𝒏𝒐

EltOf(𝑻, 𝒋): 𝑗th element in 𝑻 SetOf(𝒇, π’Œ): π‘˜th set containing 𝒇

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SLIDE 12

Sub-linear Time Set Cover

Data Access Model [NO’08,YYI’12]

  • No assumption on the order
  • Incidence list in (sub-linear) algorithms for graphs
  • Sublinear in 𝒏𝒐

Prior Results

EltOf(𝑻, 𝒋): 𝑗th element in 𝑻 SetOf(𝒇, π’Œ): π‘˜th set containing 𝒇

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SLIDE 13

Sub-linear Time Set Cover

Data Access Model [NO’08,YYI’12]

  • No assumption on the order
  • Incidence list in (sub-linear) algorithms for graphs
  • Sublinear in 𝒏𝒐

Prior Results

[Nguyen, Onak’08][Yoshida, Yamamoto, Ito’12]

  • Constant queries, if degree is constant

EltOf(𝑻, 𝒋): 𝑗th element in 𝑻 SetOf(𝒇, π’Œ): π‘˜th set containing 𝒇

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SLIDE 14

Sub-linear Time Set Cover

Data Access Model [NO’08,YYI’12]

  • No assumption on the order
  • Incidence list in (sub-linear) algorithms for graphs
  • Sublinear in 𝒏𝒐

Prior Results

[Nguyen, Onak’08][Yoshida, Yamamoto, Ito’12]

  • Constant queries, if degree is constant

[Koufogiannakis, Young’14][Grigoriadis, Kachiyan’95]:

  • Find (1 + πœ—)-approximate fractional solution, then perform

randomized rounding to achieve 𝑃(log π‘œ)-approximation EltOf(𝑻, 𝒋): 𝑗th element in 𝑻 SetOf(𝒇, π’Œ): π‘˜th set containing 𝒇

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SLIDE 15

Sub-linear Time Set Cover

Data Access Model [NO’08,YYI’12]

  • No assumption on the order
  • Incidence list in (sub-linear) algorithms for graphs
  • Sublinear in 𝒏𝒐

Prior Results

[Nguyen, Onak’08][Yoshida, Yamamoto, Ito’12]

  • Constant queries, if degree is constant

[Koufogiannakis, Young’14][Grigoriadis, Kachiyan’95]:

  • Find (1 + πœ—)-approximate fractional solution, then perform

randomized rounding to achieve 𝑃(log π‘œ)-approximation

  • 𝑃(𝑛𝑙2 + π‘œπ‘™2) (can be improved to 𝑃(𝑛 + π‘œπ‘™))

EltOf(𝑻, 𝒋): 𝑗th element in 𝑻 SetOf(𝒇, π’Œ): π‘˜th set containing 𝒇

𝒐 = number of elements 𝒏 = number of sets 𝒍 = size of the optimal solution

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SLIDE 16

Results

Problem Approximation Constraints Query Complexity

Set Cover

π›½πœ + 1 𝛽 β‰₯ 2 𝑃 𝑛 π‘œ 𝑙

1 π›½βˆ’1 + π‘œπ‘™

𝜍 + 1 βˆ’ 𝑃 π‘›π‘œ 𝑙 𝛽 𝑙 ≀ π‘œ log 𝑛

1 4𝛽+1

Ξ© 𝑛 π‘œ 𝑙

1 2𝛽

𝛽 𝛽 ≀ 1.01 𝑙 = 𝑃(π‘œ/ log 𝑛) Ξ© π‘›π‘œ 𝑙

Cover Verification

βˆ’ 𝑙 ≀ π‘œ/2 Ξ©(π‘œπ‘™) 𝝇 = approximation factor for offline Set Cover 𝒐 = number of elements 𝒏 = number of sets 𝒍 = Size of the optimal Solution

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SLIDE 17

Results

Problem Approximation Constraints Query Complexity

Set Cover

π›½πœ + 1 𝛽 β‰₯ 2 𝑃 𝑛 π‘œ 𝑙

1 π›½βˆ’1 + π‘œπ‘™

𝜍 + 1 βˆ’ 𝑃 π‘›π‘œ 𝑙 𝛽 𝑙 ≀ π‘œ log 𝑛

1 4𝛽+1

Ξ© 𝑛 π‘œ 𝑙

1 2𝛽

𝛽 𝛽 ≀ 1.01 𝑙 = 𝑃(π‘œ/ log 𝑛) Ξ© π‘›π‘œ 𝑙

Cover Verification

βˆ’ 𝑙 ≀ π‘œ/2 Ξ©(π‘œπ‘™) 𝝇 = approximation factor for offline Set Cover 𝒐 = number of elements 𝒏 = number of sets 𝒍 = Size of the optimal Solution

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SLIDE 18

Results

Problem Approximation Constraints Query Complexity

Set Cover

π›½πœ + 1 𝛽 β‰₯ 2 𝑃 𝑛 π‘œ 𝑙

1 π›½βˆ’1 + π‘œπ‘™

𝜍 + 1 βˆ’ 𝑃 π‘›π‘œ 𝑙 𝛽 𝑙 ≀ π‘œ log 𝑛

1 4𝛽+1

Ξ© 𝑛 π‘œ 𝑙

1 2𝛽

𝛽 𝛽 ≀ 1.01 𝑙 = 𝑃(π‘œ/ log 𝑛) Ξ© π‘›π‘œ 𝑙

Cover Verification

βˆ’ 𝑙 ≀ π‘œ/2 Ξ©(π‘œπ‘™) 𝝇 = approximation factor for offline Set Cover 𝒐 = number of elements 𝒏 = number of sets 𝒍 = Size of the optimal Solution Cover Verification: given a set system, verify whether a given sub-collection of sets covers the universe.

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SLIDE 19

Results

Problem Approximation Constraints Query Complexity

Set Cover

π›½πœ + 1 𝛽 β‰₯ 2 𝑃 𝑛 π‘œ 𝑙

1 π›½βˆ’1 + π‘œπ‘™

𝜍 + 1 βˆ’ 𝑃 π‘›π‘œ 𝑙 𝛽 𝑙 ≀ π‘œ log 𝑛

1 4𝛽+1

Ξ© 𝑛 π‘œ 𝑙

1 2𝛽

𝛽 𝛽 ≀ 1.01 𝑙 = 𝑃(π‘œ/ log 𝑛) Ξ© π‘›π‘œ 𝑙

Cover Verification

βˆ’ 𝑙 ≀ π‘œ/2 Ξ©(π‘œπ‘™) 𝝇 = approximation factor for offline Set Cover 𝒐 = number of elements 𝒏 = number of sets 𝒍 = Size of the optimal Solution Cover Verification: given a set system, verify whether a given sub-collection of sets covers the universe.

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SLIDE 20

Results

Problem Approximation Constraints Query Complexity

Set Cover

π›½πœ + 1 𝛽 β‰₯ 2 𝑃 𝑛 π‘œ 𝑙

1 π›½βˆ’1 + π‘œπ‘™

𝜍 + 1 βˆ’ 𝑃 π‘›π‘œ 𝑙 𝛽 𝑙 ≀ π‘œ log 𝑛

1 4𝛽+1

Ξ© 𝑛 π‘œ 𝑙

1 2𝛽

𝛽 𝛽 ≀ 1.01 𝑙 = 𝑃(π‘œ/ log 𝑛) Ξ© π‘›π‘œ 𝑙

Cover Verification

βˆ’ 𝑙 ≀ π‘œ/2 Ξ©(π‘œπ‘™) 𝝇 = approximation factor for offline Set Cover 𝒐 = number of elements 𝒏 = number of sets 𝒍 = Size of the optimal Solution Cover Verification: given a set system, verify whether a given sub-collection of sets covers the universe.

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SLIDE 21

Results

Problem Approximation Constraints Query Complexity

Set Cover

π›½πœ + 1 𝛽 β‰₯ 2 𝑃 𝑛 π‘œ 𝑙

1 π›½βˆ’1 + π‘œπ‘™

𝜍 + 1 βˆ’ 𝑃 π‘›π‘œ 𝑙 𝛽 𝑙 ≀ π‘œ log 𝑛

1 4𝛽+1

Ξ© 𝑛 π‘œ 𝑙

1 2𝛽

𝛽 𝛽 ≀ 1.01 𝑙 = 𝑃(π‘œ/ log 𝑛) Ξ© π‘›π‘œ 𝑙

Cover Verification

βˆ’ 𝑙 ≀ π‘œ/2 Ξ©(π‘œπ‘™) 𝝇 = approximation factor for offline Set Cover 𝒐 = number of elements 𝒏 = number of sets 𝒍 = Size of the optimal Solution Cover Verification: given a set system, verify whether a given sub-collection of sets covers the universe.

slide-22
SLIDE 22

Results

Problem Approximation Constraints Query Complexity

Set Cover

π›½πœ + 1 𝛽 β‰₯ 2 𝑃 𝑛 π‘œ 𝑙

1 π›½βˆ’1 + π‘œπ‘™

𝜍 + 1 βˆ’ 𝑃 π‘›π‘œ 𝑙 𝛽 𝑙 ≀ π‘œ log 𝑛

1 4𝛽+1

Ξ© 𝑛 π‘œ 𝑙

1 2𝛽

𝛽 𝛽 ≀ 1.01 𝑙 = 𝑃(π‘œ/ log 𝑛) Ξ© π‘›π‘œ 𝑙

Cover Verification

βˆ’ 𝑙 ≀ π‘œ/2 Ξ©(π‘œπ‘™) 𝝇 = approximation factor for offline Set Cover 𝒐 = number of elements 𝒏 = number of sets 𝒍 = Size of the optimal Solution Cover Verification: given a set system, verify whether a given sub-collection of sets covers the universe.

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SLIDE 23

Part one: upper bound

Theorem: There exists an algorithm that with high probability finds an O(πœπ›½)-approximate cover which uses 𝑃(π’π’πŸ/𝜷 + 𝒐𝒍) number of queries.

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SLIDE 24

Part one: upper bound

Theorem: There exists an algorithm that with high probability finds an O(πœπ›½)-approximate cover which uses 𝑃(π’π’πŸ/𝜷 + 𝒐𝒍) number of queries.

  • 1. Two simple components used for coverage problems in massive data models.
  • Set Sampling
  • Element Sampling
  • 2. The algorithm overview
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SLIDE 25

Component I: set sampling

Set Sampling: After picking β„“ sets uniformly at random, all elements with degree at least

m log π‘œ β„“

are covered w.h.p.

  • We only need to worry about low degree elements.
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SLIDE 26

Component I: set sampling

Set Sampling: After picking β„“ sets uniformly at random, all elements with degree at least

m log π‘œ β„“

are covered w.h.p.

  • We only need to worry about low degree elements.

How we use the lemma: set β„“ = 𝑃(𝑙)

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SLIDE 27

Component I: set sampling

Set Sampling: After picking β„“ sets uniformly at random, all elements with degree at least

m log π‘œ β„“

are covered w.h.p.

  • We only need to worry about low degree elements.

1,6,3,9 10,6 3,2,7 1,2,5,4 6,9,7 8,2,9,10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 β„“ = 2

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SLIDE 28

Component I: set sampling

Set Sampling: After picking β„“ sets uniformly at random, all elements with degree at least

m log π‘œ β„“

are covered w.h.p.

  • We only need to worry about low degree elements.

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 β„“ = 2 Degrees: 2 3 2 1 1 3 2 1 3 2 1,6,3,9 10,6 3,2,7 1,2,5,4 6,9,7 8,2,9,10

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SLIDE 29

Component I: set sampling

Set Sampling: After picking β„“ sets uniformly at random, all elements with degree at least

m log π‘œ β„“

are covered w.h.p.

  • We only need to worry about low degree elements.

1,6,3,9 10,6 3,2,7 1,2,5,4 6,9,7 8,2,9,10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 β„“ = 2 Degrees: 2 3 2 1 1 3 2 1 3 2

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SLIDE 30

Component I: set sampling

Set Sampling: After picking β„“ sets uniformly at random, all elements with degree at least

m log π‘œ β„“

are covered w.h.p.

  • We only need to worry about low degree elements.

10,6 1,2,5,4 8,2,9,10 4 5 8 10 6,9,7 Degrees: 2 3 2 1 1 3 2 1 3 2

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SLIDE 31

Component II: element sampling

Element Sampling: Sample a few elements and solve the set cover for the sampled elements.

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SLIDE 32

Component II: element sampling

1,6,3,9 10,6 3,2,7 1,2,5,4 6,9,7 8,2,9,10 1 2 3 4 5 6 7 8 9 10

Element Sampling: Sample a few elements and solve the set cover for the sampled elements.

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SLIDE 33

Component II: element sampling

1,6,3,9 10,6 3,2,7 1,2,5,4 6,9,7 8,2,9,10 1 2 3 4 5 6 7 8 9 10

Element Sampling: Sample a few elements and solve the set cover for the sampled elements.

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SLIDE 34

Component II: element sampling

1,6,3,9 10,6 3,2,7 1,2,5,4 6,9,7 8,2,9,10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1,3 10 3,7 1 10 7

Element Sampling: Sample a few elements and solve the set cover for the sampled elements.

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SLIDE 35

Component II: element sampling

1,3 10 3,7 1 10 7 1,6,3,9 10,6 3,2,7 1,2,5,4 6,9,7 8,2,9,10 1 2 3 4 5 6 7 8 9 10

Element Sampling: Sample a few elements and solve the set cover for the sampled elements.

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SLIDE 36

Component II: element sampling

1,6,3,9 10,6 3,2,7 1,2,5,4 6,9,7 8,2,9,10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1,3 10 3,7 1 10 7

Element Sampling: Sample a few elements and solve the set cover for the sampled elements.

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SLIDE 37

Component II: element sampling

1,6,3,9 10,6 3,2,7 1,2,5,4 6,9,7 8,2,9,10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

Element Sampling: Sample a few elements and solve the set cover for the sampled elements.

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SLIDE 38

Component II: element sampling

10,6 1,2,5,4 6,9,7 4 5

Element Sampling: Sample a few elements and solve the set cover for the sampled elements.

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SLIDE 39

Component II: element sampling

Element Sampling: Sampling Θ(

πœπ‘™ log 𝑛 πœ€

) elements uniformly at random and finding a 𝜍-approximate cover for the sampled elements, will cover (1 βˆ’ πœ€) fraction of the original elements w.h.p.

10,6 1,2,5,4 6,9,7 4 5

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SLIDE 40

Algorithm

Make a guess β„“ of the value of the optimal solution 𝑙

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SLIDE 41

Algorithm

Make a guess β„“ of the value of the optimal solution 𝑙

log π‘œ different guesses β„“ ∈ {1,2,4, … , π‘œ}

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SLIDE 42

Algorithm

Make a guess β„“ of the value of the optimal solution 𝑙  Preprocessing: perform set sampling  Sol ← sampled sets

log π‘œ different guesses β„“ ∈ {1,2,4, … , π‘œ}

slide-43
SLIDE 43

Algorithm

Make a guess β„“ of the value of the optimal solution 𝑙  Preprocessing: perform set sampling  Sol ← sampled sets

log π‘œ different guesses β„“ ∈ {1,2,4, … , π‘œ} sample β„“ sets, number of queries: π‘œβ„“ Set Sampling: After picking β„“ sets uniformly at random, all elements with degree at least

m log π‘œ β„“

are covered w.h.p.

slide-44
SLIDE 44

Algorithm

Make a guess β„“ of the value of the optimal solution 𝑙  Preprocessing: perform set sampling  Sol ← sampled sets  For 𝛽 iterations

  • Use element sampling to cover (1 βˆ’

1 π‘œ1/𝛽)-

fraction of the uncovered elements.

  • Add the sets to Sol

log π‘œ different guesses β„“ ∈ {1,2,4, … , π‘œ} sample β„“ sets, number of queries: π‘œβ„“

slide-45
SLIDE 45

Algorithm

Make a guess β„“ of the value of the optimal solution 𝑙  Preprocessing: perform set sampling  Sol ← sampled sets  For 𝛽 iterations

  • Use element sampling to cover (1 βˆ’

1 π‘œ1/𝛽)-

fraction of the uncovered elements.

  • Add the sets to Sol

log π‘œ different guesses β„“ ∈ {1,2,4, … , π‘œ} sample β„“ sets, number of queries: π‘œβ„“ Element Sampling: Sampling Θ(πœπ‘™ log 𝑛

πœ€

) elements uniformly at random and finding a 𝜍- approximate cover for the sampled elements, will cover (1 βˆ’ πœ€) fraction of the original elements w.h.p. 𝜺 = 𝟐/π’πŸ/𝜷

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SLIDE 46

Algorithm

Make a guess β„“ of the value of the optimal solution 𝑙  Preprocessing: perform set sampling  Sol ← sampled sets  For 𝛽 iterations

  • Use element sampling to cover (1 βˆ’

1 π‘œ1/𝛽)-

fraction of the uncovered elements.

  • Add the sets to Sol

log π‘œ different guesses β„“ ∈ {1,2,4, … , π‘œ} sample β„“ sets, number of queries: π‘œβ„“ sample (πœβ„“π‘œ1/𝛽 log 𝑛) elements, number of queries: 𝑃 πœβ„“π‘œ1/𝛽 log 𝑛

𝑛 log π‘œ β„“

=𝑃(πœπ‘›π‘œ1/𝛽 log 𝑛 log π‘œ) Element Sampling: Sampling Θ(πœπ‘™ log 𝑛

πœ€

) elements uniformly at random and finding a 𝜍- approximate cover for the sampled elements, will cover (1 βˆ’ πœ€) fraction of the original elements w.h.p. 𝜺 = 𝟐/π’πŸ/𝜷

slide-47
SLIDE 47

Algorithm

Make a guess β„“ of the value of the optimal solution 𝑙  Preprocessing: perform set sampling  Sol ← sampled sets  For 𝛽 iterations

  • Use element sampling to cover (1 βˆ’

1 π‘œ1/𝛽)-

fraction of the uncovered elements.

  • Add the sets to Sol
  • Update uncovered elements.

log π‘œ different guesses β„“ ∈ {1,2,4, … , π‘œ} sample β„“ sets, number of queries: π‘œβ„“ sample (πœβ„“π‘œ1/𝛽 log 𝑛) elements, number of queries: 𝑃 πœβ„“π‘œ1/𝛽 log 𝑛

𝑛 log π‘œ β„“

=𝑃(πœπ‘›π‘œ1/𝛽 log 𝑛 log π‘œ)

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SLIDE 48

Algorithm

Make a guess β„“ of the value of the optimal solution 𝑙  Preprocessing: perform set sampling  Sol ← sampled sets  For 𝛽 iterations

  • Use element sampling to cover (1 βˆ’

1 π‘œ1/𝛽)-

fraction of the uncovered elements.

  • Add the sets to Sol
  • Update uncovered elements.

log π‘œ different guesses β„“ ∈ {1,2,4, … , π‘œ} sample β„“ sets, number of queries: π‘œβ„“ number of queries: πœπ‘œβ„“ sample (πœβ„“π‘œ1/𝛽 log 𝑛) elements, number of queries: 𝑃 πœβ„“π‘œ1/𝛽 log 𝑛

𝑛 log π‘œ β„“

=𝑃(πœπ‘›π‘œ1/𝛽 log 𝑛 log π‘œ)

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SLIDE 49

Algorithm

Make a guess β„“ of the value of the optimal solution 𝑙  Preprocessing: perform set sampling  Sol ← sampled sets  For 𝛽 iterations

  • Use element sampling to cover (1 βˆ’

1 π‘œ1/𝛽)-

fraction of the uncovered elements.

  • Add the sets to Sol
  • Update uncovered elements.

 If all elements are covered, report Sol

log π‘œ different guesses β„“ ∈ {1,2,4, … , π‘œ} sample β„“ sets, number of queries: π‘œβ„“ number of queries: πœπ‘œβ„“ sample (πœβ„“π‘œ1/𝛽 log 𝑛) elements, number of queries: 𝑃 πœβ„“π‘œ1/𝛽 log 𝑛

𝑛 log π‘œ β„“

=𝑃(πœπ‘›π‘œ1/𝛽 log 𝑛 log π‘œ)

slide-50
SLIDE 50

Algorithm

Make a guess β„“ of the value of the optimal solution 𝑙  Preprocessing: perform set sampling  Sol ← sampled sets  For 𝛽 iterations

  • Use element sampling to cover (1 βˆ’

1 π‘œ1/𝛽)-

fraction of the uncovered elements.

  • Add the sets to Sol
  • Update uncovered elements.

 If all elements are covered, report Sol

log π‘œ different guesses β„“ ∈ {1,2,4, … , π‘œ} sample β„“ sets, number of queries: π‘œβ„“ number of queries: πœπ‘œβ„“ sample (πœβ„“π‘œ1/𝛽 log 𝑛) elements, number of queries: 𝑃 πœβ„“π‘œ1/𝛽 log 𝑛

𝑛 log π‘œ β„“

=𝑃(πœπ‘›π‘œ1/𝛽 log 𝑛 log π‘œ)

Theorem: There exists an algorithm that with high probability finds an O(πœπ›½)-approximate cover which uses 𝑃(π’π’πŸ/𝜷 + 𝒐𝒍) number of queries.

slide-51
SLIDE 51

Results

Problem Approximation Constraints Query Complexity

Set Cover

π›½πœ + 1 𝛽 β‰₯ 2 𝑃 𝑛 π‘œ 𝑙

1 π›½βˆ’1 + π‘œπ‘™

𝜍 + 1 βˆ’ 𝑃 π‘›π‘œ 𝑙 𝛽 𝑙 ≀ π‘œ log 𝑛

1 4𝛽+1

Ξ© 𝑛 π‘œ 𝑙

1 2𝛽

𝛽 𝛽 ≀ 1.01 𝑙 = 𝑃(π‘œ/ log 𝑛) Ξ© π‘›π‘œ 𝑙

Cover Verification

βˆ’ 𝑙 ≀ π‘œ/2 Ξ©(π‘œπ‘™) 𝝇 = approximation factor for offline Set Cover 𝒐 = number of elements 𝒏 = number of sets 𝒍 = Size of the optimal Solution Cover Verification: given a set system, verify whether a given sub-collection of sets covers the universe.

slide-52
SLIDE 52

Part two: lower bound

Theorem: Any randomized algorithm that with probability at least 2/3 distinguishes whether the minimum Set Cover size is 2 or at least 3 requires 𝛁(𝒏𝒐) number of queries.

slide-53
SLIDE 53

High Level Approach

  • 1. Construct a median instance π½βˆ—
  • Minimum Set Cover Size is 3
slide-54
SLIDE 54

High Level Approach

  • 1. Construct a median instance π½βˆ—
  • Minimum Set Cover Size is 3
  • 2. Randomized Procedure on π½βˆ— to get a modified instance 𝐽
  • Minimum Set Cover Size is 2
  • π½βˆ— and 𝐽 only differ in a few positions
  • The differences are distributed almost uniformly at

random

slide-55
SLIDE 55

High Level Approach

  • 1. Construct a median instance π½βˆ—
  • Minimum Set Cover Size is 3
  • 2. Randomized Procedure on π½βˆ— to get a modified instance 𝐽
  • Minimum Set Cover Size is 2
  • π½βˆ— and 𝐽 only differ in a few positions
  • The differences are distributed almost uniformly at

random

  • 3. Any algorithm that can detect these two cases requires to

query at least Ξ©(π‘›π‘œ) queries.

slide-56
SLIDE 56

The Median Instance

Construction: is randomized. For every 𝑇, 𝑓 the set 𝑇 contains 𝑓 with probability 1 βˆ’ π‘ž0 where π‘ž0 =

9 log 𝑛 π‘œ

slide-57
SLIDE 57

The Median Instance

Construction: is randomized. For every 𝑇, 𝑓 the set 𝑇 contains 𝑓 with probability 1 βˆ’ π‘ž0 where π‘ž0 =

9 log 𝑛 π‘œ

Properties: by Chernoff, most of such instances have the following properties:

  • 1. No 2 sets cover all the elements
  • 2. For any two sets the number of uncovered elements is 𝑃 log 𝑛
  • 3. The intersection is at least Ξ©(π‘œ)
  • 4. For each element, the number of sets not covering it is at most 6𝑛

log 𝑛 π‘œ

  • 5. For any pair of elements the number of sets containing only the first element is

at least

𝑛 9 log 𝑛 4βˆšπ‘œ

  • 6. For any three sets, the number of elements in the first two but not in the third
  • ne is at least 6 π‘œ log 𝑛
slide-58
SLIDE 58

The Median Instance

Construction: is randomized. For every 𝑇, 𝑓 the set 𝑇 contains 𝑓 with probability 1 βˆ’ π‘ž0 where π‘ž0 =

9 log 𝑛 π‘œ

Properties: by Chernoff, most of such instances have the following properties:

  • 1. No 2 sets cover all the elements
  • 2. For any two sets the number of uncovered elements is 𝑃 log 𝑛
  • 3. The intersection is at least Ξ©(π‘œ)
  • 4. For each element, the number of sets not covering it is at most 6𝑛

log 𝑛 π‘œ

  • 5. For any pair of elements the number of sets containing only the first element is

at least

𝑛 9 log 𝑛 4βˆšπ‘œ

  • 6. For any three sets, the number of elements in the first two but not in the third
  • ne is at least 6 π‘œ log 𝑛
slide-59
SLIDE 59

The Median Instance

Construction: is randomized. For every 𝑇, 𝑓 the set 𝑇 contains 𝑓 with probability 1 βˆ’ π‘ž0 where π‘ž0 =

9 log 𝑛 π‘œ

Properties: by Chernoff, most of such instances have the following properties:

Take one such instance π½βˆ— with the above properties

  • 1. No 2 sets cover all the elements
  • 2. For any two sets the number of uncovered elements is 𝑃 log 𝑛
  • 3. The intersection is at least Ξ©(π‘œ)
  • 4. For each element, the number of sets not covering it is at most 6𝑛

log 𝑛 π‘œ

  • 5. For any pair of elements the number of sets containing only the first element is

at least

𝑛 9 log 𝑛 4βˆšπ‘œ

  • 6. For any three sets, the number of elements in the first two but not in the third
  • ne is at least 6 π‘œ log 𝑛
slide-60
SLIDE 60

The Median Instance

Sets Elements

𝑓 ∈ 𝑇 𝑓 βˆ‰ 𝑇

slide-61
SLIDE 61

Generating a Modified Instance

Pick two random sets 𝑇1 and 𝑇2 and turn them into a set cover. How? 𝑽 = π’‡πŸ, π’‡πŸ‘, π’‡πŸ’, π’‡πŸ“ π‘»πŸ = π’‡πŸ‘, π’‡πŸ’ π‘»πŸ‘ = π’‡πŸ‘, π’‡πŸ“

slide-62
SLIDE 62

Generating a Modified Instance

Pick two random sets 𝑇1 and 𝑇2 and turn them into a set cover. How?

  • For each uncovered element 𝑓1 ∈ 𝑉 βˆ– 𝑇1 βˆͺ 𝑇2 ,
  • Add 𝑓1 to 𝑇2

𝑽 = π’‡πŸ, π’‡πŸ‘, π’‡πŸ’, π’‡πŸ“ π‘»πŸ = π’‡πŸ‘, π’‡πŸ’ π‘»πŸ‘ = π’‡πŸ‘, π’‡πŸ“ π’‡πŸ

slide-63
SLIDE 63

Generating a Modified Instance

Pick two random sets 𝑇1 and 𝑇2 and turn them into a set cover. How?

  • For each uncovered element 𝑓1 ∈ 𝑉 βˆ– 𝑇1 βˆͺ 𝑇2 ,
  • Add 𝑓1 to 𝑇2
  • Remove an element 𝑓2 ∈ 𝑇2 ∩ 𝑇1 from 𝑇2

𝑽 = π’‡πŸ, π’‡πŸ‘, π’‡πŸ’, π’‡πŸ“ π‘»πŸ = π’‡πŸ‘, π’‡πŸ’ π‘»πŸ‘ = π’‡πŸ‘, π’‡πŸ“ π’‡πŸ π’‡πŸ‘

slide-64
SLIDE 64

Generating a Modified Instance

Pick two random sets 𝑇1 and 𝑇2 and turn them into a set cover. How?

  • For each uncovered element 𝑓1 ∈ 𝑉 βˆ– 𝑇1 βˆͺ 𝑇2 ,
  • Add 𝑓1 to 𝑇2
  • Remove an element 𝑓2 ∈ 𝑇2 ∩ 𝑇1 from 𝑇2
  • Pick a random set 𝑇3 that contains 𝑓1 but not 𝑓2

𝑽 = π’‡πŸ, π’‡πŸ‘, π’‡πŸ’, π’‡πŸ“ π‘»πŸ = π’‡πŸ‘, π’‡πŸ’ π‘»πŸ‘ = π’‡πŸ‘, π’‡πŸ“ π‘»πŸ’ = π’‡πŸ“, π’‡πŸ

slide-65
SLIDE 65

Generating a Modified Instance

Pick two random sets 𝑇1 and 𝑇2 and turn them into a set cover. How?

  • For each uncovered element 𝑓1 ∈ 𝑉 βˆ– 𝑇1 βˆͺ 𝑇2 ,
  • Add 𝑓1 to 𝑇2
  • Remove an element 𝑓2 ∈ 𝑇2 ∩ 𝑇1 from 𝑇2
  • Pick a random set 𝑇3 that contains 𝑓1 but not 𝑓2
  • 𝑇2 and 𝑇3 swap 𝑓1 and 𝑓2

𝑽 = π’‡πŸ, π’‡πŸ‘, π’‡πŸ’, π’‡πŸ“ π‘»πŸ = π’‡πŸ‘, π’‡πŸ’ π‘»πŸ‘ = π’‡πŸ, π’‡πŸ“ π‘»πŸ’ = π’‡πŸ“, π’‡πŸ‘ π‘»πŸ = π’‡πŸ‘, π’‡πŸ’ π‘»πŸ‘ = π’‡πŸ‘, π’‡πŸ“ π‘»πŸ’ = π’‡πŸ“, π’‡πŸ

slide-66
SLIDE 66

Generating a Modified Instance

Pick two random sets 𝑇1 and 𝑇2 and turn them into a set cover. How?

  • For each uncovered element 𝑓1 ∈ 𝑉 βˆ– 𝑇1 βˆͺ 𝑇2 ,
  • Add 𝑓1 to 𝑇2
  • Remove an element 𝑓2 ∈ 𝑇2 ∩ 𝑇1 from 𝑇2
  • Pick a random set 𝑇3 that contains 𝑓1 but not 𝑓2
  • 𝑇2 and 𝑇3 swap 𝑓1 and 𝑓2

𝑽 = π’‡πŸ, π’‡πŸ‘, π’‡πŸ’, π’‡πŸ“ π‘»πŸ = π’‡πŸ‘, π’‡πŸ’ π‘»πŸ‘ = π’‡πŸ, π’‡πŸ“ π‘»πŸ’ = π’‡πŸ“, π’‡πŸ‘ π‘»πŸ = π’‡πŸ‘, π’‡πŸ’ π‘»πŸ‘ = π’‡πŸ‘, π’‡πŸ“ π‘»πŸ’ = π’‡πŸ“, π’‡πŸ

Modified instance Swap

slide-67
SLIDE 67

Generating a Modified Instance

Pick two random sets 𝑇1 and 𝑇2 and turn them into a set cover. How?

  • For each uncovered element 𝑓1 ∈ 𝑉 βˆ– 𝑇1 βˆͺ 𝑇2 ,
  • Add 𝑓1 to 𝑇2
  • Remove an element 𝑓2 ∈ 𝑇2 ∩ 𝑇1 from 𝑇2
  • Pick a random set 𝑇3 that contains 𝑓1 but not 𝑓2
  • 𝑇2 and 𝑇3 swap 𝑓1 and 𝑓2

𝑽 = π’‡πŸ, π’‡πŸ‘, π’‡πŸ’, π’‡πŸ“ π‘»πŸ = π’‡πŸ‘, π’‡πŸ’ π‘»πŸ‘ = π’‡πŸ, π’‡πŸ“ π‘»πŸ’ = π’‡πŸ“, π’‡πŸ‘ π‘»πŸ = π’‡πŸ‘, π’‡πŸ’ π‘»πŸ‘ = π’‡πŸ‘, π’‡πŸ“ π‘»πŸ’ = π’‡πŸ“, π’‡πŸ

Only four positions changes in the query access model.

Modified instance Swap

slide-68
SLIDE 68

Generating a Modified Instance

Pick two random sets 𝑇1 and 𝑇2 and turn them into a set cover. How?

  • For each uncovered element 𝑓1 ∈ 𝑉 βˆ– 𝑇1 βˆͺ 𝑇2 ,
  • Add 𝑓1 to 𝑇2
  • Remove an element 𝑓2 ∈ 𝑇2 ∩ 𝑇1 from 𝑇2
  • Pick a random set 𝑇3 that contains 𝑓1 but not 𝑓2
  • 𝑇2 and 𝑇3 swap 𝑓1 and 𝑓2

𝑽 = π’‡πŸ, π’‡πŸ‘, π’‡πŸ’, π’‡πŸ“ π‘»πŸ = π’‡πŸ‘, π’‡πŸ’ π‘»πŸ‘ = π’‡πŸ, π’‡πŸ“ π‘»πŸ’ = π’‡πŸ“, π’‡πŸ‘ π‘»πŸ = π’‡πŸ‘, π’‡πŸ’ π‘»πŸ‘ = π’‡πŸ‘, π’‡πŸ“ π‘»πŸ’ = π’‡πŸ“, π’‡πŸ

Only four positions changes in the query access model.

Modified instance Swap

Two in ElemOf oracles + Two in SetOf oracles

slide-69
SLIDE 69
  • Median Instance
  • Pick two Sets

Uniformly at Random 𝑇1 𝑇2

The Randomized Procedure

slide-70
SLIDE 70
  • Median Instance
  • Pick two Sets

Uniformly at Random

  • Find the elements

that are not covered

The Randomized Procedure

𝑇1 𝑇2

slide-71
SLIDE 71
  • Median Instance
  • Pick two Sets

Uniformly at Random

  • Find the elements that

are not covered

  • Also find the

elements that are covered by both

The Randomized Procedure

𝑇1 𝑇2

slide-72
SLIDE 72
  • Median Instance
  • Pick two Sets

Uniformly at Random

  • Find the elements that

are not covered

  • Also find the elements

that are covered by both

  • Assign one element in

the intersection to each uncovered element

The Randomized Procedure

𝑇1 𝑇2

slide-73
SLIDE 73
  • Median Instance
  • Pick two Sets

Uniformly at Random

  • Find the elements that

are not covered

  • Also find the elements

that are covered by both

  • Assign one element in

the intersection to each uncovered element

The Randomized Procedure

𝑇1 𝑇2 𝑇1 𝑇2

slide-74
SLIDE 74
  • Median Instance
  • Pick two Sets

Uniformly at Random

  • Find the elements that

are not covered

  • Also find the elements

that are covered by both

  • Assign one element in

the intersection to each uncovered element

The Randomized Procedure

𝑇1 𝑇2 𝑓1 𝑓2

slide-75
SLIDE 75
  • Median Instance
  • Pick two Sets

Uniformly at Random

  • Find the elements that

are not covered

  • Also find the elements

that are covered by both

  • Assign one element in

the intersection to each uncovered element

  • In iteration:
  • Find a candidate

set

The Randomized Procedure

𝑇1 𝑇2 𝑓1 𝑓2

slide-76
SLIDE 76
  • Median Instance
  • Pick two Sets

Uniformly at Random

  • Find the elements that

are not covered

  • Also find the elements

that are covered by both

  • Assign one element in

the intersection to each uncovered element

  • In iteration:
  • Find a candidate

set

  • swap

The Randomized Procedure

𝑇1 𝑇2 𝑓1 𝑓2

slide-77
SLIDE 77
  • Median Instance
  • Pick two Sets

Uniformly at Random

  • Find the elements that

are not covered

  • Also find the elements

that are covered by both

  • Assign one element in

the intersection to each uncovered element

  • In iteration:
  • Find a candidate

set

  • swap

The Randomized Procedure

𝑇1 𝑇2 𝑓1 𝑓2

slide-78
SLIDE 78
  • Median Instance
  • Pick two Sets

Uniformly at Random

  • Find the elements that

are not covered

  • Also find the elements

that are covered by both

  • Assign one element in

the intersection to each uncovered element

  • In iteration:
  • Find a candidate

set

  • swap

The Randomized Procedure

𝑇1 𝑇2 𝑓1 𝑓2

slide-79
SLIDE 79
  • Median Instance
  • Pick two Sets

Uniformly at Random

  • Find the elements that

are not covered

  • Also find the elements

that are covered by both

  • Assign one element in

the intersection to each uncovered element

  • In iteration:
  • Find a candidate

set

  • swap

The Randomized Procedure

𝑇1 𝑇2 𝑓1 𝑓2

slide-80
SLIDE 80
  • Median Instance
  • Pick two Sets

Uniformly at Random

  • Find the elements that

are not covered

  • Also find the elements

that are covered by both

  • Assign one element in

the intersection to each uncovered element

  • In iteration:
  • Find a candidate

set

  • swap

The Randomized Procedure

𝑇1 𝑇2 𝑓1 𝑓2

slide-81
SLIDE 81
  • Median Instance
  • Pick two Sets

Uniformly at Random

  • Find the elements that

are not covered

  • Also find the elements

that are covered by both

  • Assign one element in

the intersection to each uncovered element

  • In iteration:
  • Find a candidate

set

  • swap

The Randomized Procedure

𝑇1 𝑇2 𝑓1 𝑓2

slide-82
SLIDE 82
  • Median Instance
  • Pick two Sets

Uniformly at Random

  • Find the elements that

are not covered

  • Also find the elements

that are covered by both

  • Assign one element in

the intersection to each uncovered element

  • In iteration:
  • Find a candidate

set

  • swap

The Randomized Procedure

𝑇1 𝑇2

slide-83
SLIDE 83
  • Median Instance
  • Pick two Sets

Uniformly at Random

  • Find the elements that

are not covered

  • Also find the elements

that are covered by both

  • Assign one element in

the intersection to each uncovered element

  • In iteration:
  • Find a candidate

set

  • swap

The Randomized Procedure

𝑇1 𝑇2

slide-84
SLIDE 84
  • Median Instance
  • Pick two Sets

Uniformly at Random

  • Find the elements that

are not covered

  • Also find the elements

that are covered by both

  • Assign one element in

the intersection to each uncovered element

  • In iteration:
  • Find a candidate

set

  • swap

The Randomized Procedure

𝑇1 𝑇2

  • By Property 2 of median instance:
  • the total number of uncovered elements is

𝑃 log 𝑛

  • Thus in total only 𝑃 log 𝑛 positions have

changed.

slide-85
SLIDE 85

Overall Argument

Lemma: For any element 𝑓 and any set 𝑇, the probability that pair participate in a swap is almost uniform, i.e., 𝑃(

log 𝑛 π‘›π‘œ ).

  • Using other properties of the median instances

Input:

  • W.p. Β½ the input is the median instance π½βˆ—
  • W.p. Β½ the input is a randomly generated modified instance 𝐽
slide-86
SLIDE 86

Overall Argument

Lemma: For any element 𝑓 and any set 𝑇, the probability that pair participate in a swap is almost uniform, i.e., 𝑃(

log 𝑛 π‘›π‘œ ).

  • Using other properties of the median instances

Input:

  • W.p. Β½ the input is the median instance π½βˆ—
  • W.p. Β½ the input is a randomly generated modified instance 𝐽

Theorem: Any randomized algorithm that with probability at least 2/3 distinguishes whether the minimum Set Cover size is 2 or at least 3 requires 𝛁(𝒏𝒐) number of queries.

slide-87
SLIDE 87

Open Problems

  • Prove a lower bound of Ξ©(π‘œπ‘™) for the set cover problem as well

Problem Approximation Query Complexity

Constraints

Set Cover

π›½πœ + 1 𝑃 𝑛 π‘œ 𝑙

1 π›½βˆ’1 + π‘œπ‘™

𝛽 β‰₯ 2

𝜍 + 1 𝑃 π‘›π‘œ 𝑙

βˆ’

𝛽 Ξ© 𝑛 π‘œ 𝑙

1 2𝛽

𝑙 ≀ π‘œ log 𝑛

1 4𝛽+1

𝛽 Ξ© π‘›π‘œ 𝑙

𝛽 ≀ 1.01 𝑙 = 𝑃(π‘œ/ log 𝑛)

Cover Verification

βˆ’ Ξ©(π‘œπ‘™)

𝑙 ≀ π‘œ/2

slide-88
SLIDE 88

Open Problems

  • Prove a lower bound of Ξ©(π‘œπ‘™) for the set cover problem as well
  • Similar results for the weighted set cover?

Problem Approximation Query Complexity

Constraints

Set Cover

π›½πœ + 1 𝑃 𝑛 π‘œ 𝑙

1 π›½βˆ’1 + π‘œπ‘™

𝛽 β‰₯ 2

𝜍 + 1 𝑃 π‘›π‘œ 𝑙

βˆ’

𝛽 Ξ© 𝑛 π‘œ 𝑙

1 2𝛽

𝑙 ≀ π‘œ log 𝑛

1 4𝛽+1

𝛽 Ξ© π‘›π‘œ 𝑙

𝛽 ≀ 1.01 𝑙 = 𝑃(π‘œ/ log 𝑛)

Cover Verification

βˆ’ Ξ©(π‘œπ‘™)

𝑙 ≀ π‘œ/2

slide-89
SLIDE 89
  • Prove a lower bound of Ξ©(π‘œπ‘™) for the set cover problem as well
  • Similar results for the weighted set cover?

Open Problems

Problem Approximation Query Complexity

Constraints

Set Cover

π›½πœ + 1 𝑃 𝑛 π‘œ 𝑙

1 π›½βˆ’1 + π‘œπ‘™

𝛽 β‰₯ 2

𝜍 + 1 𝑃 π‘›π‘œ 𝑙

βˆ’

𝛽 Ξ© 𝑛 π‘œ 𝑙

1 2𝛽

𝑙 ≀ π‘œ log 𝑛

1 4𝛽+1

𝛽 Ξ© π‘›π‘œ 𝑙

𝛽 ≀ 1.01 𝑙 = 𝑃(π‘œ/ log 𝑛)

Cover Verification

βˆ’ Ξ©(π‘œπ‘™)

𝑙 ≀ π‘œ/2