MaximizingaMonotoneSubmodular
FunctionSubjecttoaCoveringanda
PackingConstraint
JoachimSpoerhase
UniversityofWroclaw,Poland
[email protected]
HighlightsofAlgorithms
10th June,2017
JointworkwithSumedhaUniyal
Messagetothe
Audience
SubmodularFunctions
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SubmodularFunctions
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SubmodularFunctions
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SubmodularFunctions
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Capturesmanycombinatorialoptimizationproblemslike
- Max-k-Coverage
- Max-Cut
- MaxFacilityLocation…
𝐴
𝑒
𝐵
𝓤
Asetfunctiononallsubsetsof𝒰, 𝑓: 2|𝒰| → ℝA suchthat∀𝐴 ⊆ 𝐵 ⊆ 𝒰, 𝑒 ∈ 𝒰\B,
𝑓 𝐴 ∪ 𝑒 − 𝑓 𝐴 ≥ 𝑓 𝐵 ∪ 𝑒 − 𝑓(𝐵) (Submodular)
𝑓(𝐵) ≥ 𝑓(𝐴) (Monotone)
CoveringandPackingConstraints
W=50k＄
3 30k＄
6
4
25k＄
4
2
20k＄
2
1 50k＄
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Foreachelemente ∈ 𝒰,wearegivesomeweightaprofitfunction𝑝: 𝒰 → ℕ,a
weightfunction𝑤: 𝒰 → ℕ ,aprofitrequirement𝑃 andabudget𝐵.
Feasibility NP-Hard
Goal:max𝒇(𝑨) forsome𝐴 ⊆ 𝒰 suchthat
(Subset-Sum Problem)
𝑝(𝐴) ≥ 𝑃and𝑤(𝐴) ≤ 𝑊
Asetfunctiononallsubsetsof𝒰, 𝑓: 2|𝒰| → ℝA suchthat∀𝐴 ⊆ 𝐵 ⊆ 𝒰, 𝑒 ∈ 𝒰\B,
𝑓 𝐴 ∪ 𝑒 − 𝑓 𝐴 ≥ 𝑓 𝐵 ∪ 𝑒 − 𝑓(𝐵) (Submodular)
𝑓(𝐵) ≥ 𝑓(𝐴) (Monotone)
CoveringandPackingConstraints
W=50k＄
3 30k＄
6
2
20k＄
2
Foreachelemente ∈ 𝒰,wearegivesomeweightaprofitfunction𝑝: 𝒰 → ℕ,a
weightfunction𝑤: 𝒰 → ℕ ,aprofitrequirement𝑃 andabudget𝐵.
Feasibility NP-Hard
Goal:max𝒇(𝑨) forsome𝐴 ⊆ 𝒰 suchthat
(Subset-Sum Problem)
𝑝(𝐴) ≥ 𝑃and𝑤(𝐴) ≤ 𝑊
Asetfunctiononallsubsetsof𝒰, 𝑓: 2|𝒰| → ℝA suchthat∀𝐴 ⊆ 𝐵 ⊆ 𝒰, 𝑒 ∈ 𝒰\B,
𝑓 𝐴 ∪ 𝑒 − 𝑓 𝐴 ≥ 𝑓 𝐵 ∪ 𝑒 − 𝑓(𝐵) (Submodular)
𝑓(𝐵) ≥ 𝑓(𝐴) (Monotone)
StateoftheArtResults
•
MonotoneSubmodularMaximization
– UnderCardinalityConstraint
• Greedy1-1/e– apx [Nemhauser etal.,Math.Program.’78]
• Hardness:1-1/e [Nemhauser etal.,Math.O.R..,1978],[Feige,STOC’96]
– UnderPackingConstraint
• Greedy1-1/e– apx [Sviridenko,O.R.Lett.’04]
– O(1)PackingConstraints
• Multilinearrelaxation:1-1/e– ε – apx [Kulik etal.,SODA’09]
– MatroidConstraints
• Multilinearrelaxation:0.309– apx [Vondrák,STOC’08],[Calinescu etal.,IPCO’07]
•
Capacitatedk– MedianProblem
– Hardness:1.736 (followsfromtheuncapacitated case)
– Approximationfactor:O(logn)[Folklore]
– O(1)approximationalgorithmsundercapacity/cardinalityviolations.
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Algorithms
• Agreedybaseddynamicprogram.
• Outputsanapproximatesolution whichisbestofallpossiblegreedy
W‘
chains.
Greedy chain may get stuck
P‘
Bestgreedy chain
Weights/profits as in
optimum solution
Tableentry:𝑇[𝑙, 𝑃′, 𝑊′] stores𝑙-elementapproximatesolutionwith
profit𝑃′andweight𝑊′.
Tocompute𝑇[𝑙, 𝑃′, 𝑊′] pickbestwaytoextendanentry
𝑇[𝑙 − 1, 𝑃′′, 𝑊′′] byoneelementinafeasiblemanner.
†
…
Output:Bestentry𝑇[𝑙, 𝑃′, 𝑊′]with𝑃 ≥ ‡ and𝑊’ ≤ 𝑊.
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Algorithms
• Agreedybaseddynamicprogram.
• Outputsanapproximatesolution whichisbestofallpossiblegreedy
chains.
ForbiddenSets
Crucialtoreducetheprofitviolationfrom2 + 𝜀 downto 1 + 𝜀.
Worksforonecoveringandonepackingconstraint.
Idea:Foreach𝑇[𝑃′, 𝑊′],forbidthesuffixset𝐹‹… oforderedsetbynonŒ(•)
increasingŽ(•) suchthat
p(𝑇 𝑃… , 𝑊 … ∪ 𝐹‹… ) ≥ 𝑃 andw(𝑇 𝑃 … , 𝑊 … ∪ 𝐹‹… ) ≤ 𝑊.
Tableentry:𝑇[𝑙, 𝑃′, 𝑊′] stores𝑙-elementapproximatesolutionwith
profit𝑃′andweight𝑊′.
Tocompute𝑇[𝑙, 𝑃′, 𝑊′] pickbestwaytoextendanentry
𝑇[𝑙 − 1, 𝑃′′, 𝑊′′] byoneelementinafeasiblemanner.
†
…
Output:Bestentry𝑇[𝑙, 𝑃′, 𝑊′]with𝑃 ≥ ‡ and𝑊’ ≤ 𝑊.
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MainTheorems
Theorem1.Thereisanalgorithmformaximizingamonotonesubmodular
functionsubjecttoO(1)coveringandonepackingconstraintthatoutputsfor
any𝜺 > 𝟎in𝒏𝑶(𝟏/𝜺) timea4-approximatesolutionwithprofitatleast
𝟏
− 𝜺 𝑷andwithweightatmost 𝟏 + 𝜺 𝑾.
𝟐
Theorem2.Thereisanalgorithmformaximizingamonotonesubmodular
functionsubjecttoonecoveringandonepackingconstraintthatoutputsforany
𝜺 > 𝟎in𝒏𝑶(𝟏/𝜺) timea4-approximatesolutionwithprofitatleast 𝟏 − 𝜺 𝑷and
withweightatmost 𝟏 + 𝜺 𝑾.
Corollary.Thereisa2.6- approximationalgorithmfork-medianproblemwith
non-uniformandhardcapacitiesiftheunderlyingmetricspacehasonlytwo
possibledistances.
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Factor-revealingLP
𝐦𝐢𝐧 𝒂𝒏 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑡𝑜
(LP) 𝐦𝐚𝐱 ∑𝒏 𝒊 𝒚 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑡𝑜
𝒊¯𝟏 𝒏 𝒊
𝑖
𝑎¥ ≥ 1 − 𝑜¥ ;
𝑛
(DUAL)
𝑥© + 𝑦© − 𝑥©A¥ ≤ 0∀𝑖 ∈ 𝑛 − 1 ;
𝑖
𝑎© ≥ 𝑎©ª¥ + 1 − 𝑜© ∀𝑖 ∈ 𝑛 \{1};
𝑛
𝑥· + 𝑦· ≤ 1;
©
𝑖
𝑎© ≥
1 − - 𝑜® ∀𝑖 ∈ 𝑛 ;
𝑛
®¯¥
∑·®¯© 𝑦® − 1 − 𝑥© ≤ 0∀𝑖 ∈ 𝑛 ;
·
·
𝑎© ≥ 0,𝑜© ≥ 0∀𝑖 ∈ 𝑛 .
𝑥© ≥ 0,𝑦© ≥ 0∀𝑖 ∈ 𝑛 .
®
©
Theorem3.Thereisanalgorithmformaximizingamonotonesubmodular
functionsubjecttoO(1)coveringandonepackingconstraintthatoutputsfor
any𝜺 > 𝟎in𝒏𝑶(𝟏/𝜺) timeae-approximatesolutionwithprofitatleast
𝟏
− 𝜺 𝑷andwithweightatmost 𝟏 + 𝜺 𝑾.
𝟐
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OpenProblems
• Canwegetatight1 − 1/𝑒 approximationfor1 cardinalityand1
coveringconstraintwithoutanyviolationortheproblembecomes
harder?
• Canwegeta𝑂(1)approximationfor1 coveringand1 packing
constraintbyviolatingonlyoneconstraint?
• WhichotherproblemscanbenefitfromourgreedyDPtechnique?
14
Thankyou