An Improved FPTAS for 0-1 Knapsack Ce Jin Tsinghua University 0-1 - PowerPoint PPT Presentation

An Improved FPTAS for 0-1 Knapsack Ce Jin Tsinghua University

0-1 Knapsack Problem Given: Knapsack capacity 𝑋 > 0 𝑜 items Each item 𝑗 has weight 0 < 𝑥 𝑗 ≤ 𝑋 and profit 𝑞 𝑗 > 0 Find a subset of items 𝐽 ⊆ 𝑜 such that: • 𝑥 𝐽 ≔ σ 𝑗∈𝐽 𝑥 𝑗 ≤ 𝑋 • 𝑞 𝐽 ≔ σ 𝑗∈𝐽 𝑞 𝑗 is maximized

0-1 Knapsack Problem Given: Knapsack capacity 𝑋 > 0 𝑜 items Each item 𝑗 has weight 0 < 𝑥 𝑗 ≤ 𝑋 and profit 𝑞 𝑗 > 0 Find a subset of items 𝐽 ⊆ 𝑜 such that: • 𝑥 𝐽 ≔ σ 𝑗∈𝐽 𝑥 𝑗 ≤ 𝑋 • 𝑞 𝐽 ≔ σ 𝑗∈𝐽 𝑞 𝑗 is maximized A well-known NP-hard problem

FPTAS for 0-1 Knapsack OPT = optimal total profit 𝑞(𝐽) For 𝜁 > 0 , find a subset 𝐽 ⊆ 𝑜 such that: • 𝑥 𝐽 ≤ 𝑋 OPT • 𝑞 𝐽 ≥ 1+𝜁 1 Solvable in poly 𝑜, 𝜁 time

Prior Work FPTAS for 0-1 Knapsack: • ෨ 𝑃(𝑜 3 /𝜁) (textbook algorithm) • ෨ 𝑃(𝑜 + 𝜁 −4 ) [Ibarra and Kim, 1975] • ෨ 𝑃(𝑜 + 𝜁 −3 ) [Kellerer and Pferschy, 2004] • ෨ 𝑃(𝑜 + 𝜁 −2.5 ) [Rhee, 2015] • ෩ 𝑷(𝒐 + 𝜻 −𝟑.𝟓 ) [Chan, 2018]

Prior Work FPTAS for 0-1 Knapsack: • ෨ 𝑃(𝑜 3 /𝜁) (textbook algorithm) • ෨ 𝑃(𝑜 + 𝜁 −4 ) [Ibarra and Kim, 1975] • ෨ 𝑃(𝑜 + 𝜁 −3 ) [Kellerer and Pferschy, 2004] • ෨ 𝑃(𝑜 + 𝜁 −2.5 ) [Rhee, 2015] • ෩ 𝑷(𝒐 + 𝜻 −𝟑.𝟓 ) [Chan, 2018] • ෩ 𝑷(𝒐 + 𝜻 −𝟑.𝟑𝟔 ) (this work) Conditional Lower Bound: 𝑜 + 𝜁 −1 2−𝜀 No 𝑃 time algorithm, unless 𝐧𝐣𝐨, + convolution has truly subquadratic algo [Cygan, Mucha, W ę grzycki, and W ł odarczyk, 2017]

A Special Case FPTAS for Subset Sum ( 𝑞 𝑗 = 𝑥 𝑗 ): • ෨ 𝑃 min 𝒐 + 𝜻 −𝟑 , 𝑜𝜁 −1 [Kellerer, Mansini, Pferschy, and Speranza, 2003]

A Special Case FPTAS for Subset Sum ( 𝑞 𝑗 = 𝑥 𝑗 ): • ෨ 𝑃 min 𝒐 + 𝜻 −𝟑 , 𝑜𝜁 −1 [Kellerer, Mansini, Pferschy, and Speranza, 2003] Our 0-1 Knapsack algorithm utilizes this result

• Our ෩ 𝑷(𝒐 + 𝜻 −𝟑.𝟑𝟔 ) algorithm builds on Chan’s ෩ 𝑷(𝒐 + 𝜻 −𝟑.𝟓 ) algorithm • Two new ideas 1. Extending Chan’s number-theoretic technique from two levels to multiple levels.

• Our ෩ 𝑷(𝒐 + 𝜻 −𝟑.𝟑𝟔 ) algorithm builds on Chan’s ෩ 𝑷(𝒐 + 𝜻 −𝟑.𝟓 ) algorithm • Two new ideas 1. Extending Chan’s number-theoretic technique from two levels to multiple levels. 2. A greedy argument ⇒ less computation spent on cheap items (small unit profit 𝑞 𝑗 /𝑥 𝑗 )

• Our ෩ 𝑷(𝒐 + 𝜻 −𝟑.𝟑𝟔 ) algorithm builds on Chan’s ෩ 𝑷(𝒐 + 𝜻 −𝟑.𝟓 ) algorithm • Two new ideas 1. Extending Chan’s number-theoretic technique from two levels to multiple levels. 2. A greedy argument ⇒ less computation spent on cheap items (small unit profit 𝑞 𝑗 /𝑥 𝑗 ) This talk

• Our ෩ 𝑷(𝒐 + 𝜻 −𝟑.𝟑𝟔 ) algorithm builds on Chan’s ෩ 𝑷(𝒐 + 𝜻 −𝟑.𝟓 ) algorithm • Two new ideas 1. Extending Chan’s number-theoretic technique from two levels to multiple levels. 2. A greedy argument ⇒ less computation spent on cheap items (small unit profit 𝑞 𝑗 /𝑥 𝑗 ) Preliminaries (based on [Chan, 2018]) This talk Using Chan’s Lemmas as blackboxes ෩ 𝑷(𝒐 + 𝜻 −𝟑.𝟒𝟒𝟒 ) algo

Preliminaries • Assume 𝑜 ≤ poly 𝜁 −1 . 𝜁 • Too cheap items (𝑞 𝑗 < 𝑜 max 𝑞 𝑘 ) are 𝑘 discarded at the beginning (loss ≤ 𝜁 ⋅ OPT ) max 𝑞 𝑘 min 𝑞 𝑘 ≤ poly 𝜁 −1 So 𝑜 items, capacity = 𝑋 weight 0 < 𝑥 𝑗 ≤ 𝑋 and profit 𝑞 𝑗 > 0

Preliminaries •“Profit function” (defined over real 𝑦 ≥ 0 ) 𝑔 𝐽 𝑦 = max 𝑞 𝐾 : 𝐾 ⊆ 𝐽, 𝑥 𝐾 ≤ 𝑦 𝑜 ≤ poly 𝜁 −1 items profit max 𝑞 𝑘 /min 𝑞 𝑘 ≤ poly 𝜁 −1 weight 0 < 𝑥 𝑗 ≤ 𝑋

Preliminaries •“Profit function” (defined over real 𝑦 ≥ 0 ) 𝑔 𝐽 𝑦 = max 𝑞 𝐾 : 𝐾 ⊆ 𝐽, 𝑥 𝐾 ≤ 𝑦 • Task: compute a 1 + 𝜁 -approximation of profit function 𝑔 𝐽 , ෩ 𝐽 𝑦 ≤ 1 + 𝜁 ෩ 𝑔 𝐽 𝑦 ≤ 𝑔 𝑔 𝐽 𝑦 , ∀𝑦 ≥ 0 𝑜 ≤ poly 𝜁 −1 items profit max 𝑞 𝑘 /min 𝑞 𝑘 ≤ poly 𝜁 −1 weight 0 < 𝑥 𝑗 ≤ 𝑋

Preliminaries •“Profit function” (defined over real 𝑦 ≥ 0 ) 𝑔 𝐽 𝑦 = max 𝑞 𝐾 : 𝐾 ⊆ 𝐽, 𝑥 𝐾 ≤ 𝑦 • Task: compute a 1 + 𝜁 -approximation of profit function 𝑔 𝐽 , ෩ 𝐽 𝑦 ≤ 1 + 𝜁 ෩ 𝑔 𝐽 𝑦 ≤ 𝑔 𝑔 𝐽 𝑦 , ∀𝑦 ≥ 0 • For disjoint sets 𝐽, 𝐾 of items, 𝑔 𝐽∪𝐾 𝑦 = 𝑔 𝐽 ⊕ 𝑔 𝑦 ≔ max 0≤𝑧≤𝑦 𝑔 𝐽 𝑧 + 𝑔 𝐾 𝑦 − 𝑧 𝐾 𝑜 ≤ poly 𝜁 −1 items profit max 𝑞 𝑘 /min 𝑞 𝑘 ≤ poly 𝜁 −1 weight 0 < 𝑥 𝑗 ≤ 𝑋

Preliminaries •“Profit function” (defined over real 𝑦 ≥ 0 ) 𝑔 𝐽 𝑦 = max 𝑞 𝐾 : 𝐾 ⊆ 𝐽, 𝑥 𝐾 ≤ 𝑦 • Task: compute a 1 + 𝜁 -approximation of profit function 𝑔 𝐽 , ෩ 𝐽 𝑦 ≤ 1 + 𝜁 ෩ 𝑔 𝐽 𝑦 ≤ 𝑔 𝑔 𝐽 𝑦 , ∀𝑦 ≥ 0 • For disjoint sets 𝐽, 𝐾 of items, 𝑔 𝐽∪𝐾 𝑦 = 𝑔 𝐽 ⊕ 𝑔 𝑦 ≔ max 0≤𝑧≤𝑦 𝑔 𝐽 𝑧 + 𝑔 𝐾 𝑦 − 𝑧 𝐾 • ෩ 𝐽 ⊕ ෩ 𝑔 𝑔 𝐾 is a 1 + 𝜁 -approximation of 𝑔 𝐽∪𝐾 𝑜 ≤ poly 𝜁 −1 items profit max 𝑞 𝑘 /min 𝑞 𝑘 ≤ poly 𝜁 −1 weight 0 < 𝑥 𝑗 ≤ 𝑋

Preliminaries • A nondecreasing step function 𝑔 has a (1 + 𝜁) -approx. with only ෨ 𝑃 1/𝜁 steps (by rounding down to powers of (1 + 𝜁) ) 𝑜 ≤ poly 𝜁 −1 items profit max 𝑞 𝑘 /min 𝑞 𝑘 ≤ poly 𝜁 −1

Preliminaries • A nondecreasing step function 𝑔 has a (1 + 𝜁) -approx. with only ෨ 𝑃 1/𝜁 steps (by rounding down to powers of (1 + 𝜁) ) •“ Merging Lemma ”: Computing (a 1 + 𝜁 - 𝑛 takes ෨ 𝑃 𝑛/𝜁 2 approx. of) 𝑔 1 ⊕ ⋯ ⊕ 𝑔 time. ( log 𝑛 depth binary tree. 𝜁 ′ ≔ 𝜁/ log 𝑛 ) 𝑔 ⊕ 𝑕 ≔ max 0≤𝑧≤𝑦 𝑔 𝑧 + 𝑕 𝑦 − 𝑧

Preliminaries • Divide items into 𝑃 log 1/𝜁 groups (group 𝑙 : 𝑞 𝑗 ∈ [2 𝑙 , 2 𝑙+1 ] ) 𝑜 ≤ poly 𝜁 −1 items profit max 𝑞 𝑘 /min 𝑞 𝑘 ≤ poly 𝜁 −1 weight 0 < 𝑥 𝑗 ≤ 𝑋 𝑃 𝑛/𝜁 2 time. 𝑛 : ෨ Merging 𝑔 1 ⊕ ⋯ ⊕ 𝑔

Preliminaries • Divide items into 𝑃 log 1/𝜁 groups (group 𝑙 : 𝑞 𝑗 ∈ [2 𝑙 , 2 𝑙+1 ] ) • Compute all 𝑔 𝑙 and merge them in ෨ 𝑃(1/𝜁 2 ) time 𝑜 ≤ poly 𝜁 −1 items profit max 𝑞 𝑘 /min 𝑞 𝑘 ≤ poly 𝜁 −1 weight 0 < 𝑥 𝑗 ≤ 𝑋 𝑃 𝑛/𝜁 2 time. 𝑛 : ෨ Merging 𝑔 1 ⊕ ⋯ ⊕ 𝑔

Preliminaries • Divide items into 𝑃 log 1/𝜁 groups (group 𝑙 : 𝑞 𝑗 ∈ [2 𝑙 , 2 𝑙+1 ] ) • Compute all 𝑔 𝑙 and merge them in ෨ 𝑃(1/𝜁 2 ) time • Now assume 𝑞 𝑗 ∈ [1,2] 𝑜 ≤ poly 𝜁 −1 items profit max 𝑞 𝑘 /min 𝑞 𝑘 ≤ poly 𝜁 −1 weight 0 < 𝑥 𝑗 ≤ 𝑋 𝑃 𝑛/𝜁 2 time. 𝑛 : ෨ Merging 𝑔 1 ⊕ ⋯ ⊕ 𝑔

Preliminaries 𝑞 1 𝑞 2 • Simple greedy (sort by unit profit 𝑥 1 ≥ 𝑥 2 ≥ ⋯ ) approximates with additive error 𝑒 ≤ max 𝑞 𝑗 = 𝑃(1) profit 𝑞 𝑗 ∈ 1,2 weight 0 < 𝑥 𝑗 ≤ 𝑋

Preliminaries 𝑞 1 𝑞 2 • Simple greedy (sort by unit profit 𝑥 1 ≥ 𝑥 2 ≥ ⋯ ) approximates with additive error 𝑒 ≤ max 𝑞 𝑗 = 𝑃(1) • For 𝑔 𝑥 ≥ Ω(𝜁 −1 ) , this is 1 + 𝑃(𝜁) multiplicative approx. profit 𝑞 𝑗 ∈ 1,2 weight 0 < 𝑥 𝑗 ≤ 𝑋

Preliminaries 𝑞 1 𝑞 2 • Simple greedy (sort by unit profit 𝑥 1 ≥ 𝑥 2 ≥ ⋯ ) approximates with additive error 𝑒 ≤ max 𝑞 𝑗 = 𝑃(1) • For 𝑔 𝑥 ≥ Ω(𝜁 −1 ) , this is 1 + 𝑃(𝜁) multiplicative approx. • Only need to 1 + 𝑃(𝜁) approximate for 𝑪 = 𝑷 𝜻 −𝟐 ! 𝐧𝐣𝐨 𝑪, 𝒈 𝑱 profit 𝑞 𝑗 ∈ 1,2 weight 0 < 𝑥 𝑗 ≤ 𝑋

Preliminaries • Round 𝑞 𝑗 down to 1,1 + 𝜁, 1 + 2𝜁, … , 2 − 𝜁 1 + 𝜁 multiplicative error • Only 𝑛 = 𝑃(1/𝜁) different 𝑞 𝑗 ’s ! 𝑞 𝑗 ∈ 1,2

Preliminaries • Round 𝑞 𝑗 down to 1,1 + 𝜁, 1 + 2𝜁, … , 2 − 𝜁 1 + 𝜁 multiplicative error • Only 𝑛 = 𝑃(1/𝜁) different 𝑞 𝑗 ’s ! • Collect all items with the same profit 𝒒 . Then 𝑔 𝑞 can be computed by simple greedy (sort 𝑥 1 ≤ 𝑥 2 ≤ ⋯ ) 𝑞 𝑗 ∈ 1,2

Recap 𝑞 𝑗 ∈ 1,2 Profit functions 𝑔 1 , … , 𝑔 𝑛 obtained by simple greedy (one for every 𝑞 𝑗 ) ( 𝑛 = 𝑃(1/𝜁) ) Task: 1 + 𝑃 𝜁 approximate ( 𝐶 = 𝑃(𝜁 −1 ) ) min 𝐶, 𝑔 1 ⊕ ⋯ ⊕ 𝑔 𝑛 𝑃 𝑛/𝜁 2 time. 𝑛 in ෨ Lemma: Merging 𝑔 1 ⊕ ⋯ ⊕ 𝑔 (Immediately gives ෨ 𝑃(𝑜 + 𝜁 −3 ) algo)