Competitive Evaluation of Threshold Functions and Game ...
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Competitive Evaluation ofThreshold Functions and Game Trees
in the Priced Information Model
Ferdinando Cicalese Martin Milanic
University of Salerno Bielefeld University
DIMACS-RUTCOR Workshop on Boolean and Pseudo-Boolean FunctionsIn Memory of Peter L. Hammer
January 19, 2009
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The function evaluation problem
Input:
a function f over the variables x1, . . . , xn
each variable has a positive cost of reading its value
an unknown assignment x1 = a1, . . . , xn = an
Goal:
Determine f (a1, . . . an)
adaptively reading the values of the variablesincurring little cost
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The function evaluation problem
f = (x and y) or (x and z)
x , y , z: binary variables
for some inputs it is possible to evaluate f without readingall variables
Example:
(x , y , z) = (0, 1, 1)
It is enough to know the value of x
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The function evaluation problem
f = (x and y) or (x and z)
x , y , z: binary variables
for some inputs it is possible to evaluate f without readingall variables
Example:
(x , y , z) = (0, 1, 1)
It is enough to know the value of x
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Algorithms for evaluating f
• Dynamically select the next variable based on the values ofthe variables read so far
• Stop when the value of f is determined
y
z x
x
0 1
0 10
1010
10
10
f = (x and y) or (x and z)
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Evaluation measure
Evasive Functions
• For any possible algorithm, all the variables must be readin the worst case.
• f = (x and y) or (x and z)
• Some important functions are evasive (e.g. game trees,AND/OR trees and threshold trees).
Worst case analysis cannot distinguish among theperformance of different algorithms.
Instead, we use competitive analysis (Charikar etal. 2002)
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Evaluation measure
Evasive Functions
• For any possible algorithm, all the variables must be readin the worst case.
• f = (x and y) or (x and z)
• Some important functions are evasive (e.g. game trees,AND/OR trees and threshold trees).
Worst case analysis cannot distinguish among theperformance of different algorithms.
Instead, we use competitive analysis (Charikar etal. 2002)
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Cost of evaluation
f = (x and y) or (x and z)cost(x) = 3, cost(y) = 4, cost(z) = 5
Asignment (x , y , z) Value of f Cheapest Proof Cost(0,0,0) 0 {x} 3(1,1,0) 1 {x,y} 3+4=7
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The competitive ratio
(x, y , z) f (x, y , z) Cost of Algorithm Ratio
Cheapest Cost
Proof
(0,0,0) 0 3 9 3(1,0,0) 0 9 9 1(0,1,0) 0 3 7 7/3(0,0,1) 0 3 12 4(1,1,0) 1 7 7 1(1,0,1) 1 8 12 3/2(0,1,1) 0 3 7 7/3(1,1,1) 1 7 7 1
y
z x
x
0 1
0 10
1010
10
10
4
5
9
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The competitive ratio
(x, y , z) f (x, y , z) Cost of Algorithm Ratio
Cheapest Cost
Proof
(0,0,0) 0 3 9 3(1,0,0) 0 9 9 1(0,1,0) 0 3 7 7/3(0,0,1) 0 3 12 4(1,1,0) 1 7 7 1(1,0,1) 1 8 12 3/2(0,1,1) 0 3 7 7/3(1,1,1) 1 7 7 1
y
z x
x
0 1
0 10
1010
10
10
4
3
7
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The competitive ratio
(x, y , z) f (x, y , z) Cost of Algorithm Ratio
Cheapest Cost
Proof
(0,0,0) 0 3 9 3(1,0,0) 0 9 9 1(0,1,0) 0 3 7 7/3(0,0,1) 0 3 12 4(1,1,0) 1 7 7 1(1,0,1) 1 8 12 3/2(0,1,1) 0 3 7 7/3(1,1,1) 1 7 7 1
y
z x
x
0 1
0 10
1010
10
10
4
5
3
3
9 7
12
7
12
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Measures of algorithm’s performance
Competitive ratio of algorithm A for (f , c):
maxall assignments σ
cost of A to evaluate f on σ
cost of cheapest proof of f on σ
In this talk:Extremal competitive ratio of A for f :
maxall assignments σ,all cost vectors c
cost of A to evaluate f on (σ, c)
cost of cheapest proof of f on (σ, c)
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Measures of algorithm’s performance
Competitive ratio of algorithm A for (f , c):
maxall assignments σ
cost of A to evaluate f on σ
cost of cheapest proof of f on σ
In this talk:Extremal competitive ratio of A for f :
maxall assignments σ,all cost vectors c
cost of A to evaluate f on (σ, c)
cost of cheapest proof of f on (σ, c)
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Measure of function’s complexity
Extremal competitive ratio of f :
minall deterministic algorithms A
that evaluate f
{
extremal competitive ratio of A for f}
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The function evaluation problem
Given:a function f over the variables x1, x2, . . . , xn
Combinatorial Goal:
• Determine the extremal competitive ratio of f
Algorithmic Goal:• Devise an algorithm for evaluating f that:
1. achieves the optimal (or close to optimal) extremalcompetitive ratio
2. is efficient (runs in time polynomial in the size of f )
The algorithm knows the reading costs.
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The function evaluation problem
Given:a function f over the variables x1, x2, . . . , xn
Combinatorial Goal:
• Determine the extremal competitive ratio of f
Algorithmic Goal:• Devise an algorithm for evaluating f that:
1. achieves the optimal (or close to optimal) extremalcompetitive ratio
2. is efficient (runs in time polynomial in the size of f )
The algorithm knows the reading costs.
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Applications
Applications of the function evaluation problem:
Reliability testing / diagnosisTelecommunications: testing connectivity of networks
Manufacturing: testing machines before shipping
DatabasesQuery optimization
Artificial IntelligenceFinding optimal derivation strategies in knowledge-based systems
Decision-making strategies (AND-OR trees)
Computer-aided game playing for two-player zero-sum games with
perfect information, e.g. chess (game trees)
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Related work - other models/measures
Non-uniform costs & competitive analysisCharikar et al. [STOC 2000, JCSS 2002]
Unknown costsCicalese and Laber [SODA 2006]
Restricted costs (selection and sorting)Gupta and Kumar [FOCS 2001], Kannan and Khanna [SODA 2003]
Randomized algorithmsSnir [TCS 1985], Saks and Wigderson [FOCS 1986], Laber [STACS 2004]
Stochastic modelsRandom input, uniform probabilities
Tarsi [JACM 1983], Boros and Unluyurt [AMAI 1999]Charikar et al. [STOC 2000, JCSS 2002], Greiner et al. [AI 2005]
Random input, arbitrary probabilitiesKaplan et al. [STOC 2005]
Random costsAngelov et al. [LATIN 2008]
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How to evaluate general functions?
Good algorithms are expected to test. . .
• cheap variables
• important variables ???
We usea linear program that captures the impact of the variables
(C.-Laber 2008)
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How to evaluate general functions?
Good algorithms are expected to test. . .
• cheap variables
• important variables ???
We usea linear program that captures the impact of the variables
(C.-Laber 2008)
Cicalese–Milani c Threshold Functions and Game Trees
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The minimal proofs
f - a function over V = {x1, . . . , xn}
R - range of f
DefinitionLet r ∈ R. A minimal proof for f (x) = r is a minimal set ofvariables P ⊆ V such that there is an assignment σ of values tothe variables in P such that fσ is constantly equal to r .
Example: f (x1, x2, x3) = (x1 and x2) or (x1 and x3), R = {0, 1}
minimal proofs for f (x) = 1: {{x1, x2}, {x1, x3}}
minimal proofs for f (x) = 0: {{x1}, {x2, x3}}
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The minimal proofs
f - a function over V = {x1, . . . , xn}
R - range of f
DefinitionLet r ∈ R. A minimal proof for f (x) = r is a minimal set ofvariables P ⊆ V such that there is an assignment σ of values tothe variables in P such that fσ is constantly equal to r .
Example: f (x1, x2, x3) = (x1 and x2) or (x1 and x3), R = {0, 1}
minimal proofs for f (x) = 1: {{x1, x2}, {x1, x3}}
minimal proofs for f (x) = 0: {{x1}, {x2, x3}}
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The Linear Program (C.-Laber 2008)
LP(f )
Minimize∑
x∈V s(x)s.t.∑
x∈P s(x) ≥ 1 for every minimal proof P of fs(x) ≥ 0 for every x ∈ V
Intuitively, s(x) measures the impact of variable x .
The feasible solutions to the LP(f ) are precisely the fractionalhitting sets of the set of minimal proofs of f .
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The Linear Program (C.-Laber 2008)
LP(f )
Minimize∑
x∈V s(x)s.t.∑
x∈P s(x) ≥ 1 for every minimal proof P of fs(x) ≥ 0 for every x ∈ V
Intuitively, s(x) measures the impact of variable x .
The feasible solutions to the LP(f ) are precisely the fractionalhitting sets of the set of minimal proofs of f .
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LP(f ): example
f (x1, x2, x3) = (x1 and x2) or (x1 and x3)
minimal proofs for f = 1: {{x1, x2}, {x1, x3}}
minimal proofs for f = 0: {{x1}, {x2, x3}}
LP(f )
Minimize s1 + s2 + s3
s.t.s1 + s2 ≥ 1s1 + s3 ≥ 1
s1 ≥ 1s2 + s3 ≥ 1
s1, s2, s3 ≥ 0
Optimal solution: s = (1, 1/2, 1/2)
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The Linear Programming Approach
LPA(f : function)
While the value of f is unknown
Select a feasible solution s() of LP(f )
Read the variable u which minimizes c(x)/s(x)(cost/impact)
c(x) = c(x)− s(x)c(u)/s(u)
f ← restriction of f after reading u
End While
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The LPA bounds the extremal competitive ratio
The selection of solution s determines both the computationalefficiency and the performance (extremal competitive ratio) ofthe algorithm.
Key Lemma (C.-Laber 2008)
Let K be a positive number. If
ObjectiveFunctionValue(s) ≤ K ,
for every selected solution s
then
ExtremalCompetitiveRatio(f ) ≤ K .
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Cross-intersecting families
A
B
• cross-intersecting: A ∈ A, B ∈ B ⇒ A ∩ B 6= ∅
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Minimal proofs and cross-intersecting families
f : S1 × · · · × Sn → S, a function over V = {x1, . . . , xn}
R - range of f
For r ∈ R, let P(r) denote the set of minimal proofs forf (x) = r .
Then:
for every r 6= r ′, the families P(r) and P(r ′) arecross-intersecting
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The Linear Program and cross-intersection
LP(f )
Minimize∑
x∈V s(x)s.t.∑
x∈P s(x) ≥ 1 for every P ∈ Ps(x) ≥ 0 for every x ∈ V
P = ∪r∈RP(r)
union of pairwise cross-intersecting families
For every function f : S1 × · · · × Sn → S, the LP(f ) seeks aminimal fractional hitting set of a union of pairwisecross-intersecting families.
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Cross-intersecting lemma
Cross-Intersecting Lemma (C.-Laber 2008)
Let A and B be two non-empty cross-intersecting families ofsubsets of V .Then, there is a fractional hitting set s of A ∪ B such that
‖s‖1 =∑
x∈V
s(x) ≤ max{|P| : P ∈ A ∪ B} .
• geometric proof
• generalizes to any number of pairwise cross-intersectingfamilies
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Applications of the cross-intersecting lemma
1 f : S1 × · · · × Sn → S, nonconstant:ExtremalCompetitiveRatio(f ) ≤ PROOF (f )
(PROOF (f ) = size of the largest minimal proof of f )
2 Monotone Boolean functions:ExtremalCompetitiveRatio(f ) = PROOF (f )
3 Game trees: ExtremalCompetitiveRatio(f ) ≤ TBA
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Applications of the cross-intersecting lemma
1 f : S1 × · · · × Sn → S, nonconstant:ExtremalCompetitiveRatio(f ) ≤ PROOF (f )
(PROOF (f ) = size of the largest minimal proof of f )
2 Monotone Boolean functions:ExtremalCompetitiveRatio(f ) = PROOF (f )
3 Game trees: ExtremalCompetitiveRatio(f ) ≤ TBA
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Applications of the cross-intersecting lemma
1 f : S1 × · · · × Sn → S, nonconstant:ExtremalCompetitiveRatio(f ) ≤ PROOF (f )
(PROOF (f ) = size of the largest minimal proof of f )
2 Monotone Boolean functions:ExtremalCompetitiveRatio(f ) = PROOF (f )
3 Game trees: ExtremalCompetitiveRatio(f ) ≤ TBA
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Game trees
x3
x1 x5x4
x2
x9
x8
x6
x7
MIN MIN
MAX
MAX
MAX
MIN
game tree
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Game trees
x3
x1 x5x4
x2
x9
x8
x6
x7
MIN MIN
MAX
MAX
MAX
MIN
2
8 6
5 7 4 5
91
game tree
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Game trees
x3
x1 x5x4
x2
x9
x8
x6
x7
MIN MIN
MAX
MAX
MAX
MIN
2 8
8 6
5 7 4 59
91
game tree
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Game trees
x3
x1 x5x4
x2
x9
x8
x6
x7
MIN MIN
MAX
MAX
MAX
MIN5 42
2 8
8 6
5 7 4 59
91
game tree
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Game trees
x3
x1 x5x4
x2
x9
x8
x6
x7
MIN MIN
MAX
MAX
MAX
MIN
5
5 42
2 8
8 6
5 7 4 59
91
game tree
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Minterms of game trees
Minterm : minimal set A ⊆ V of variables such that
value of f ≥ value of A := min value of variables in A
Minterms can prove a lower bound for the value of f .
x3
x1 x5x4
x2
x9
x8
x6
x7
MIN MIN
MAX
MAX
MAX
MIN
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Minterms of game trees
Minterm : minimal set A ⊆ V of variables such that
value of f ≥ value of A := min value of variables in A
Minterms can prove a lower bound for the value of f .
x3
x1 x5x4
x2
x9
x8
x6
x7
MIN MIN
MAX
MAX
MAX
MIN
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Minterms of game trees
Minterm : minimal set A ⊆ V of variables such that
value of f ≥ value of A := min value of variables in A
Minterms can prove a lower bound for the value of f .
x3
x1 x5x4
x2
x9
x8
x6
x7
MIN MIN
MAX
MAX
MAX
MIN
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Maxterms of game trees
Maxterm : minimal set B ⊆ V of variables such that
value of f ≤ value of B := max value of variables in B
Maxterms can prove an upper bound for the value of f .
x3
x1 x5x4
x2
x9
x8
x6
x7
MIN MIN
MAX
MAX
MAX
MIN
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Maxterms of game trees
Maxterm : minimal set B ⊆ V of variables such that
value of f ≤ value of B := max value of variables in B
Maxterms can prove an upper bound for the value of f .
x3
x1 x5x4
x2
x9
x8
x6
x7
MIN MIN
MAX
MAX
MAX
MIN
Cicalese–Milani c Threshold Functions and Game Trees
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Lower bound for the extremal competitive ratio
• k(f ) = max{|A| : A minterm of f}
• l(f ) = max{|B| : B maxterm of f}
Theorem (Cicalese-Laber 2005)
Let f be a game tree with no minterms or maxterms of size 1.Then,
ExtremalCompetitiveRatio(f ) ≥ max{k(f ), l(f )} .
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Minimal proofs of game trees
To prove that the value of f is b, we need:
• a minterm of value b [proves f ≥ b]
• a maxterm of value b [proves f ≤ b]
Every minimal proof = union of a minterm and a maxterm
A - minterm
B - maxterm
x9
x4
x1
x7 x6
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A first upper bound
It follows that
PROOF (f ) = size of the largest minimal proof of f= k(f ) + l(f )− 1.
Theorem (Cicalese-Laber 2008)
f : S1 × · · · × Sn → S, nonconstant:ExtremalCompetitiveRatio(f ) ≤ PROOF (f )
For a game tree f ,ExtremalCompetitiveRatio(f ) ≤ k(f ) + l(f )− 1.
Lower bound: max{k(f ), l(f )}
Cicalese–Milani c Threshold Functions and Game Trees
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A first upper bound
It follows that
PROOF (f ) = size of the largest minimal proof of f= k(f ) + l(f )− 1.
Theorem (Cicalese-Laber 2008)
f : S1 × · · · × Sn → S, nonconstant:ExtremalCompetitiveRatio(f ) ≤ PROOF (f )
For a game tree f ,ExtremalCompetitiveRatio(f ) ≤ k(f ) + l(f )− 1.
Lower bound: max{k(f ), l(f )}
Cicalese–Milani c Threshold Functions and Game Trees
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A first upper bound
It follows that
PROOF (f ) = size of the largest minimal proof of f= k(f ) + l(f )− 1.
Theorem (Cicalese-Laber 2008)
f : S1 × · · · × Sn → S, nonconstant:ExtremalCompetitiveRatio(f ) ≤ PROOF (f )
For a game tree f ,ExtremalCompetitiveRatio(f ) ≤ k(f ) + l(f )− 1.
Lower bound: max{k(f ), l(f )}
Cicalese–Milani c Threshold Functions and Game Trees
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Can we close the gap?
Yes:
Claim
For every restriction f ′ of f , there is a fractional hitting set s ofthe set of minimal proofs of f ′ such that
‖s‖1 ≤ max{k(f ), l(f )} .
By the Key Lemma ,
ExtremalCompetitiveRatio(f ) ≤ max{k(f ), l(f )}
Cicalese–Milani c Threshold Functions and Game Trees
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Can we close the gap?
Yes:
Claim
For every restriction f ′ of f , there is a fractional hitting set s ofthe set of minimal proofs of f ′ such that
‖s‖1 ≤ max{k(f ), l(f )} .
By the Key Lemma ,
ExtremalCompetitiveRatio(f ) ≤ max{k(f ), l(f )}
Cicalese–Milani c Threshold Functions and Game Trees
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Restrictions of game trees
x3
7 x5
6
x9
x8
15
x7
MIN MIN
MAX
MAX
MAX
MIN
3
restriction of f
f ′(x3, x5, x7, x8, x9)
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Restrictions of game trees
x3
7 x5
6
x9
x8
15
x7
MIN MIN
MAX
MAX
MAX
MIN
3
f (x3, x5, x7, x8, x9) ≥ 6
Cicalese–Milani c Threshold Functions and Game Trees
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Restrictions of game trees
x3
7 x5
6
x9
x8
15
x7
MIN MIN
MAX
MAX
MAX
MIN
3
f (x3, x5, x7, x8, x9) ≤ 15
Cicalese–Milani c Threshold Functions and Game Trees
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Restrictions of game trees
x3
7 x5
6
x9
x8
15
x7
MIN MIN
MAX
MAX
MAX
MIN
3
f (x3, x5, x7, x8, x9) ∈ [6, 15] = [LB, UB]
Cicalese–Milani c Threshold Functions and Game Trees
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Minimal proofs of a restriction of a game tree
minterm
maxterm
15
7
3
x7 x9
minterm
maxterm
x7 x9
x8
3
7
15
maxterm proofs minterm proofs
combined proofs
b = LB b = UB
minterm
maxterm
7
x9
x3
33
minterm
maxterm
6 7
x3
x9
3
LB ≤ b ≤ UB
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Case 1: No maxterm has been fully evaluated yet
minterm
maxterm
15
7
3
x7 x9
minterm
maxterm
x7 x9
x8
3
7
15
maxterm proofs minterm proofs
combined proofs
minterm
maxterm
7
x9
x3
33
minterm
maxterm
6 7
x3
x9
3
Cicalese–Milani c Threshold Functions and Game Trees
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Case 1: No maxterm has been fully evaluated yet
minterm
maxterm
15
7
3
x7 x9
minterm
maxterm
x7 x9
x8
3
7
15
maxterm proofs minterm proofs
combined proofs
x7 x9
x8
3
7
15
minterm
maxterm
7
x9
x3
33
minterm
maxterm
6 7
3
x7
x8
x7
x8
x7
x8
x3
x9
x7
x8
x7
x8
Let s be (the characteristic vector of) a minimal hitting set ofthe shaded sets .
‖s‖1 ≤ k(f ) ≤ max{k(f ), l(f )}
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Case 2: There is a fully evaluated maxterm
minterm
maxterm
15
7
3
x7 x9
minterm
maxterm
x7 x9
x8
3
7
15
maxterm proofs minterm proofs
combined proofs
minterm
maxterm
7
x9
x3
33
minterm
maxterm
6 7
x3
x9
3
Cicalese–Milani c Threshold Functions and Game Trees
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Case 2: There is a fully evaluated maxterm
minterm
maxterm
x7 x9
x8
3
7
15
maxterm proofs minterm proofs
combined proofs
x7 x9
x8
3
7
15
minterm
maxterm
7
x9
minterm
maxterm
15
7
3
x7 x9
x3
33
minterm
maxterm
6 7
3
x7
x8
x3
x9
x7
x8
x7
x8
R: the family of the minterm proofs
B: the family of the maxterm parts of the non-minterm proofs
Cicalese–Milani c Threshold Functions and Game Trees
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Case 2: There is a fully evaluated maxterm
R and B are non-empty sets
R and B are cross-intersecting
every minimal proof contains a member of R ∪ B
By the Cross-intersecting lemma , there exists a feasiblesolution s to the LP(f′) such that
‖s‖1 ≤ max{|P| : P ∈ R ∪ B} ≤ max{k(f ), l(f )} .
Cicalese–Milani c Threshold Functions and Game Trees
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Case 2: There is a fully evaluated maxterm
R and B are non-empty sets
R and B are cross-intersecting
every minimal proof contains a member of R ∪ B
By the Cross-intersecting lemma , there exists a feasiblesolution s to the LP(f′) such that
‖s‖1 ≤ max{|P| : P ∈ R ∪ B} ≤ max{k(f ), l(f )} .
Cicalese–Milani c Threshold Functions and Game Trees
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The extremal competitive ratio for game trees
TheoremLet f be a game tree with no minterms or maxterms of size 1.Then,
ExtremalCompetitiveRatio(f ) = max{k(f ), l(f )} .
Cicalese–Milani c Threshold Functions and Game Trees
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Value dependent costs
Suppose that the cost of reading a variable can depend onthe variable’s value:
c(x) =
{
50, if x = 0;1000, if x = 1.
TheoremLet f be a monotone Boolean function or a game tree. Then,
ExtremalCompetitiveRatio(f , r) = r · ECR(f )− r + 1 ,
where
r = maxx∈V
cmax (x)
cmin(x).
Cicalese–Milani c Threshold Functions and Game Trees
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Value dependent costs
Suppose that the cost of reading a variable can depend onthe variable’s value:
c(x) =
{
50, if x = 0;1000, if x = 1.
TheoremLet f be a monotone Boolean function or a game tree. Then,
ExtremalCompetitiveRatio(f , r) = r · ECR(f )− r + 1 ,
where
r = maxx∈V
cmax (x)
cmin(x).
Cicalese–Milani c Threshold Functions and Game Trees
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LPA has a very broad applicability
LPA does not depend on the structure of f
It can be used to derive upper bounds on the extremalcompetitive ratios of very different functions :
f = minimum of a list:ExtremalCompetitiveRatio(f ) ≤ n − 1 [Cicalese-Laber 2005]
f = the sorting function: ExtremalCompetitiveRatio(f ) ≤ n − 1[Cicalese-Laber 2008]
f : S1 × · · · × Sn → S, nonconstant:ExtremalCompetitiveRatio(f ) ≤ PROOF (f ) [Cicalese-Laber 2008]
f = monotone Boolean function:ExtremalCompetitiveRatio(f ) = PROOF (f ) [Cicalese-Laber 2008]
f = game tree: ExtremalCompetitiveRatio(f ) ≤ max{k(f ), l(f )}
Cicalese–Milani c Threshold Functions and Game Trees
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Summary
We have seen:
the Linear Programming Approach for the development ofcompetitive algorithms for the function evaluation problem,
the extremal competitive ratio for game trees ,
the more general model of value dependent costs.
Cicalese–Milani c Threshold Functions and Game Trees
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Part II
Threshold functionsand
Extended threshold tree functions
Cicalese–Milani c Threshold Functions and Game Trees
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Algorithmic issues: the state of the art
Are there efficient algorithms with optimal competitiveness?
game trees: there is a polynomial-time algorithm
monotone Boolean functions ??? OPEN QUESTIONsubclasses of monotone Boolean functions:
AND/OR trees = game trees with 0-1 values[Charikar et al. 2002]threshold tree functions [Cicalese-Laber 2005]
This talk:threshold functions (a quadratic algorithm )extended threshold tree functions(a pseudo-polynomial algorithm )
Cicalese–Milani c Threshold Functions and Game Trees
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Algorithmic issues: the state of the art
Are there efficient algorithms with optimal competitiveness?
game trees: there is a polynomial-time algorithm
monotone Boolean functions ??? OPEN QUESTIONsubclasses of monotone Boolean functions:
AND/OR trees = game trees with 0-1 values[Charikar et al. 2002]threshold tree functions [Cicalese-Laber 2005]
This talk:threshold functions (a quadratic algorithm )extended threshold tree functions(a pseudo-polynomial algorithm )
Cicalese–Milani c Threshold Functions and Game Trees
![Page 71: Competitive Evaluation of Threshold Functions and Game ...](https://reader030.fdocuments.in/reader030/viewer/2022020622/61ede60b530f9e4e7917cf63/html5/thumbnails/71.jpg)
Algorithmic issues: the state of the art
Are there efficient algorithms with optimal competitiveness?
game trees: there is a polynomial-time algorithm
monotone Boolean functions ??? OPEN QUESTIONsubclasses of monotone Boolean functions:
AND/OR trees = game trees with 0-1 values[Charikar et al. 2002]threshold tree functions [Cicalese-Laber 2005]
This talk:threshold functions (a quadratic algorithm )extended threshold tree functions(a pseudo-polynomial algorithm )
Cicalese–Milani c Threshold Functions and Game Trees
![Page 72: Competitive Evaluation of Threshold Functions and Game ...](https://reader030.fdocuments.in/reader030/viewer/2022020622/61ede60b530f9e4e7917cf63/html5/thumbnails/72.jpg)
Algorithmic issues: the state of the art
Are there efficient algorithms with optimal competitiveness?
game trees: there is a polynomial-time algorithm
monotone Boolean functions ??? OPEN QUESTIONsubclasses of monotone Boolean functions:
AND/OR trees = game trees with 0-1 values[Charikar et al. 2002]threshold tree functions [Cicalese-Laber 2005]
This talk:threshold functions (a quadratic algorithm )extended threshold tree functions(a pseudo-polynomial algorithm )
Cicalese–Milani c Threshold Functions and Game Trees
![Page 73: Competitive Evaluation of Threshold Functions and Game ...](https://reader030.fdocuments.in/reader030/viewer/2022020622/61ede60b530f9e4e7917cf63/html5/thumbnails/73.jpg)
Algorithmic issues: the state of the art
Are there efficient algorithms with optimal competitiveness?
game trees: there is a polynomial-time algorithm
monotone Boolean functions ??? OPEN QUESTIONsubclasses of monotone Boolean functions:
AND/OR trees = game trees with 0-1 values[Charikar et al. 2002]threshold tree functions [Cicalese-Laber 2005]
This talk:threshold functions (a quadratic algorithm )extended threshold tree functions(a pseudo-polynomial algorithm )
Cicalese–Milani c Threshold Functions and Game Trees
![Page 74: Competitive Evaluation of Threshold Functions and Game ...](https://reader030.fdocuments.in/reader030/viewer/2022020622/61ede60b530f9e4e7917cf63/html5/thumbnails/74.jpg)
Algorithmic issues: the state of the art
Are there efficient algorithms with optimal competitiveness?
game trees: there is a polynomial-time algorithm
monotone Boolean functions ??? OPEN QUESTIONsubclasses of monotone Boolean functions:
AND/OR trees = game trees with 0-1 values[Charikar et al. 2002]threshold tree functions [Cicalese-Laber 2005]
This talk:threshold functions (a quadratic algorithm )extended threshold tree functions(a pseudo-polynomial algorithm )
Cicalese–Milani c Threshold Functions and Game Trees
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Threshold Functions
f : {0, 1}n → {0, 1} is a threshold function if ∃w1, . . . , wn, tintegers s.t.
f (x1, . . . , xn) =
{
1 if∑
i xiwi ≥ t0 otherwise
Separating structure
(w1, . . . , wn; t) is a separating structure of f
We assume that 1 ≤ wi ≤ t for all i .
Cicalese–Milani c Threshold Functions and Game Trees
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Threshold Functions
f : {0, 1}n → {0, 1} is a threshold function if ∃w1, . . . , wn, tintegers s.t.
f (x1, . . . , xn) =
{
1 if∑
i xiwi ≥ t0 otherwise
Separating structure
(w1, . . . , wn; t) is a separating structure of f
We assume that 1 ≤ wi ≤ t for all i .
Cicalese–Milani c Threshold Functions and Game Trees
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Applications
Switching networks
Automatic diagnosis
Mutually exclusive mechanisms
Decision-making strategies
Neural networks
Weighted majority games
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Outline
Separating structures for f and feasible solutions for LP(f )Quadratic γ(f )-competitive algo for threshold functions
γ(f ) = extremal competitive ratio of f
The range of separating structures of f
Extended threshold tree functionsPseudo-polytime γ(f )-competitive algo for extendedthreshold tree functionsThe HLPf for studying function evaluation
Cicalese–Milani c Threshold Functions and Game Trees
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Outline
Separating structures for f and feasible solutions for LP(f )Quadratic γ(f )-competitive algo for threshold functions
γ(f ) = extremal competitive ratio of f
The range of separating structures of f
Extended threshold tree functionsPseudo-polytime γ(f )-competitive algo for extendedthreshold tree functionsThe HLPf for studying function evaluation
Cicalese–Milani c Threshold Functions and Game Trees
![Page 80: Competitive Evaluation of Threshold Functions and Game ...](https://reader030.fdocuments.in/reader030/viewer/2022020622/61ede60b530f9e4e7917cf63/html5/thumbnails/80.jpg)
Outline
Separating structures for f and feasible solutions for LP(f )Quadratic γ(f )-competitive algo for threshold functions
γ(f ) = extremal competitive ratio of f
The range of separating structures of f
Extended threshold tree functionsPseudo-polytime γ(f )-competitive algo for extendedthreshold tree functionsThe HLPf for studying function evaluation
Cicalese–Milani c Threshold Functions and Game Trees
![Page 81: Competitive Evaluation of Threshold Functions and Game ...](https://reader030.fdocuments.in/reader030/viewer/2022020622/61ede60b530f9e4e7917cf63/html5/thumbnails/81.jpg)
Outline
Separating structures for f and feasible solutions for LP(f )Quadratic γ(f )-competitive algo for threshold functions
γ(f ) = extremal competitive ratio of f
The range of separating structures of f
Extended threshold tree functionsPseudo-polytime γ(f )-competitive algo for extendedthreshold tree functionsThe HLPf for studying function evaluation
Cicalese–Milani c Threshold Functions and Game Trees
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Certificates of a threshold function
f threshold function with separating structure (w1, . . . , wn; t).
X is a minterm (prime implicant) of f∑
X
wi ≥ t and∑
X\{j}
wi < t for every j ∈ X
X is a maxterm (prime implicate) of f∑
V\X
wi < t and∑
V\X∪{j}
wi ≥ t for every j ∈ X
Technical assumption
t ≤ 12(
∑
i wi + 1), i.e., f is dual majorthen every maxterm contains a minterm
Cicalese–Milani c Threshold Functions and Game Trees
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Certificates of a threshold function
f threshold function with separating structure (w1, . . . , wn; t).
X is a minterm (prime implicant) of f∑
X
wi ≥ t and∑
X\{j}
wi < t for every j ∈ X
X is a maxterm (prime implicate) of f∑
V\X
wi < t and∑
V\X∪{j}
wi ≥ t for every j ∈ X
Technical assumption
t ≤ 12(
∑
i wi + 1), i.e., f is dual majorthen every maxterm contains a minterm
Cicalese–Milani c Threshold Functions and Game Trees
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Certificates of a threshold function
f threshold function with separating structure (w1, . . . , wn; t).
X is a minterm (prime implicant) of f∑
X
wi ≥ t and∑
X\{j}
wi < t for every j ∈ X
X is a maxterm (prime implicate) of f∑
V\X
wi < t and∑
V\X∪{j}
wi ≥ t for every j ∈ X
Technical assumption
t ≤ 12(
∑
i wi + 1), i.e., f is dual majorthen every maxterm contains a minterm
Cicalese–Milani c Threshold Functions and Game Trees
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Certificates of a threshold function
f threshold function with separating structure (w1, . . . , wn; t).
X is a minterm (prime implicant) of f∑
X
wi ≥ t and∑
X\{j}
wi < t for every j ∈ X
X is a maxterm (prime implicate) of f∑
V\X
wi < t and∑
V\X∪{j}
wi ≥ t for every j ∈ X
Technical assumption
t ≤ 12(
∑
i wi + 1), i.e., f is dual majorthen every maxterm contains a minterm
Cicalese–Milani c Threshold Functions and Game Trees
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Certificates of a threshold function
f threshold function with separating structure (w1, . . . , wn; t).
X is a minterm (prime implicant) of f∑
X
wi ≥ t and∑
X\{j}
wi < t for every j ∈ X
X is a maxterm (prime implicate) of f∑
V\X
wi < t and∑
V\X∪{j}
wi ≥ t for every j ∈ X
Technical assumption
t ≤ 12(
∑
i wi + 1), i.e., f is dual majorthen every maxterm contains a minterm
Cicalese–Milani c Threshold Functions and Game Trees
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Separating Structure and LP (f )
LemmaEvery separating structure (w1, . . . , wn; t) for f induces afeasible solution s = (s(x1), . . . , s(xn)) for LP(f ).
s(xi ) = wi/t
X is a minterm of f∑
i∈X
s(xi ) =∑
i∈X
wi
t≥ 1.
Y is a maxterm of f
∃X ⊆ Y s.t. X is a minterm ⇒∑
i∈Y
s(xi) ≥∑
i∈X
s(xi ) ≥ 1.
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Separating Structure and LP (f )
LemmaEvery separating structure (w1, . . . , wn; t) for f induces afeasible solution s = (s(x1), . . . , s(xn)) for LP(f ).
s(xi ) = wi/t
X is a minterm of f∑
i∈X
s(xi ) =∑
i∈X
wi
t≥ 1.
Y is a maxterm of f
∃X ⊆ Y s.t. X is a minterm ⇒∑
i∈Y
s(xi) ≥∑
i∈X
s(xi ) ≥ 1.
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Separating Structure and LP (f )
LemmaEvery separating structure (w1, . . . , wn; t) for f induces afeasible solution s = (s(x1), . . . , s(xn)) for LP(f ).
s(xi ) = wi/t
X is a minterm of f∑
i∈X
s(xi ) =∑
i∈X
wi
t≥ 1.
Y is a maxterm of f
∃X ⊆ Y s.t. X is a minterm ⇒∑
i∈Y
s(xi) ≥∑
i∈X
s(xi ) ≥ 1.
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Separating Structure and LP (f )
LemmaEvery separating structure (w1, . . . , wn; t) for f induces afeasible solution s for LP(f ) s.t.
‖s‖1 = val(w; t) = 1/t∑
wi .
How bad can the best separating structure be for the purposeof the LPA?
Cicalese–Milani c Threshold Functions and Game Trees
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Separating Structure and LP (f )
LemmaEvery separating structure (w1, . . . , wn; t) for f induces afeasible solution s for LP(f ) s.t.
‖s‖1 = val(w; t) = 1/t∑
wi .
How bad can the best separating structure be for the purposeof the LPA?
Cicalese–Milani c Threshold Functions and Game Trees
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Separating Structure and LP (f )
TheoremFor any thr. func. f , there exists sep. str. (w; t) s.t.val(w; t) =
∑
wi/t ≤ max{k(f ), l(f )}.
induces an optimal implementation of the LPA
Theorem
Every pair of separating structures (w; t), (w′; t ′) satisfies
max{ ∑
wi/t∑
w ′i /t ′
,
∑
w ′i /t ′
∑
wi/t
}
≤ 2
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Separating Structure and LP (f )
TheoremFor any thr. func. f , there exists sep. str. (w; t) s.t.val(w; t) =
∑
wi/t ≤ max{k(f ), l(f )}.
induces an optimal implementation of the LPA
Theorem
Every pair of separating structures (w; t), (w′; t ′) satisfies
max{ ∑
wi/t∑
w ′i /t ′
,
∑
w ′i /t ′
∑
wi/t
}
≤ 2
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Separating Structure and LP (f )
TheoremFor any thr. func. f , there exists sep. str. (w; t) s.t.val(w; t) =
∑
wi/t ≤ max{k(f ), l(f )}.
induces an optimal implementation of the LPA
Theorem
Every pair of separating structures (w; t), (w′; t ′) satisfies
max{ ∑
wi/t∑
w ′i /t ′
,
∑
w ′i /t ′
∑
wi/t
}
≤ 2
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Separating Structure optimal for the LP (f )
TheoremFor any thr. func. f , there exists sep. str. (w; t) s.t.∑
wi/t ≤ max{k(f ), l(f )}.
Let f : (w; t) and suppose that∑
wi/t > max{k(f ), l(f )}
Every maxterm has size l(f ) ≥ k(f )if not
∑
V wi =∑
P wi +∑
V\P wi ≤ (l(f )− 1)t + t ≤ l(f )t(contradiction)
Every pair of maxterms P1, P2 satisfies |P1 ∩ P2| = l(f )− 1if not
∑
V wi ≤∑
P1∩P2wi +
∑
V\P1wi +
∑
V\P2wi ≤
(l(f )− 2)t + 2t = l(f )t(contradiction)
Fix a maxterm P. Then ∀P ′ (maxterm) P ′ = P \ {x} ∪ {y}for some x ∈ P, y 6∈ P.
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Separating Structure optimal for the LP (f )
TheoremFor any thr. func. f , there exists sep. str. (w; t) s.t.∑
wi/t ≤ max{k(f ), l(f )}.
Let f : (w; t) and suppose that∑
wi/t > max{k(f ), l(f )}
Every maxterm has size l(f ) ≥ k(f )if not
∑
V wi =∑
P wi +∑
V\P wi ≤ (l(f )− 1)t + t ≤ l(f )t(contradiction)
Every pair of maxterms P1, P2 satisfies |P1 ∩ P2| = l(f )− 1if not
∑
V wi ≤∑
P1∩P2wi +
∑
V\P1wi +
∑
V\P2wi ≤
(l(f )− 2)t + 2t = l(f )t(contradiction)
Fix a maxterm P. Then ∀P ′ (maxterm) P ′ = P \ {x} ∪ {y}for some x ∈ P, y 6∈ P.
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Separating Structure optimal for the LP (f )
TheoremFor any thr. func. f , there exists sep. str. (w; t) s.t.∑
wi/t ≤ max{k(f ), l(f )}.
Let f : (w; t) and suppose that∑
wi/t > max{k(f ), l(f )}
Every maxterm has size l(f ) ≥ k(f )if not
∑
V wi =∑
P wi +∑
V\P wi ≤ (l(f )− 1)t + t ≤ l(f )t(contradiction)
Every pair of maxterms P1, P2 satisfies |P1 ∩ P2| = l(f )− 1if not
∑
V wi ≤∑
P1∩P2wi +
∑
V\P1wi +
∑
V\P2wi ≤
(l(f )− 2)t + 2t = l(f )t(contradiction)
Fix a maxterm P. Then ∀P ′ (maxterm) P ′ = P \ {x} ∪ {y}for some x ∈ P, y 6∈ P.
Cicalese–Milani c Threshold Functions and Game Trees
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Separating Structure optimal for the LP (f )
TheoremFor any thr. func. f , there exists sep. str. (w; t) s.t.∑
wi/t ≤ max{k(f ), l(f )}.
Let f : (w; t) and suppose that∑
wi/t > max{k(f ), l(f )}
Every maxterm has size l(f ) ≥ k(f )if not
∑
V wi =∑
P wi +∑
V\P wi ≤ (l(f )− 1)t + t ≤ l(f )t(contradiction)
Every pair of maxterms P1, P2 satisfies |P1 ∩ P2| = l(f )− 1if not
∑
V wi ≤∑
P1∩P2wi +
∑
V\P1wi +
∑
V\P2wi ≤
(l(f )− 2)t + 2t = l(f )t(contradiction)
Fix a maxterm P. Then ∀P ′ (maxterm) P ′ = P \ {x} ∪ {y}for some x ∈ P, y 6∈ P.
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Separating Structure optimal for the LP (f )
TheoremFor any thr. func. f , there exists sep. str. (w; t) s.t.∑
wi/t ≤ max{k(f ), l(f )}.
Let f : (w; t) and suppose that∑
wi/t > max{k(f ), l(f )}
Every maxterm has size l(f ) ≥ k(f )if not
∑
V wi =∑
P wi +∑
V\P wi ≤ (l(f )− 1)t + t ≤ l(f )t(contradiction)
Every pair of maxterms P1, P2 satisfies |P1 ∩ P2| = l(f )− 1if not
∑
V wi ≤∑
P1∩P2wi +
∑
V\P1wi +
∑
V\P2wi ≤
(l(f )− 2)t + 2t = l(f )t(contradiction)
Fix a maxterm P. Then ∀P ′ (maxterm) P ′ = P \ {x} ∪ {y}for some x ∈ P, y 6∈ P.
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Separating Structure optimal for the LP (f )
TheoremFor any thr. func. f , there exists sep. str. (w; t) s.t.∑
wi/t ≤ max{k(f ), l(f )}.
Let f : (w; t) and suppose that∑
wi/t > max{k(f ), l(f )
Every maxterm has size l(f ) ≥ k(f )
Every two maxterms P1, P2 satisfy |P1 ∩ P2| = l(f )− 1
Fix a maxterm P. Then ∀P ′ (maxterm) P ′ = P \ {x} ∪ {y}
for some x ∈ P, y 6∈ P.
x
y
z
u
P
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Separating Structure optimal for the LP (f )
TheoremFor any thr. func. f , there exists sep. str. (w; t) s.t.∑
wi/t ≤ max{k(f ), l(f )}.
Let f : (w; t) and suppose that∑
wi/t > max{k(f ), l(f )
Every maxterm has size l(f ) ≥ k(f )
Every maxterms P1, P2 satisfy |P1 ∩ P2| = l(f )− 1
Fix a maxterm P. There are two possible cases:
x
y1
z
u
P
y2
y2 � y1
z
u
P
x
y1
z
u
P
y2
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Separating Structure optimal for the LP (f )
TheoremFor any thr. func. f , there exists sep. str. (w; t) s.t.∑
wi/t ≤ max{k(f ), l(f )}.
Fix a maxterm P, we have two cases
1. One variable of P is substitutable
x
y1
z
u
P
y2
y2
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Separating Structure optimal for the LP (f )
TheoremFor any thr. func. f , there exists sep. str. (w; t) s.t.∑
wi/t ≤ max{k(f ), l(f )}.
Fix a maxterm P, we have two cases
Case 2: Only one variable outside P
x
y1
z
u
P
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Linear algorithm for the opt. Separating Structure
f : given by (w; t)
Verify if f belongs to one of the two above cases(can be done in linear time).
If so: construct an explicit sep. str. (w′; t ′)
If not: the given sep. str. (w; t) is optimal for LP(f ).
Combined with the LPA framework:quadratic γ(f )-competitive algorithm for threshold functions
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Linear algorithm for the opt. Separating Structure
f : given by (w; t)
Verify if f belongs to one of the two above cases(can be done in linear time).
If so: construct an explicit sep. str. (w′; t ′)
If not: the given sep. str. (w; t) is optimal for LP(f ).
Combined with the LPA framework:quadratic γ(f )-competitive algorithm for threshold functions
Cicalese–Milani c Threshold Functions and Game Trees
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Linear algorithm for the opt. Separating Structure
f : given by (w; t)
Verify if f belongs to one of the two above cases(can be done in linear time).
If so: construct an explicit sep. str. (w′; t ′)
If not: the given sep. str. (w; t) is optimal for LP(f ).
Combined with the LPA framework:quadratic γ(f )-competitive algorithm for threshold functions
Cicalese–Milani c Threshold Functions and Game Trees
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Linear algorithm for the opt. Separating Structure
f : given by (w; t)
Verify if f belongs to one of the two above cases(can be done in linear time).
If so: construct an explicit sep. str. (w′; t ′)
If not: the given sep. str. (w; t) is optimal for LP(f ).
Combined with the LPA framework:quadratic γ(f )-competitive algorithm for threshold functions
Cicalese–Milani c Threshold Functions and Game Trees
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Any separating structure guarantees 2γ(f )
Theorem
Every pair of separating structures (w; t), (w′; t ′) satisfies
max{
val(w; t)val(w′; t ′)
,val(w′; t ′)val(w; t)
}
≤ 2
τ∗(H) ≤ val(w; t) ≤ χ∗(H) ≤ χ(H) ≤ 2τ∗(H)
This bound is sharp:
(t , t ; t + 1) for t ≥ 1 all define the same f
for t = 1, we get val(w; t) = 1
val(w; t)→ 2 as t →∞
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Any separating structure guarantees 2γ(f )
Theorem
Every pair of separating structures (w; t), (w′; t ′) satisfies
max{
val(w; t)val(w′; t ′)
,val(w′; t ′)val(w; t)
}
≤ 2
τ∗(H) ≤ val(w; t) ≤ χ∗(H) ≤ χ(H) ≤ 2τ∗(H)
This bound is sharp:
(t , t ; t + 1) for t ≥ 1 all define the same f
for t = 1, we get val(w; t) = 1
val(w; t)→ 2 as t →∞
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Any separating structure guarantees 2γ(f )
Theorem
Every pair of separating structures (w; t), (w′; t ′) satisfies
max{
val(w; t)val(w′; t ′)
,val(w′; t ′)val(w; t)
}
≤ 2
τ∗(H) ≤ val(w; t) ≤ χ∗(H) ≤ χ(H) ≤ 2τ∗(H)
This bound is sharp:
(t , t ; t + 1) for t ≥ 1 all define the same f
for t = 1, we get val(w; t) = 1
val(w; t)→ 2 as t →∞
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Any separating structure guarantees 2γ(f )
Theorem
Every pair of separating structures (w; t), (w′; t ′) satisfies
max{
val(w; t)val(w′; t ′)
,val(w′; t ′)val(w; t)
}
≤ 2
τ∗(H) ≤ val(w; t) ≤ χ∗(H) ≤ χ(H) ≤ 2τ∗(H)
This bound is sharp:
(t , t ; t + 1) for t ≥ 1 all define the same f
for t = 1, we get val(w; t) = 1
val(w; t)→ 2 as t →∞
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Extended threshold tree functions
x3
x1
x2
x8
(1, 2, 2, 3; 5)
(1, 1; 2)
x11x10
x5x4x9
(1, 1; 2)(1, 1; 2)
(1, 1; 1)
(3, 2, 2, 1; 6)
x7
x6
Threshold tree functions: w = (1, . . . , 1) at every nodeExtended threshold tree functions: no such restriction
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Extended threshold tree functions
x3
x1
x2
x8
(1, 2, 2, 3; 5)
(1, 1; 2)
x11x10
x5x4x9
(1, 1; 2) (1, 1; 2)
(1, 1; 1)
(3, 2, 2, 1; 6)
x7
x6 1
1 01
1
10
1
1
10
Threshold tree functions: w = (1, . . . , 1) at every nodeExtended threshold tree functions: no such restriction
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Extended threshold tree functions
x3
x1
x2
x8
(1, 2, 2, 3; 5)
(1, 1; 2)
x11x10
x5x4x9
(1, 1; 2) (1, 1; 2)
(1, 1; 1)
(3, 2, 2, 1; 6)
x7
x6 1
1
1 01
1
11 0
1
1
10
Threshold tree functions: w = (1, . . . , 1) at every nodeExtended threshold tree functions: no such restriction
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Extended threshold tree functions
x3
x1
x2
x8
(1, 2, 2, 3; 5)
(1, 1; 2)
x11x10
x5x4x9
(1, 1; 2) (1, 1; 2)
(1, 1; 1)
(3, 2, 2, 1; 6)
x7
x61 0 11
1
1 01
1
11 0
1
1
10
Threshold tree functions: w = (1, . . . , 1) at every nodeExtended threshold tree functions: no such restriction
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Extended threshold tree functions
x3
x1
x2
x8
(1, 2, 2, 3; 5)
(1, 1; 2)
x11x10
x5x4x9
(1, 1; 2) (1, 1; 2)
(1, 1; 1)
(3, 2, 2, 1; 6)
x7
x6
1
1 0 11
1
1 01
1
11 0
1
1
10
Threshold tree functions: w = (1, . . . , 1) at every nodeExtended threshold tree functions: no such restriction
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Relation to threshold functionsand threshold tree functions
This class properly contains the classes of threshold functionsand threshold tree functions:
Threshold tree functions
Extended threshold tree functions
Threshold functions
x1x2 ∨ x3x4(2,2,3,1;5)
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Pseudo-polynomial algorithm based on LPA
LPA(f : function)
While the value of f is unknown
Select a feasible solution s() of LP(f )
Read the variable u which minimizes c(x)/s(x)(cost/impact)
c(x) = c(x)− s(x)c(u)/s(u)
f ← restriction of f after reading u
End While
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Constructing s
Ti
ki, li, ci, di
si
g : (w1, . . . , wr; t)
T
si : the solution to the LPTi
ki , li : maximum size of a minterm (maxterm) in Ti
ci , di ≥ 1
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Constructing s
Ti
ki, li, ci, di
si
g : (w1, . . . , wr; t)
T
s(x) =zi · si(x)
δ
δ > 0: scaling factorz: a nontrivial solution of HLPg
Cicalese–Milani c Threshold Functions and Game Trees
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Constructing s (main steps)
1. compute k(T ) and l(T ): dynamic programming,using (w; t), the ki ’s and the the li ’s
2. compute a solution z 6= 0 to the following system HLPg:
HLPg∑
ciki zi ≤ k(T ) ·∑
P cizi ∀maxterm P of g,∑
di lizi ≤ l(T ) ·∑
P dizi ∀minterm P of g,zi ≥ 0 ∀ i = 1, . . . , r
dynamic prog. algorithm for the separation problem(linearly many knapsack problems)
3. compute s:
s(x) =zi · si(x)
δ
Cicalese–Milani c Threshold Functions and Game Trees
![Page 122: Competitive Evaluation of Threshold Functions and Game ...](https://reader030.fdocuments.in/reader030/viewer/2022020622/61ede60b530f9e4e7917cf63/html5/thumbnails/122.jpg)
Constructing s (main steps)
1. compute k(T ) and l(T ): dynamic programming,using (w; t), the ki ’s and the the li ’s
2. compute a solution z 6= 0 to the following system HLPg:
HLPg∑
ciki zi ≤ k(T ) ·∑
P cizi ∀maxterm P of g,∑
di lizi ≤ l(T ) ·∑
P dizi ∀minterm P of g,zi ≥ 0 ∀ i = 1, . . . , r
dynamic prog. algorithm for the separation problem(linearly many knapsack problems)
3. compute s:
s(x) =zi · si(x)
δ
Cicalese–Milani c Threshold Functions and Game Trees
![Page 123: Competitive Evaluation of Threshold Functions and Game ...](https://reader030.fdocuments.in/reader030/viewer/2022020622/61ede60b530f9e4e7917cf63/html5/thumbnails/123.jpg)
Constructing s (main steps)
1. compute k(T ) and l(T ): dynamic programming,using (w; t), the ki ’s and the the li ’s
2. compute a solution z 6= 0 to the following system HLPg:
HLPg∑
ciki zi ≤ k(T ) ·∑
P cizi ∀maxterm P of g,∑
di lizi ≤ l(T ) ·∑
P dizi ∀minterm P of g,zi ≥ 0 ∀ i = 1, . . . , r
dynamic prog. algorithm for the separation problem(linearly many knapsack problems)
3. compute s:
s(x) =zi · si(x)
δ
Cicalese–Milani c Threshold Functions and Game Trees
![Page 124: Competitive Evaluation of Threshold Functions and Game ...](https://reader030.fdocuments.in/reader030/viewer/2022020622/61ede60b530f9e4e7917cf63/html5/thumbnails/124.jpg)
Generalization
The algorithm can be generalized to work for any tree functionwhose node connectives are monotone Boolean functions.
G: a class of functions closed under restrictions
Suppose that all functions at the nodes belong to G.
The complexity of the algo depends on the complexity of:
Finding a cheapest minterm (maxterm) of a given g ∈ G.
Computing a restriction of a given g ∈ G.
Testing whether g ≡ 0 (g ≡ 1) for a given g ∈ G.
Cicalese–Milani c Threshold Functions and Game Trees
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Generalization
The algorithm can be generalized to work for any tree functionwhose node connectives are monotone Boolean functions.
G: a class of functions closed under restrictions
Suppose that all functions at the nodes belong to G.
The complexity of the algo depends on the complexity of:
Finding a cheapest minterm (maxterm) of a given g ∈ G.
Computing a restriction of a given g ∈ G.
Testing whether g ≡ 0 (g ≡ 1) for a given g ∈ G.
Cicalese–Milani c Threshold Functions and Game Trees
![Page 126: Competitive Evaluation of Threshold Functions and Game ...](https://reader030.fdocuments.in/reader030/viewer/2022020622/61ede60b530f9e4e7917cf63/html5/thumbnails/126.jpg)
Generalization
The algorithm can be generalized to work for any tree functionwhose node connectives are monotone Boolean functions.
G: a class of functions closed under restrictions
Suppose that all functions at the nodes belong to G.
The complexity of the algo depends on the complexity of:
Finding a cheapest minterm (maxterm) of a given g ∈ G.
Computing a restriction of a given g ∈ G.
Testing whether g ≡ 0 (g ≡ 1) for a given g ∈ G.
Cicalese–Milani c Threshold Functions and Game Trees
![Page 127: Competitive Evaluation of Threshold Functions and Game ...](https://reader030.fdocuments.in/reader030/viewer/2022020622/61ede60b530f9e4e7917cf63/html5/thumbnails/127.jpg)
The HLPf for studying function evaluation
f : a monotone Boolean function over {x1, . . . , xn}Consider the homogeneous linear system HLPf :
HLPf∑
zi ≤ k(f ) ·∑
P zi ∀maxterm P of f ,∑
zi ≤ l(f ) ·∑
P zi ∀minterm P of f ,zi ≥ 0 ∀ i = 1, . . . , n
always has a nontrivial solution
Corollary:
For every monotone Boolean function f :
There exists a γ(f )-competitive implementation ofthe LPA.
γ(f ) = max{k(f ), l(f )}.
Cicalese–Milani c Threshold Functions and Game Trees
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The HLPf for studying function evaluation
f : a monotone Boolean function over {x1, . . . , xn}Consider the homogeneous linear system HLPf :
HLPf∑
zi ≤ k(f ) ·∑
P zi ∀maxterm P of f ,∑
zi ≤ l(f ) ·∑
P zi ∀minterm P of f ,zi ≥ 0 ∀ i = 1, . . . , n
always has a nontrivial solution
Corollary:
For every monotone Boolean function f :
There exists a γ(f )-competitive implementation ofthe LPA.
γ(f ) = max{k(f ), l(f )}.
Cicalese–Milani c Threshold Functions and Game Trees
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The HLPf for studying function evaluation
f : a monotone Boolean function over {x1, . . . , xn}Consider the homogeneous linear system HLPf :
HLPf∑
zi ≤ k(f ) ·∑
P zi ∀maxterm P of f ,∑
zi ≤ l(f ) ·∑
P zi ∀minterm P of f ,zi ≥ 0 ∀ i = 1, . . . , n
always has a nontrivial solution
Corollary:
For every monotone Boolean function f :
There exists a γ(f )-competitive implementation ofthe LPA.
γ(f ) = max{k(f ), l(f )}.
Cicalese–Milani c Threshold Functions and Game Trees
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Some Open Questions
Can threshold functions be optimally evaluated in lineartime?
Is the extremal competitive ratio always integer?
Find the extremal comp. ratio of general Boolean functions.
Is there a polynomial algorithm with optimal extremalcomp. ratio for evaluating monotone Boolean functions(given by an oracle/by the list of minterms)?
Cicalese–Milani c Threshold Functions and Game Trees
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Some Open Questions
Can threshold functions be optimally evaluated in lineartime?
Is the extremal competitive ratio always integer?
Find the extremal comp. ratio of general Boolean functions.
Is there a polynomial algorithm with optimal extremalcomp. ratio for evaluating monotone Boolean functions(given by an oracle/by the list of minterms)?
Cicalese–Milani c Threshold Functions and Game Trees
![Page 132: Competitive Evaluation of Threshold Functions and Game ...](https://reader030.fdocuments.in/reader030/viewer/2022020622/61ede60b530f9e4e7917cf63/html5/thumbnails/132.jpg)
Some Open Questions
Can threshold functions be optimally evaluated in lineartime?
Is the extremal competitive ratio always integer?
Find the extremal comp. ratio of general Boolean functions.
Is there a polynomial algorithm with optimal extremalcomp. ratio for evaluating monotone Boolean functions(given by an oracle/by the list of minterms)?
Cicalese–Milani c Threshold Functions and Game Trees
![Page 133: Competitive Evaluation of Threshold Functions and Game ...](https://reader030.fdocuments.in/reader030/viewer/2022020622/61ede60b530f9e4e7917cf63/html5/thumbnails/133.jpg)
Some Open Questions
Can threshold functions be optimally evaluated in lineartime?
Is the extremal competitive ratio always integer?
Find the extremal comp. ratio of general Boolean functions.
Is there a polynomial algorithm with optimal extremalcomp. ratio for evaluating monotone Boolean functions(given by an oracle/by the list of minterms)?
Cicalese–Milani c Threshold Functions and Game Trees
![Page 134: Competitive Evaluation of Threshold Functions and Game ...](https://reader030.fdocuments.in/reader030/viewer/2022020622/61ede60b530f9e4e7917cf63/html5/thumbnails/134.jpg)
The end
THANK YOU
Cicalese–Milani c Threshold Functions and Game Trees