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

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 ℱ of sets 𝑇 1 , … , 𝑇 𝑛 , each a subset of 𝒱 = {1, … , 𝑜} 4 2 5 1 3

Set Cover Problem Input: Collection ℱ of sets 𝑇 1 , … , 𝑇 𝑛 , each a subset of 𝒱 = {1, … , 𝑜} 4 Output: a subset 𝒟 of ℱ such that: • 𝒟 covers 𝒱 2 • | 𝒟 | is minimized 5 1 3

Set Cover Problem Input: Collection ℱ of sets 𝑇 1 , … , 𝑇 𝑛 , each a subset of 𝒱 = {1, … , 𝑜} 4 Output: a subset 𝒟 of ℱ such that: • 𝒟 covers 𝒱 2 • | 𝒟 | is minimized 5 1 Complexity : • NP-hard 3

Set Cover Problem Input: Collection ℱ of sets 𝑇 1 , … , 𝑇 𝑛 , each a subset of 𝒱 = {1, … , 𝑜} 4 Output: a subset 𝒟 of ℱ such that: • 𝒟 covers 𝒱 2 • | 𝒟 | is minimized 5 1 Complexity : • NP-hard 3 • Greedy (ln 𝑜) -approximation algorithm

Set Cover Problem Input: Collection ℱ of sets 𝑇 1 , … , 𝑇 𝑛 , each a subset of 𝒱 = {1, … , 𝑜} 4 Output: a subset 𝒟 of ℱ such that: • 𝒟 covers 𝒱 2 • | 𝒟 | is minimized 5 1 Complexity : • NP-hard 3 • Greedy (ln 𝑜) -approximation algorithm • Can’t do better unless P=NP [LY91][RS97][Fei98][AMS06][DS14]

Set Cover Problem Input: Collection ℱ of sets 𝑇 1 , … , 𝑇 𝑛 , each a subset of 𝒱 = {1, … , 𝑜} 4 Output: a subset 𝒟 of ℱ such that: • 𝒟 covers 𝒱 2 • | 𝒟 | is minimized 5 1 Complexity : • NP-hard 3 • Greedy (ln 𝑜) -approximation algorithm • Can’t do better unless P=NP [LY91][RS97][Fei98][AMS06][DS14] “Is it possible to solve minimum set cover in sub-linear time ?”

Sub-linear Time Set Cover Data Access Model ?

Sub-linear Time Set Cover Data Access Model [NO’08,YYI’12] EltOf (𝑻, 𝒋) : 𝑗 th element in 𝑻 SetOf (𝒇, 𝒌) : 𝑘 th set containing 𝒇

Sub-linear Time Set Cover Data Access Model [NO’08,YYI’12] EltOf (𝑻, 𝒋) : 𝑗 th element in 𝑻 • No assumption on the order SetOf (𝒇, 𝒌) : 𝑘 th set containing 𝒇 • Incidence list in (sub-linear) algorithms for graphs

Sub-linear Time Set Cover Data Access Model [NO’08,YYI’12] EltOf (𝑻, 𝒋) : 𝑗 th element in 𝑻 • No assumption on the order SetOf (𝒇, 𝒌) : 𝑘 th set containing 𝒇 • Incidence list in (sub-linear) algorithms for graphs • Sublinear in 𝒏𝒐

Sub-linear Time Set Cover Data Access Model [NO’08,YYI’12] EltOf (𝑻, 𝒋) : 𝑗 th element in 𝑻 • No assumption on the order SetOf (𝒇, 𝒌) : 𝑘 th set containing 𝒇 • Incidence list in (sub-linear) algorithms for graphs • Sublinear in 𝒏𝒐 Prior Results

Sub-linear Time Set Cover Data Access Model [NO’08,YYI’12] EltOf (𝑻, 𝒋) : 𝑗 th element in 𝑻 • No assumption on the order SetOf (𝒇, 𝒌) : 𝑘 th set containing 𝒇 • 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

Sub-linear Time Set Cover Data Access Model [NO’08,YYI’12] EltOf (𝑻, 𝒋) : 𝑗 th element in 𝑻 • No assumption on the order SetOf (𝒇, 𝒌) : 𝑘 th set containing 𝒇 • 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 + 𝜗) -approximate fractional solution , then perform randomized rounding to achieve 𝑃(log 𝑜) -approximation

Sub-linear Time Set Cover Data Access Model [NO’08,YYI’12] EltOf (𝑻, 𝒋) : 𝑗 th element in 𝑻 • No assumption on the order SetOf (𝒇, 𝒌) : 𝑘 th set containing 𝒇 • 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 + 𝜗) -approximate fractional solution , then perform randomized rounding to achieve 𝑃(log 𝑜) -approximation  𝑃(𝑛𝑙 2 + 𝑜𝑙 2 ) (can be improved to 𝑃(𝑛 + 𝑜𝑙) ) 𝒐 = number of elements 𝒏 = number of sets 𝒍 = size of the optimal solution

Results Problem Approximation Constraints Query Complexity 1 𝑜 𝛽−1 + 𝑜𝑙 𝛽𝜍 + 1 𝛽 ≥ 2 𝑃 𝑛 𝑙 𝑃 𝑛𝑜 𝜍 + 1 − 𝑙 Set Cover 1 1 𝑜 4𝛽+1 𝑜 2𝛽 𝛽 𝑙 ≤ Ω 𝑛 log 𝑛 𝑙 Ω 𝑛𝑜 𝛽 ≤ 1.01 𝛽 𝑙 = 𝑃(𝑜/ log 𝑛) 𝑙 Cover Verification − 𝑙 ≤ 𝑜/2 Ω(𝑜𝑙) 𝝇 = approximation factor for offline Set Cover 𝒐 = number of elements 𝒏 = number of sets 𝒍 = Size of the optimal Solution

Results Problem Approximation Constraints Query Complexity 1 𝑜 𝛽−1 + 𝑜𝑙 𝛽𝜍 + 1 𝛽 ≥ 2 𝑃 𝑛 𝑙 𝑃 𝑛𝑜 𝜍 + 1 − 𝑙 Set Cover 1 1 𝑜 4𝛽+1 𝑜 2𝛽 𝛽 𝑙 ≤ Ω 𝑛 log 𝑛 𝑙 Ω 𝑛𝑜 𝛽 ≤ 1.01 𝛽 𝑙 = 𝑃(𝑜/ log 𝑛) 𝑙 Cover Verification − 𝑙 ≤ 𝑜/2 Ω(𝑜𝑙) 𝝇 = approximation factor for offline Set Cover 𝒐 = number of elements 𝒏 = number of sets 𝒍 = Size of the optimal Solution

Results Problem Approximation Constraints Query Complexity 1 𝑜 𝛽−1 + 𝑜𝑙 𝛽𝜍 + 1 𝛽 ≥ 2 𝑃 𝑛 𝑙 𝑃 𝑛𝑜 𝜍 + 1 − 𝑙 Set Cover 1 1 𝑜 4𝛽+1 𝑜 2𝛽 𝛽 𝑙 ≤ Ω 𝑛 log 𝑛 𝑙 Ω 𝑛𝑜 𝛽 ≤ 1.01 𝛽 𝑙 = 𝑃(𝑜/ log 𝑛) 𝑙 Cover Verification − 𝑙 ≤ 𝑜/2 Ω(𝑜𝑙) Cover Verification : given a set system, verify whether a given sub-collection of sets covers the universe. 𝝇 = approximation factor for offline Set Cover 𝒐 = number of elements 𝒏 = number of sets 𝒍 = Size of the optimal Solution

Results Problem Approximation Constraints Query Complexity 1 𝑜 𝛽−1 + 𝑜𝑙 𝛽𝜍 + 1 𝛽 ≥ 2 𝑃 𝑛 𝑙 𝑃 𝑛𝑜 𝜍 + 1 − 𝑙 Set Cover 1 1 𝑜 4𝛽+1 𝑜 2𝛽 𝛽 𝑙 ≤ Ω 𝑛 log 𝑛 𝑙 Ω 𝑛𝑜 𝛽 ≤ 1.01 𝛽 𝑙 = 𝑃(𝑜/ log 𝑛) 𝑙 Cover Verification − 𝑙 ≤ 𝑜/2 Ω(𝑜𝑙) Cover Verification : given a set system, verify whether a given sub-collection of sets covers the universe. 𝝇 = approximation factor for offline Set Cover 𝒐 = number of elements 𝒏 = number of sets 𝒍 = Size of the optimal Solution

Results Problem Approximation Constraints Query Complexity 1 𝑜 𝛽−1 + 𝑜𝑙 𝛽𝜍 + 1 𝛽 ≥ 2 𝑃 𝑛 𝑙 𝑃 𝑛𝑜 𝜍 + 1 − 𝑙 Set Cover 1 1 𝑜 4𝛽+1 𝑜 2𝛽 𝛽 𝑙 ≤ Ω 𝑛 log 𝑛 𝑙 Ω 𝑛𝑜 𝛽 ≤ 1.01 𝛽 𝑙 = 𝑃(𝑜/ log 𝑛) 𝑙 Cover Verification − 𝑙 ≤ 𝑜/2 Ω(𝑜𝑙) Cover Verification : given a set system, verify whether a given sub-collection of sets covers the universe. 𝝇 = approximation factor for offline Set Cover 𝒐 = number of elements 𝒏 = number of sets 𝒍 = Size of the optimal Solution

Results Problem Approximation Constraints Query Complexity 1 𝑜 𝛽−1 + 𝑜𝑙 𝛽𝜍 + 1 𝛽 ≥ 2 𝑃 𝑛 𝑙 𝑃 𝑛𝑜 𝜍 + 1 − 𝑙 Set Cover 1 1 𝑜 4𝛽+1 𝑜 2𝛽 𝛽 𝑙 ≤ Ω 𝑛 log 𝑛 𝑙 Ω 𝑛𝑜 𝛽 ≤ 1.01 𝛽 𝑙 = 𝑃(𝑜/ log 𝑛) 𝑙 Cover Verification − 𝑙 ≤ 𝑜/2 Ω(𝑜𝑙) Cover Verification : given a set system, verify whether a given sub-collection of sets covers the universe. 𝝇 = approximation factor for offline Set Cover 𝒐 = number of elements 𝒏 = number of sets 𝒍 = Size of the optimal Solution

Results Problem Approximation Constraints Query Complexity 1 𝑜 𝛽−1 + 𝑜𝑙 𝛽𝜍 + 1 𝛽 ≥ 2 𝑃 𝑛 𝑙 𝑃 𝑛𝑜 𝜍 + 1 − 𝑙 Set Cover 1 1 𝑜 4𝛽+1 𝑜 2𝛽 𝛽 𝑙 ≤ Ω 𝑛 log 𝑛 𝑙 Ω 𝑛𝑜 𝛽 ≤ 1.01 𝛽 𝑙 = 𝑃(𝑜/ log 𝑛) 𝑙 Cover Verification − 𝑙 ≤ 𝑜/2 Ω(𝑜𝑙) Cover Verification : given a set system, verify whether a given sub-collection of sets covers the universe. 𝝇 = approximation factor for offline Set Cover 𝒐 = number of elements 𝒏 = number of sets 𝒍 = Size of the optimal Solution

Part one: upper bound Theorem: There exists an algorithm that with high probability 𝑃(𝒏𝒐 𝟐/𝜷 + 𝒐𝒍) finds an O(𝜍𝛽) -approximate cover which uses number of queries.

Part one: upper bound Theorem: There exists an algorithm that with high probability 𝑃(𝒏𝒐 𝟐/𝜷 + 𝒐𝒍) finds an O(𝜍𝛽) -approximate cover which uses number of queries. 1. Two simple components used for coverage problems in massive data models. • Set Sampling • Element Sampling 2. The algorithm overview

Component I: set sampling Set Sampling: After picking ℓ sets uniformly at random, all m log 𝑜 elements with degree at least are covered w.h.p. ℓ • We only need to worry about low degree elements.