Randomized Strategies for Cardinality Robustness in the Knapsack - PowerPoint PPT Presentation

Randomized Strategies for Cardinality Robustness in the Knapsack Problem Yusuke Kobayashi University of Tsukuba Kenjiro Takazawa Kyoto University ANALCO, Arlington, Virginia, USA Jan 11, 2016

Knapsack Problem 2 • Item set: 𝐹 𝑥 𝑓 • Profit: 𝑞 𝑓 ≥ 0 ( 𝑓 ∈ 𝐹 ) 𝑞 𝑓 = 100 • Weight: 𝑥 𝑓 ≥ 0 ( 𝑓 ∈ 𝐹 ) 20 60 80 • Capacity: 𝐷 ≥ 0 70 50 20 50  Family of feas. sets ℱ = {𝑌 ⊆ 𝐹: 𝑥(𝑌) ≤ 𝐷} 𝐷 𝑥 𝑌 = ∑ 𝑓∈𝑌 𝑥 𝑓  Problem 80 70 𝑌 = 𝑞 𝑌 = ∑ 𝑓∈𝑌 𝑞 𝑓 maximize 𝑞(𝑌) 20 𝑍 = 100 subject to 𝑌 ∈ ℱ 60 50 50  NP-hard 𝑎 =  FPTAS

Cardinality Robustness 3  Cardinality constraint |𝒀| ≤ 𝒍 is given after choosing 𝑌  𝑌 𝑙 : expensive ≤ 𝑙 items in 𝑌  OPT 𝑙 : optimal sol. Def 𝑌 ∈ ℱ, 0 < 𝛽 ≤ 1 100  𝑌 is 𝜷 - robust def ∀𝒍, 𝒒(𝒀 𝒍 ) ≥ 𝜷 ∙ 𝒒 OPT 𝒍 20 60 80 70 50 𝒒 𝒀 𝒍  robustness ≝ 𝐧𝐣𝐨 20 50 𝒒 𝐏𝐐𝐔 𝒍 𝒍 𝐷 𝑞(𝑌 1 ) = 80 𝑞 OPT 1 = 100 80 70 𝑌 = 𝑞 OPT 2 = 150 𝑞(𝑌 2 ) = 150 𝑍 = 100 20 𝑞 OPT 3 = 160 𝑞(𝑌 3 ) = 150 60 50 50 … 𝑎 = …  Robustness = 0.8

Contents 4  Introduction : Robust knapsack problem  Related Work  Hassin, Rubinstein [2002]: Robust matching  Kakimura, Makino [2013]: Robust independence system  Matuschke, Skutella, Soto [2015]: Mixed strategy  Our Result : Mixed strategy for robust knapsack problem  Upper/Lower bound for robustness  Better than pure strategy  Concluding Remarks

Matching / Matroid Intersection 5 3  Hassin, Rubinstein [2002]  Matroid : greedy alg.  1-robust 8 6 4 𝟑  𝟐  Matching : maximizing ∑ 𝒇∈𝒀 𝒒 𝒇 𝟑 -robust 5 𝟐 𝟑 is best possible 𝑞 OPT 1 = 8  𝑞(OPT 2 ) = 10 1 1 2 𝑌 : 0.8-robust 𝑍 : 0.75-robust  Fujita, K, Makino [2013] 𝟑  𝟐  Matroid Intersection : maximizing ∑ 𝒇∈𝒀 𝒒 𝒇 𝟑 -robust  Computation of max robustness: NP-hard

Robust Independence System 6 𝑍 Def ∅ ∈ ℱ, def 𝐹, ℱ : independence system 𝑌 𝑌 ⊆ 𝑍, 𝑍 ∈ ℱ ⇒ 𝑌 ∈ ℱ  Kakimura, Makino [2013] 𝟑  𝟐  Ind. system : maximizing ∑ 𝒇∈𝒀 𝒒 𝒇 𝝂(ℱ) -robust 𝟐 𝝂(𝓖) is best possible  Def [Mestre 2006] 𝝂(𝓖) ≜ min. integer 𝜈 satisfying 𝑌 𝑍 𝑌, 𝑍 ∈ ℱ, 𝑓 ∈ 𝑍 − 𝑌 𝑓 ⇒ ∃𝑎 ⊆ 𝑌 − 𝑍 𝑎 s.t. 𝑎 ≤ 𝜈, 𝑌 − 𝑎 + 𝑓 ∈ ℱ

7 𝝂(𝓖) : Tractability of Independence System 𝟑  𝟐  Kakimura, Makino [2013]: max. ∑ 𝒇∈𝒀 𝒒 𝒇 𝝂(ℱ) -robust 𝑌 𝑍  Matroid: 𝜈 ℱ = 1  Matching: 𝜈 ℱ ≤ 2 𝑓 𝑎  Intersection of 𝑛 matroids: 𝜈 ℱ ≤ 𝑛  Feasible sets of Knapsack Problem  𝜈 ℱ = 𝑁 (arbitrarily large) 𝐷 𝑌 = 𝑓 1 , … , 𝑓 𝑁 𝑥 𝑓 𝑗 = 𝐷 𝑁 𝑌 = 𝑍 = 𝑓 0 (𝑥 𝑓 0 = 𝐷) 𝑍 =  Kakimura, Makino, Seimi [2012]  Robust Knapsack Problem: weakly NP-hard + FPTAS

Mixed (or Randomized) Strategy 8  Matuschke, Skutella, Soto [2015] : Zero-Sum game Alice: Choose 𝑌 ∈ ℱ Bob: Choose 𝑙 (knowing 𝑌 ) 𝑞(𝑌(𝑙))  Alice’s payoff = 𝑞(OPT 𝑙 )  Mixed Strategy = Distribution on ℱ  Choose 𝑌 𝑗 with probability 𝜇 𝑗  robustness : 1 1 2 𝑞 𝑌 𝑙 ∑ 𝑗 𝜇 𝑗 𝑞(𝑌 𝑗 (𝑙)) min 𝐅 𝑞(OPT 𝑙 ) = min 𝑞 OPT 1 = 2 𝑞(OPT 𝑙 ) 𝑙 𝑙 𝑞(OPT 2 ) = 2  Ex. Choose 𝑌 or 𝑍 with prob. ½ Robustness of 𝑌 , 𝑍 1 2 ⋅1+ 1 1 2 ⋅2+ 1 2 ⋅ 2 2 ⋅ 2 2+ 2 min , = = 0.8535 … 1 2 = 0.7071 … 2 2 4

Mixed (or Randomized) Strategy 9  Matuschke, Skutella, Soto [2015] 1. Choose 𝑦 in [0,1] uniformly at random ′ ≔ 𝟑 𝒓 𝒇 −𝒚 , and 2. For each 𝑓 , set 𝑟 𝑓 ≔ log 2 𝑞 𝑓 , 𝒒 𝒇 find 𝑌 ∈ ℱ maximizing 𝑞′(𝑌) Round value 𝑞 to power of two Thm [MSS 15] 𝟐 The above mixed strategy is 𝐦𝐨 𝟓 -robust for  Matching 0.7213 …  Matroid intersection  Strongly base orderable matroid parity etc. 1 cf. 2 = 0.7071 …

Contents 10  Introduction : Robust knapsack problem  Related Work  Hassin, Rubinstein [2002]: Robust matching  Kakimura, Makino [2013]: Robust independence system  Matuschke, Skutella, Soto [2015]: Mixed strategy  Our Result : Mixed strategy for robust knapsack problem  Upper/Lower bound for robustness  Better than pure strategy  Concluding Remarks

Mixed Strategy for Robust Knapsack Problem 11 𝟐  Robustness of pure strategy: 𝝂(𝓖) [Kakimura, Makino 13] 𝜈(ℱ) : arbitrarily large  Robustness of mixed strategy [Our result] 𝝇(𝓖) : another 𝐦𝐩𝐡 𝐦𝐩𝐡 𝝂(𝓖) 𝐦𝐩𝐡 𝐦𝐩𝐡 𝝇(𝓖) 1. Upper bound 𝐏 , 𝐏 parameter of ind. sys. 𝐦𝐩𝐡 𝝂(𝓖) 𝐦𝐩𝐡 𝝇(𝓖) 𝟐 𝟐 2. Lower bound 𝛁 𝐦𝐩𝐡 𝝂(𝓖) , 𝛁 𝐦𝐩𝐡 𝝇(𝓖) : Design a strategy 𝟐 𝟐 𝟐 Extend to ind. sys. : 𝐏 𝐦𝐩𝐡 𝝂(𝓖) , 𝐏 𝐦𝐩𝐡 𝝇(𝓖) , 𝛁 𝐦𝐩𝐡 𝝇(𝓖)

Result 1. Upper Bound: Hard Instance 12 𝐷 = 𝑁 2𝑈 Type Number Total profit 𝑥 𝑓 𝑞 𝑓 𝑞 𝑓 𝑥 𝑓 𝑁 2𝑈 𝑁 2𝑈 𝑁 2𝑈 0 1 1 𝑁 2𝑈−2 𝑁 2𝑈−1 𝑁 2 𝑁 2𝑈+1 1 𝑁 ∶ ∶ ∶ ∶ ∶ ∶ … 𝑁 2𝑈−2𝑗 𝑁 2𝑈−𝑗 𝑁 2𝑗 𝑁 𝑗 𝑁 2𝑈+𝑗 𝑗 ∶ ∶ ∶ ∶ ∶ ∶ 𝑁 𝑈 𝑁 2𝑈 𝑁 𝑈 𝑁 3𝑈 𝑈 1 = 𝑁 2𝑈 , 𝑞 𝑃𝑄𝑈 𝑁 2𝑈 = 𝑁 3𝑈 𝑞 OPT 1 Thm [Our result] 𝟐 𝟑 For any mixed strategy, robustness ≤ 𝑼+𝟐 + 𝑵  No mixed strategy can achieve constant robustness

Result 1. Upper Bound: Hard Instance 13 𝐷 = 𝑁 2𝑈 Type Number Total profit 𝑥 𝑓 𝑞 𝑓 𝑞 𝑓 𝑥 𝑓 𝑁 2𝑈 𝑁 2𝑈 𝑁 2𝑈 0 1 1 ∶ ∶ ∶ ∶ ∶ ∶ … 𝑁 2𝑈−2𝑗 𝑁 2𝑈−𝑗 𝑁 2𝑗 𝑁 𝑗 𝑁 2𝑈+𝑗 𝑗 ∶ ∶ ∶ ∶ ∶ ∶ 𝑁 𝑈 𝑁 2𝑈 𝑁 𝑈 𝑁 3𝑈 𝑈 1  𝝂 𝓖 = 𝑵 𝟑𝑼 𝑌 𝑍 𝑓 Thm [Our result] 𝑎 𝟐 𝟑 For any mixed strategy, robustness ≤ 𝑼+𝟐 + 𝑵

Result 1. Upper Bound: Hard Instance 14 𝐷 = 𝑁 2𝑈 Type Number Total profit 𝑥 𝑓 𝑞 𝑓 𝑞 𝑓 𝑥 𝑓 𝑁 2𝑁 𝑁 2𝑁 𝑁 2𝑁 0 1 1 ∶ ∶ ∶ ∶ ∶ ∶ … 𝑁 2𝑁−2𝑗 𝑁 2𝑁−𝑗 𝑁 2𝑗 𝑁 𝑗 𝑁 2𝑁+𝑗 𝑗 ∶ ∶ ∶ ∶ ∶ ∶ 𝑁 𝑁 𝑁 2𝑁 𝑁 𝑁 𝑁 3𝑁 𝑁 1 log 𝑁 2𝑁 = Θ 𝑁 log 𝑁  𝝂 𝓖 = 𝑵 𝟑𝑵 log log 𝑁 2𝑁 = Θ log 𝑁 Thm [Our result] 𝟒 For any mixed strategy, robustness ≤ 𝑵

Result 1. Upper Bound: Hard Instance 15 𝐷 = 𝑁 2𝑈 Type Number Total profit 𝑥 𝑓 𝑞 𝑓 𝑞 𝑓 𝑥 𝑓 𝑁 2𝑁 𝑁 2𝑁 𝑁 2𝑁 0 1 1 ∶ ∶ ∶ ∶ ∶ ∶ … 𝑁 2𝑁−2𝑗 𝑁 2𝑁−𝑗 𝑁 2𝑗 𝑁 𝑗 𝑁 2𝑁+𝑗 𝑗 ∶ ∶ ∶ ∶ ∶ ∶ 𝑁 𝑁 𝑁 2𝑁 𝑁 𝑁 𝑁 3𝑁 𝑁 1 log 𝑁 2𝑁 = Θ 𝑁 log 𝑁  𝝂 𝓖 = 𝑵 𝟑𝑵 log log 𝑁 2𝑁 = Θ log 𝑁 Thm [Our result] 𝟒 𝐦𝐩𝐡 𝐦𝐩𝐡 𝝂 𝓖 For any mixed strategy, robustness ≤ 𝑵 = 𝐏 𝐦𝐩𝐡 𝝂 𝓖

16 Result 2. Lower Bound: 𝛁 𝟐 𝐦𝐩𝐡 𝝂(𝓖) 𝑢 𝑡) 𝑛 ≔ log ( Strategy (A) 𝟐  ∀𝑗 ∈ 0,1, … , 𝑛 , choose 𝒀 𝒋 = 𝐏𝐐𝐔 𝟑 𝒋 ⋅𝒕 with prob. 𝒏+𝟐 𝐹 Thm [Our result] 𝑢 OPT 2 𝑛 ⋅𝑡 𝟐 Robustness ≥ 𝒏+𝟐 : 𝑡 𝑛 = O log 𝜈(ℱ) ???  NO OPT 2 0 ⋅𝑡 𝓖 ∅ 𝐷 𝒖 ≔ max{ 𝑌 : 𝑌 ∈ ℱ} 𝒕 ≔ min 𝑌 : 𝑌 ∉ ℱ − 1 Idea 𝜈(ℱ) =1 Choose small items in advance 𝑛 : large 𝐷 /2

17 Result 2. Lower Bound: 𝛁 𝟐 𝐦𝐩𝐡 𝝂(𝓖) Strategy (B) 𝑢 𝑌 ∗ 1. 𝑌 ∗ : optimal sol., 𝑍 : heaviest 𝑡 elements 2. 𝑌 0 ⊆ 𝑌 ∗ : 𝑥 𝑌 0 ≤ 𝐷 − 𝑥(𝑍) with max size 𝑍 𝑡 𝓖 log 𝑌 ∗ −𝑌 0 𝐷 ′ ≔ 𝐷 − 𝑥(𝑌 0 ) , 𝐹 ′ ≔ 𝐹 − 𝑌 0 , 𝑛 ′ ≔ 3. 𝑡 𝟐 4. ∀𝑗 ∈ 0,1, … , 𝑛′ , choose 𝐏𝐐𝐔′ 𝟑 𝒋 ⋅𝒕 ∪ 𝒀 𝟏 with prob. 𝒏 ′ +𝟐 Thm [Our result] 𝐷 𝟐 𝟐 Robustness ≥ 𝑌 ∗ 𝟓(𝒏 ′ +𝟐) = 𝛁 𝐦𝐩𝐡 𝝂 𝓖 𝑌 0 𝑍 𝟐 cf. pure strategy: 𝝂(𝓖) [Kakimura, Makino 13]

Contents 18  Introduction : Robust knapsack problem  Related Work  Hassin, Rubinstein [2002]: Robust matching  Kakimura, Makino [2013]: Robust independence system  Matuschke, Skutella, Soto [2015]: Mixed strategy  Our Result : Mixed strategy for robust knapsack problem  Upper/Lower bound for robustness  Better than pure strategy  Concluding Remarks