A Computational Analysis of Minimal Unidirectional...
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A Computational Analysis of Minimal Unidire tionalCovering SetsDorothea Baumeister, Felix Brandt,Felix Fis her, Jan Ho�mann, and Jörg RotheEstoril, 10. April 2010
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Solution Con epts Unidire tional Covering Results SummaryOutline1 Solution Con epts2 Unidire tional Covering3 Results4 Summary
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Solution Con epts Unidire tional Covering Results SummarySolution Con eptsBinary dominan e relationsIdentify the �most desirable� elements in a pairwise majorityrelation:game theoryso ial hoi e theoryargumentation theorysports tournaments...Natural on ept: Choose the maximal element.
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Solution Con epts Unidire tional Covering Results SummaryExampleyx
z
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Solution Con epts Unidire tional Covering Results SummaryExampleyx
zMaximal elementz is the winner.
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Solution Con epts Unidire tional Covering Results SummaryExampleyx
zMaximal elementz is the winner.yx
z
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Solution Con epts Unidire tional Covering Results SummaryExampleyx
zMaximal elementz is the winner.?
yx
zMaximal elementThere is no winner!Condor et's Paradox renders maximality useless⇒ solution on epts
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Solution Con epts Unidire tional Covering Results SummarySolution Con ept: Minimal Unidire tional Covering SetsUnidire tional CoveringLet A be a �nite set of alternatives, B ⊆ A, ≻⊆ A× A adominan e relation, and let x , y ∈ B .x upward overs y (xCuy) if x ≻ y and for all z ∈ B , z ≻ ximplies z ≻ y .xCuy , zCux , and zCuy yx
z
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Solution Con epts Unidire tional Covering Results SummarySolution Con ept: Minimal Unidire tional Covering SetsUnidire tional CoveringLet A be a �nite set of alternatives, B ⊆ A, ≻⊆ A× A adominan e relation, and let x , y ∈ B .x upward overs y (xCuy) if x ≻ y and for all z ∈ B , z ≻ ximplies z ≻ y .x downward overs y (xCdy) if x ≻ y and for all z ∈ B , y ≻ zimplies x ≻ z .xCuy , zCux , and zCuyzCdx , zCdy , and xCdy yx
z
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Solution Con epts Unidire tional Covering Results SummarySolution Con ept: Minimal Unidire tional Covering SetsUn overed SetLet A be a �nite set of alternatives, B ⊆ A, ≻⊆ A× A adominan e relation, and let C be a overing relation on A. Theun overed set of B with respe t to C is:UCC (B) = {x ∈ B | yCx for no y ∈ B}.UCu({x , y , z}) = {z}UCd({x , y , z}) = {z} yx
z
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Solution Con epts Unidire tional Covering Results SummarySolution Con ept: Minimal Unidire tional Covering SetsMinimal Covering SetLet A be a �nite set of alternatives, ≻⊆ A× A a dominan erelation, and C a overing relation. B ⊆ A is a overing set for Aunder C , if:UCC (B) = B (internal stability), andfor all x ∈ A− B , x 6∈ UCC (B ∪ {x}) (external stability).Su h a B is minimal if no B ′ ⊂ B is a overing set for A under C .Minimal upward overing sets:B1 = {a, } and B2 = {b, d}Minimal downward overing set:B3 = {a, b, , d} a
cd
b
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Solution Con epts Unidire tional Covering Results SummaryMinimal Upward Covering Set MemberDe�nitionName: Minimal Upward Covering Set Member (MCu-Member).Instan e: A set A of alternatives, a dominan e relation ≻ on A,and a distinguished element d ∈ A.Question: Is d ontained in some minimal upward overing setfor A?A = {x , y , z}≻ = {(z , x), (z , y), (x , y)}
(A,≻, z) ∈ MCu-Member(A,≻, x) 6∈ MCu-Member
yx
z
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Solution Con epts Unidire tional Covering Results SummaryUnidire tional Covering Set ProblemsMCu-Size: Given a set A of alternatives, a dominan e relation≻ on A, and a positive integer k , does there exist someminimal upward overing set for A ontaining at most kalternatives?MCu-Member-All: Given a set A of alternatives, a dominan erelation ≻ on A, and a distinguished element d ∈ A, is d ontained in all minimal upward overing sets for A?MCu-Unique: Given a set A of alternatives and a dominan erelation ≻ on A, does there exist a unique minimal upward overing set for A?MCu-Test: Given a set A of alternatives, a dominan e relation≻ on A, and a subset M ⊆ A, is M a minimal upward overingset for A?MCu-Find: Given a set A of alternatives and a dominan erelation ≻ on A, �nd a minimal upward overing set for A.
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Solution Con epts Unidire tional Covering Results SummaryMinimality versus Minimum SizeSet-in lusion Minimality versus Minimum Cardinality ardinality: lassi al problems (maximum-size independentset, minimum-size dominating set, et .)set in lusion: minimal upward overing set member.⇒ Standard te hniques are not dire tly appli able.Upward overing sets:S = {a, , e}T = {b, d}set in lusion minimal: S and T ardinality minimal: only T
a b e
d c
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Solution Con epts Unidire tional Covering Results SummaryLower BoundApproa h for proving Θp2-hardnessPSfrag repla ements NP-hardness oNP-hardnessDP-hardness
Θp2-hardness
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Solution Con epts Unidire tional Covering Results SummaryNP-HardnessRedu tion from SAT to MCu-MemberThere is a satisfying assignment for ϕ⇔there is a minimal upward overing set that ontains d .
ϕ = (u ∨ v ∨ w) ∧ (u ∨ w )
qp
d
u u
u’ u’
v v
v’v’
w w
w’w’
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Solution Con epts Unidire tional Covering Results SummaryExample: NP-HardnessRedu tion from SAT to MCu-MemberThere is a satisfying assignment for ϕ⇔there is a minimal upward overing set that ontains d .
ϕ = (u ∨ v ∨ w) ∧ (u ∨ w )
d
p q
u u
u’ u’
v v
v’v’
w w
w’w’
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Solution Con epts Unidire tional Covering Results SummaryExample: NP-HardnessRedu tion from SAT to MCu-MemberThere is a satisfying assignment for ϕ⇔there is a minimal upward overing set that ontains d .
ϕ = (u ∨ v ∨ w) ∧ (u ∨ w )
p q
d
u u
u’ u’
v v
v’v’
w w
w’w’
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Solution Con epts Unidire tional Covering Results SummaryExample: NP-HardnessRedu tion from SAT to MCu-MemberThere is a satisfying assignment for ϕ⇔there is a minimal upward overing set that ontains d .
ϕ = (u ∨ v ∨ w) ∧ (u ∨ w ), satisfying assignment: u = v = w = 1u u
u’ u’
v v
v’v’
p q
d
w w
w’w’
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Solution Con epts Unidire tional Covering Results Summary oNP-HardnessThe lass oNPClass of sets whose omplements are in NP.Redu tion from SAT to the omplement of MCu-MemberThere is a satisfying assignment for ψ⇔there is no minimal upward overing set that ontains e.Additionally: e is ontained in all minimal upward overing sets ifand only if there is no satisfying assignment for ψ.
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Solution Con epts Unidire tional Covering Results SummaryExample: oNP-Hardnessψ = (a ∨ b) ∧ (b ∨ )
e e’
f f’
x y
z
b b
b’ b’
c c
c’
a a
a’a’ c’
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Solution Con epts Unidire tional Covering Results SummaryExample: oNP-Hardnessψ = (a ∨ b) ∧ (b ∨ )
x y
z
f f’
e e’
b b
b’ b’
c c
c’
a a
a’a’ c’
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Solution Con epts Unidire tional Covering Results SummaryExample: oNP-Hardnessψ = (a ∨ b) ∧ (b ∨ )
x y
z
f f’
e e’
b b
b’ b’
c c
c’
a a
a’a’ c’
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Solution Con epts Unidire tional Covering Results SummaryExample: oNP-Hardnessψ = (a ∨ b) ∧ (b ∨ )
x y
z
f f’
e e’
b b
b’ b’
c c
c’
a a
a’a’ c’
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Solution Con epts Unidire tional Covering Results SummaryExample: oNP-Hardnessψ = (a ∨ b) ∧ (b ∨ )
f f’
e e’
z
yx
b b
b’ b’
c c
c’
a a
a’a’ c’
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Solution Con epts Unidire tional Covering Results SummaryExample: oNP-Hardnessψ = (a ∨ b) ∧ (b ∨ ), satisfying assignment: a = b = = 1
yx
z
e’
b b
b’ b’
c c
c’
f f’
a a
a’a’ c’
e
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Solution Con epts Unidire tional Covering Results SummaryDP-HardnessThe lass DPThe lass of di�eren es of two NP sets: DP = {A−B |A,B ∈ NP}.NP ∪ oNP ⊆ DP.Wagner's Lemma for DP-HardnessLet A be some NP- omplete problem, let B be an arbitrary problem.If there exists a polynomial-time omputable fun tion f su h that,for all strings x1, x2 satsifying that if x2 ∈ A then x1 ∈ A, it holds:(x1 ∈ A and x2 6∈ A) ⇔ f (x1, x2) ∈ B ,then B is DP-hard.Constru tionThere is a satisfying assignment for ϕ, and none for ψ
⇔there is a minimal upward overing set that ontains d .
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Solution Con epts Unidire tional Covering Results SummaryProof Sket h: DP-HardnessCombination of the previously presented NP and oNP redu tions.construction
NP coNPconstruction
d
a’ a’
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Solution Con epts Unidire tional Covering Results SummaryΘ
p2-HardnessThe lass Θp2Θp2 (also known as PNP|| ) is the lass of problems solvable by apolynomial-time algorithm having parallel a ess to an NP ora le.NP ∪ oNP ⊆ DP ⊆ Θp2 .Wagner's Lemma for Θp2-HardnessLet A be some NP- omplete problem, and let B be an arbitraryproblem. If there exists a polynomial-time omputable fun tion fsu h that, for all m ≥ 1 and all strings x1, x2, . . . , x2m satisfyingthat if xj ∈ A then xj−1 ∈ A, 1 < j ≤ 2m, it holds that
||{i | xi ∈ A}|| is odd ⇔ f (x1, x2, . . . , x2m) ∈ B ,then B is Θp2-hard.
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Solution Con epts Unidire tional Covering Results SummaryProof Sket h: Θp2-HardnessCon atenation of the onstru tion used to show DP-hardness.
satisfiable formulas not satisfiable formulas
NP coNP NP coNP NP coNPconstr. constr. constr. constr. constr. constr.
d d d
a’ a’ a’ a’ a’ a’
There is some odd i su h that ϕi ∈ SAT and ϕi+1 6∈ SAT⇔there is a minimal upward overing set that ontains d .
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Solution Con epts Unidire tional Covering Results SummarySummary of ResultsProblem MCu , MCd MSCu MSCdSize NP- omplete NP- omplete NP- ompleteMember Θ
p2-hard, in Σp2 Θ
p2- omplete oNP-hard, in Θp2Member-All oNP- omplete Θ
p2- omplete oNP-hard, in Θp2Unique oNP-hard, in Σ
p2 oNP-hard, in Θp2 oNP-hard, in Θ
p2Test oNP- omplete oNP- omplete oNP- ompleteFind not in polynomial time unless P = NP
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Solution Con epts Unidire tional Covering Results SummaryThank you for your attention!The Complexity of Computing Minimal Unidire tionalCovering Sets, D. Baumeister, F. Brandt, F. Fis her, J. Ho�mann,and J. Rothe, to appear in the Pro eedings of CIAC 2010.