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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:

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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

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2

Knapsack Problem

  • Item set: 𝐹
  • Profit: π‘žπ‘“ β‰₯ 0 (𝑓 ∈ 𝐹)
  • Weight: π‘₯𝑓 β‰₯ 0 (𝑓 ∈ 𝐹)
  • Capacity: 𝐷 β‰₯ 0

 Family of feas. sets β„± = {π‘Œ βŠ† 𝐹: π‘₯(π‘Œ) ≀ 𝐷} maximize π‘ž(π‘Œ) subject to π‘Œ ∈ β„± π‘₯ π‘Œ = βˆ‘π‘“βˆˆπ‘Œ π‘₯𝑓 π‘ž π‘Œ = βˆ‘π‘“βˆˆπ‘Œ π‘žπ‘“

  • NP-hard
  • FPTAS

𝐷  Problem π‘Œ =

80 70

π‘Ž =

60 50 50 100 20

𝑍 =

π‘žπ‘“ = 100 80 70 20 60 50 20 50

π‘₯𝑓

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3

Cardinality Robustness

 Cardinality constraint |𝒀| ≀ 𝒍 is given after choosing π‘Œ

  • π‘Œ 𝑙 : expensive ≀ 𝑙 items in π‘Œ
  • OPT𝑙 : optimal sol.

π‘ž OPT

1 = 100

π‘ž OPT2 = 150 π‘ž OPT3 = 160 𝐷

100 80 70 20 60 50 20 50

π‘Œ =

80 70

π‘Ž =

60 50 50 100 20

𝑍 = π‘ž(π‘Œ 1 ) = 80 π‘ž(π‘Œ 2 ) = 150 π‘ž(π‘Œ 3 ) = 150 … …  Robustness = 0.8 π‘Œ ∈ β„±, 0 < 𝛽 ≀ 1

  • π‘Œ is 𝜷-robust defβˆ€π’, 𝒒(𝒀 𝒍 ) β‰₯ 𝜷 βˆ™ 𝒒 OPT𝒍
  • robustness ≝ 𝐧𝐣𝐨

𝒍 𝒒 𝒀 𝒍 𝒒 𝐏𝐐𝐔𝒍

Def

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4

Contents

 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

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5

Matching / Matroid Intersection

 Hassin, Rubinstein [2002] 3 4 6 5 8 π‘ž OPT

1 = 8

π‘ž(OPT2) = 10 π‘Œ: 0.8-robust 𝑍: 0.75-robust

  • Matching: maximizing βˆ‘π’‡βˆˆπ’€ 𝒒𝒇

πŸ‘  𝟐 πŸ‘-robust

 Fujita, K, Makino [2013]

  • Matroid Intersection: maximizing βˆ‘π’‡βˆˆπ’€ 𝒒𝒇

πŸ‘  𝟐 πŸ‘-robust

  • Computation of max robustness: NP-hard

1 1 2

  • 𝟐

πŸ‘ is best possible

  • Matroid: greedy alg.  1-robust
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6

Robust Independence System

 Kakimura, Makino [2013]

  • Ind. system: maximizing βˆ‘π’‡βˆˆπ’€ 𝒒𝒇

πŸ‘  𝟐 𝝂(β„±)-robust

  • 𝟐

𝝂(𝓖) is best possible

𝝂(𝓖) β‰œ min. integer 𝜈 satisfying π‘Œ, 𝑍 ∈ β„±, 𝑓 ∈ 𝑍 βˆ’ π‘Œ β‡’ βˆƒπ‘Ž βŠ† π‘Œ βˆ’ 𝑍 s.t. π‘Ž ≀ 𝜈, π‘Œ βˆ’ π‘Ž + 𝑓 ∈ β„± π‘Œ 𝑍 𝑓 π‘Ž 𝑍 π‘Œ 𝐹, β„± : independence system

def

βˆ… ∈ β„±, π‘Œ βŠ† 𝑍, 𝑍 ∈ β„± β‡’ π‘Œ ∈ β„± Def Def [Mestre 2006]

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𝝂(𝓖) : Tractability of Independence System

  • Matroid: 𝜈 β„± = 1
  • Matching: 𝜈 β„± ≀ 2
  • Intersection of 𝑛 matroids: 𝜈 β„± ≀ 𝑛

π‘Œ 𝑍 𝑓 π‘Ž

  • Feasible sets of Knapsack Problem

 𝜈 β„± = 𝑁 (arbitrarily large) π‘Œ = 𝑓1, … , 𝑓𝑁 π‘₯𝑓𝑗 = 𝐷 𝑁 𝑍 = 𝑓0 (π‘₯𝑓0 = 𝐷)  Kakimura, Makino [2013]: max. βˆ‘π’‡βˆˆπ’€ 𝒒𝒇

πŸ‘  𝟐 𝝂(β„±)-robust

 Kakimura, Makino, Seimi [2012]

  • Robust Knapsack Problem: weakly NP-hard + FPTAS

𝐷 𝑍 = π‘Œ =

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Mixed (or Randomized) Strategy

 Matuschke, Skutella, Soto [2015]: Zero-Sum game  Mixed Strategy = Distribution on β„±  Choose π‘Œπ‘— with probability πœ‡π‘—  robustness: min 𝐅

π‘ž π‘Œ 𝑙 π‘ž(OPT𝑙) = min βˆ‘π‘— πœ‡π‘— π‘ž(π‘Œπ‘—(𝑙)) π‘ž(OPT𝑙)

𝑙 𝑙

Alice: Choose π‘Œ ∈ β„± Bob: Choose 𝑙 (knowing π‘Œ)  Alice’s payoff =

π‘ž(π‘Œ(𝑙)) π‘ž(OPT𝑙)

1 1 2

π‘ž OPT

1 =

2 π‘ž(OPT2) = 2 Robustness of π‘Œ,𝑍

1 2 = 0.7071 …

  • Ex. Choose π‘Œ or 𝑍 with prob. Β½

min

1 2β‹…1+1 2β‹… 2

2

,

1 2β‹…2+1 2β‹… 2

2

=

2+ 2 4

= 0.8535 …

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Mixed (or Randomized) Strategy

cf.

1 2 = 0.7071 …

  • 1. Choose 𝑦 in [0,1] uniformly at random
  • 2. For each 𝑓, set π‘Ÿπ‘“ ≔ log2 π‘žπ‘“ , 𝒒𝒇

β€² ≔ πŸ‘ π’“π’‡βˆ’π’š , and

find π‘Œ ∈ β„± maximizing π‘žβ€²(π‘Œ)

Round value π‘ž to power of two

The above mixed strategy is

𝟐 𝐦𝐨 πŸ“-robust for

Thm [MSS 15] 0.7213 …  Matuschke, Skutella, Soto [2015]

  • Matching
  • Matroid intersection
  • Strongly base orderable matroid parity etc.
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Contents

 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

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Mixed Strategy for Robust Knapsack Problem

  • 1. Upper bound 𝐏

𝐦𝐩𝐑 𝐦𝐩𝐑 𝝂(𝓖) 𝐦𝐩𝐑 𝝂(𝓖)

, 𝐏

𝐦𝐩𝐑 𝐦𝐩𝐑 𝝇(𝓖) 𝐦𝐩𝐑 𝝇(𝓖)

  • 2. Lower bound 𝛁

𝟐 𝐦𝐩𝐑 𝝂(𝓖) , 𝛁 𝟐 𝐦𝐩𝐑 𝝇(𝓖) : Design a strategy

Extend to ind. sys.: 𝐏

𝟐 𝐦𝐩𝐑 𝝂(𝓖) , 𝐏 𝟐 𝐦𝐩𝐑 𝝇(𝓖) , 𝛁 𝟐 𝐦𝐩𝐑 𝝇(𝓖)

 Robustness of pure strategy:

𝟐 𝝂(𝓖) [Kakimura, Makino 13]

 Robustness of mixed strategy [Our result]

𝜈(β„±): arbitrarily large

𝝇(𝓖): another parameter of ind. sys.

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Result 1. Upper Bound: Hard Instance

Type

π‘₯𝑓 π‘žπ‘“

Number

π‘žπ‘“ π‘₯𝑓

Total profit

𝑁2π‘ˆ 𝑁2π‘ˆ 1 1 𝑁2π‘ˆ 1 𝑁2π‘ˆβˆ’2 𝑁2π‘ˆβˆ’1 𝑁2 𝑁 𝑁2π‘ˆ+1 ∢ ∢ ∢ ∢ ∢ ∢ 𝑗 𝑁2π‘ˆβˆ’2𝑗 𝑁2π‘ˆβˆ’π‘— 𝑁2𝑗 𝑁𝑗 𝑁2π‘ˆ+𝑗 ∢ ∢ ∢ ∢ ∢ ∢ π‘ˆ 1 π‘π‘ˆ 𝑁2π‘ˆ π‘π‘ˆ 𝑁3π‘ˆ π‘ž OPT 1 = 𝑁2π‘ˆ, π‘ž π‘ƒπ‘„π‘ˆ 𝑁2π‘ˆ = 𝑁3π‘ˆ

𝐷 = 𝑁2π‘ˆ

… For any mixed strategy, robustness ≀

𝟐 𝑼+𝟐 + πŸ‘ 𝑡

Thm [Our result]

  • No mixed strategy can achieve constant robustness
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Result 1. Upper Bound: Hard Instance

𝐷 = 𝑁2π‘ˆ

…  𝝂 𝓖 = π‘΅πŸ‘π‘Ό π‘Œ 𝑍 𝑓 π‘Ž

Type

π‘₯𝑓 π‘žπ‘“

Number

π‘žπ‘“ π‘₯𝑓

Total profit

𝑁2π‘ˆ 𝑁2π‘ˆ 1 1 𝑁2π‘ˆ ∢ ∢ ∢ ∢ ∢ ∢ 𝑗 𝑁2π‘ˆβˆ’2𝑗 𝑁2π‘ˆβˆ’π‘— 𝑁2𝑗 𝑁𝑗 𝑁2π‘ˆ+𝑗 ∢ ∢ ∢ ∢ ∢ ∢ π‘ˆ 1 π‘π‘ˆ 𝑁2π‘ˆ π‘π‘ˆ 𝑁3π‘ˆ For any mixed strategy, robustness ≀

𝟐 𝑼+𝟐 + πŸ‘ 𝑡

Thm [Our result]

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Result 1. Upper Bound: Hard Instance

𝐷 = 𝑁2π‘ˆ

…  𝝂 𝓖 = π‘΅πŸ‘π‘΅ log 𝑁2𝑁 = Θ 𝑁 log 𝑁 log log 𝑁2𝑁 = Θ log 𝑁

Type

π‘₯𝑓 π‘žπ‘“

Number

π‘žπ‘“ π‘₯𝑓

Total profit

𝑁2𝑁 𝑁2𝑁 1 1 𝑁2𝑁 ∢ ∢ ∢ ∢ ∢ ∢ 𝑗 𝑁2π‘βˆ’2𝑗 𝑁2π‘βˆ’π‘— 𝑁2𝑗 𝑁𝑗 𝑁2𝑁+𝑗 ∢ ∢ ∢ ∢ ∢ ∢ 𝑁 1 𝑁𝑁 𝑁2𝑁 𝑁𝑁 𝑁3𝑁 For any mixed strategy, robustness ≀

πŸ’ 𝑡

Thm [Our result]

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Result 1. Upper Bound: Hard Instance

𝐷 = 𝑁2π‘ˆ

…  𝝂 𝓖 = π‘΅πŸ‘π‘΅ log 𝑁2𝑁 = Θ 𝑁 log 𝑁 log log 𝑁2𝑁 = Θ log 𝑁

Type

π‘₯𝑓 π‘žπ‘“

Number

π‘žπ‘“ π‘₯𝑓

Total profit

𝑁2𝑁 𝑁2𝑁 1 1 𝑁2𝑁 ∢ ∢ ∢ ∢ ∢ ∢ 𝑗 𝑁2π‘βˆ’2𝑗 𝑁2π‘βˆ’π‘— 𝑁2𝑗 𝑁𝑗 𝑁2𝑁+𝑗 ∢ ∢ ∢ ∢ ∢ ∢ 𝑁 1 𝑁𝑁 𝑁2𝑁 𝑁𝑁 𝑁3𝑁 For any mixed strategy, robustness ≀

πŸ’ 𝑡 = 𝐏 𝐦𝐩𝐑 𝐦𝐩𝐑 𝝂 𝓖 𝐦𝐩𝐑 𝝂 𝓖

Thm [Our result]

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Result 2. Lower Bound: 𝛁

𝟐 𝐦𝐩𝐑 𝝂(𝓖)

 βˆ€π‘— ∈ 0,1, … , 𝑛 , choose 𝒀𝒋 = πππ”πŸ‘π’‹β‹…π’• with prob.

𝟐 𝒏+𝟐

Strategy (A) 𝑛 ≔ log ( 𝑒 𝑑) βˆ… 𝐹 Robustness β‰₯

𝟐 𝒏+𝟐

Thm [Our result]

𝑛 = O log 𝜈(β„±) ???  NO

𝓖 𝑒 𝑑 OPT2𝑛⋅𝑑 OPT20⋅𝑑 : 𝐷 𝐷/2 𝜈(β„±)=1 𝑛: large Choose small items in advance Idea

𝒕 ≔ min π‘Œ : π‘Œ βˆ‰ β„± βˆ’ 1 𝒖 ≔ max{ π‘Œ : π‘Œ ∈ β„±}

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π‘Œ0 1. π‘Œβˆ—: optimal sol., 𝑍: heaviest 𝑑 elements 2. π‘Œ0 βŠ† π‘Œβˆ—: π‘₯ π‘Œ0 ≀ 𝐷 βˆ’ π‘₯(𝑍) with max size 3. 𝐷′ ≔ 𝐷 βˆ’ π‘₯(π‘Œ0), 𝐹′ ≔ 𝐹 βˆ’ π‘Œ0, 𝑛′ ≔

log π‘Œβˆ—βˆ’π‘Œ0 𝑑

4. βˆ€π‘— ∈ 0,1, … , 𝑛′ , choose πππ”β€²πŸ‘π’‹β‹…π’• βˆͺ π’€πŸ with prob.

𝟐 𝒏′+𝟐

Result 2. Lower Bound: 𝛁

𝟐 𝐦𝐩𝐑 𝝂(𝓖)

Robustness β‰₯

𝟐 πŸ“(𝒏′+𝟐) = 𝛁 𝟐 𝐦𝐩𝐑 𝝂 𝓖

Thm [Our result] Strategy (B) 𝓖 𝑒 𝑑 𝑍 π‘Œβˆ— 𝐷 𝑍 π‘Œβˆ—

  • cf. pure strategy:

𝟐 𝝂(𝓖) [Kakimura, Makino 13]

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Contents

 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

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Summary・Future work

 Our result: mixed strategy for robust knapsack problem  Future work 1. Close the gap between upper and lower bounds 2. Ω

1 log 𝜈(β„±) -robust strategy for general ind. sys.

3. Evaluation with rank quotient 𝑠(β„±) 𝑠 β„± : = min

π‘ŒβŠ†πΉ

min{|maximal sol in π‘Œ|} max{|maximal sol in π‘Œ|}

  • 1. Upper bound 𝐏

𝐦𝐩𝐑 𝐦𝐩𝐑 𝝂(𝓖) 𝐦𝐩𝐑 𝝂(𝓖)

, 𝐏

𝐦𝐩𝐑 𝐦𝐩𝐑 𝝇(𝓖) 𝐦𝐩𝐑 𝝇(𝓖)

  • 2. Lower bound 𝛁

𝟐 𝐦𝐩𝐑 𝝂(𝓖) , 𝛁 𝟐 𝐦𝐩𝐑 𝝇(𝓖) : Design a strategy

Extend to ind. sys.: 𝐏

𝟐 𝐦𝐩𝐑 𝝂(𝓖) , 𝐏 𝟐 𝐦𝐩𝐑 𝝇(𝓖) , 𝛁 𝟐 𝐦𝐩𝐑 𝝇(𝓖)