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Multidimensional Binary Vector Assignment problem: standard, structural and above guarantee parameterizations Rmi Watrigant 1 joint work with Marin Bougeret 2 , Guillerme Duvilli 2 , Rodolphe Giroudeau 2 1 Hong Kong Polytechnic University,

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SLIDE 1

Multidimensional Binary Vector Assignment problem: standard, structural and above guarantee parameterizations

Rémi Watrigant1

joint work with Marin Bougeret2, Guillerme Duvillié2, Rodolphe Giroudeau2

1 Hong Kong Polytechnic University, Hong Kong 2 LIRMM, Montpellier, France

FCT 2015, Gdansk, Poland. August 17-19 2015

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 1/10

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SLIDE 2

Contents

1

Applications, definitions and related works

2

First observations

3

Above guarantee parameterization

4

Lower bounds

5

Conclusion

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 2/10

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SLIDE 3

Applications

Yield maximization in wafer-to-wafer 3D chip integration. Multidimensional Binary Vector Assignment a wafer = a binary vector of good/bad dies (1/0)

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 3/10

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SLIDE 4

Applications

Yield maximization in wafer-to-wafer 3D chip integration. Multidimensional Binary Vector Assignment a wafer = a binary vector of good/bad dies (1/0) a stack = superposition of several wafers

stack of wafers resulting vector

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 3/10

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SLIDE 5

Applications

Yield maximization in wafer-to-wafer 3D chip integration. Multidimensional Binary Vector Assignment a wafer = a binary vector of good/bad dies (1/0) a stack = superposition of several wafers Input: m sets of n wafers (p-dimensional binary vectors)

n p-dimensional vectors m sets wafers

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 3/10

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SLIDE 6

Applications

Yield maximization in wafer-to-wafer 3D chip integration. Multidimensional Binary Vector Assignment a wafer = a binary vector of good/bad dies (1/0) a stack = superposition of several wafers Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks

n p-dimensional vectors m sets wafers

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 3/10

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SLIDE 7

Applications

Yield maximization in wafer-to-wafer 3D chip integration. Multidimensional Binary Vector Assignment a wafer = a binary vector of good/bad dies (1/0) a stack = superposition of several wafers Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total

n p-dimensional vectors m sets wafers n stacks cost = number of bad dies

= 11

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 3/10

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SLIDE 8

Related works

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total Previous results: NP-hard even when m = 3 (reduction from 3D matching)

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 4/10

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SLIDE 9

Related works

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total Previous results: NP-hard even when m = 3 (reduction from 3D matching) Approximating the maximization version (at least k good dies):

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 4/10

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SLIDE 10

Related works

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total Previous results: NP-hard even when m = 3 (reduction from 3D matching) Approximating the maximization version (at least k good dies):

◮ f (m)-approximation

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 4/10

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SLIDE 11

Related works

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total Previous results: NP-hard even when m = 3 (reduction from 3D matching) Approximating the maximization version (at least k good dies):

◮ f (m)-approximation ◮ O(p1−ǫ) and O(m1−ǫ) inapproximability unless P = NP

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 4/10

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SLIDE 12

Related works

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total Previous results: NP-hard even when m = 3 (reduction from 3D matching) Approximating the maximization version (at least k good dies):

◮ f (m)-approximation ◮ O(p1−ǫ) and O(m1−ǫ) inapproximability unless P = NP ◮

p c -approximation for any c ∈ N

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 4/10

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SLIDE 13

Related works

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total Previous results: NP-hard even when m = 3 (reduction from 3D matching) Approximating the maximization version (at least k good dies):

◮ f (m)-approximation ◮ O(p1−ǫ) and O(m1−ǫ) inapproximability unless P = NP ◮

p c -approximation for any c ∈ N

◮ FPT parameterized by p

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 4/10

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SLIDE 14

Related works

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total Previous results: NP-hard even when m = 3 (reduction from 3D matching) Approximating the maximization version (at least k good dies):

◮ f (m)-approximation ◮ O(p1−ǫ) and O(m1−ǫ) inapproximability unless P = NP ◮

p c -approximation for any c ∈ N

◮ FPT parameterized by p ◮ W [1]-hard for standard parameter (maximization version)

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 4/10

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SLIDE 15

Parameterized algorithms

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 5/10

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SLIDE 16

Parameterized algorithms

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 5/10

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SLIDE 17

Parameterized algorithms

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total For an instance I of the problem, choose a parameter κ(I)

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 5/10

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SLIDE 18

Parameterized algorithms

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total For an instance I of the problem, choose a parameter κ(I)= m

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 5/10

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SLIDE 19

Parameterized algorithms

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total For an instance I of the problem, choose a parameter κ(I)= n

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 5/10

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SLIDE 20

Parameterized algorithms

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total For an instance I of the problem, choose a parameter κ(I)= p

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 5/10

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SLIDE 21

Parameterized algorithms

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total For an instance I of the problem, choose a parameter κ(I)= k

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 5/10

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SLIDE 22

Parameterized algorithms

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total For an instance I of the problem, choose a parameter κ(I)=?

Fixed-Parameter Tractability

A problem is FPT if there is an algorithm solving any instance I in time O( f (κ(I)) poly(|I|) )

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 5/10

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SLIDE 23

Parameterized algorithms

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total For an instance I of the problem, choose a parameter κ(I)=?

Fixed-Parameter Tractability

A problem is FPT if there is an algorithm solving any instance I in time O( f (κ(I)) poly(|I|) ) Corresponding lower bounds:

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 5/10

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SLIDE 24

Parameterized algorithms

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total For an instance I of the problem, choose a parameter κ(I)=?

Fixed-Parameter Tractability

A problem is FPT if there is an algorithm solving any instance I in time O( f (κ(I)) poly(|I|) ) Corresponding lower bounds: W [1], W [2]-hardness: we suppose FPT = W [1], W [2]

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 5/10

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SLIDE 25

Parameterized algorithms

Multidimensional Binary Vector Assignment Input: m sets of n wafers (p-dimensional binary vectors) Output: pick one wafer from each set to form n stacks Goal: obtain at most k bad dies in total For an instance I of the problem, choose a parameter κ(I)=?

Fixed-Parameter Tractability

A problem is FPT if there is an algorithm solving any instance I in time O( f (κ(I)) poly(|I|) ) Corresponding lower bounds: W [1], W [2]-hardness: we suppose FPT = W [1], W [2] Exponential Time Hypothesis: we suppose that 3-SAT cannot be solved in time O∗(2o(n)) (n = number of variables)

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 5/10

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SLIDE 26

First observations

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Warmup: is it alway safe to create "full-good stacks" ? set 1 set 2 set 3

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 6/10

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SLIDE 27

First observations

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Warmup: is it alway safe to create "full-good stacks" ? set 1 set 2 set 3

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 6/10

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SLIDE 28

First observations

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Warmup: is it alway safe to create "full-good stacks" ? set 1 set 2 set 3

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 6/10

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SLIDE 29

First observations

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Warmup: is it alway safe to create "full-good stacks" ? set 1 set 2 set 3 cost = 4

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 6/10

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SLIDE 30

First observations

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Warmup: is it alway safe to create "full-good stacks" ? No! set 1 set 2 set 3 cost = 3

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 6/10

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SLIDE 31

First observations

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

is k a good parameter ? Let us show that we can suppose n ≤ k

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 6/10

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SLIDE 32

First observations

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

is k a good parameter ? Let us show that we can suppose n ≤ k Suppose n > k

m sets n wafers p dimensions

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 6/10

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SLIDE 33

First observations

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

is k a good parameter ? Let us show that we can suppose n ≤ k Suppose n > k

m sets n wafers p dimensions

  • each set must have a full-good wafer (otherwise cost ≥ n > k)
  • R. Watrigant

Multidimensional Binary Vector Assignment problem 6/10

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SLIDE 34

First observations

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

is k a good parameter ? Let us show that we can suppose n ≤ k Suppose n > k

m sets n wafers p dimensions

  • each set must have a full-good wafer (otherwise cost ≥ n > k)
  • any solution must have a full-good stack (same argument)
  • R. Watrigant

Multidimensional Binary Vector Assignment problem 6/10

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SLIDE 35

First observations

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

is k a good parameter ? Let us show that we can suppose n ≤ k Suppose n > k

m sets n wafers p dimensions

  • each set must have a full-good wafer (otherwise cost ≥ n > k)
  • any solution must have a full-good stack (same argument)

⇒ we can arbitrarily form a full-good stack remove it, and continue until n ≤ k

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 6/10

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SLIDE 36

First observations

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

is k a good parameter ? Let us show that we can suppose n ≤ k Let us show that we can suppose p ≤ k as well

m sets n wafers p dimensions

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 6/10

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SLIDE 37

First observations

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

is k a good parameter ? Let us show that we can suppose n ≤ k Let us show that we can suppose p ≤ k as well

m sets n wafers p dimensions

  • if there is a component with good dies everywhere
  • R. Watrigant

Multidimensional Binary Vector Assignment problem 6/10

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SLIDE 38

First observations

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

is k a good parameter ? Let us show that we can suppose n ≤ k Let us show that we can suppose p ≤ k as well

m sets n wafers p dimensions

  • if there is a component with good dies everywhere

⇒ any solution will also have this good die

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 6/10

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SLIDE 39

First observations

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

is k a good parameter ? Let us show that we can suppose n ≤ k Let us show that we can suppose p ≤ k as well

m sets n wafers p dimensions

  • if there is a component with good dies everywhere

⇒ any solution will also have this good die

  • we can remove the component from the instance
  • R. Watrigant

Multidimensional Binary Vector Assignment problem 6/10

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SLIDE 40

First observations

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

is k a good parameter ? Let us show that we can suppose n ≤ k Let us show that we can suppose p ≤ k as well

m sets n wafers p dimensions

  • if there is a component with good dies everywhere

⇒ any solution will also have this good die

  • we can remove the component from the instance

⇒ the cost of any solution will be at least p

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 6/10

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SLIDE 41

First observations

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

is k a good parameter ? Let us show that we can suppose n ≤ k Let us show that we can suppose p ≤ k as well

m sets n wafers p dimensions

Theorem

  • From any instance, we can produce in polynomial time an equivalent one of size

at most O(k2m) ⇒ kernel of size O(k2m)

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 6/10

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SLIDE 42

First observations

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

is k a good parameter ? Let us show that we can suppose n ≤ k Let us show that we can suppose p ≤ k as well

m sets n wafers p dimensions

Theorem

  • From any instance, we can produce in polynomial time an equivalent one of size

at most O(k2m) ⇒ kernel of size O(k2m)

  • No kernel polynomial in p even for m = 3 (unless NP ⊆ coNP/poly)
  • R. Watrigant

Multidimensional Binary Vector Assignment problem 6/10

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SLIDE 43

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

We just proved that we can suppose n, p ≤ k ⇒ k is a too large parameter Idea: substracting a lower bound to the objective function

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

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SLIDE 44

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

We just proved that we can suppose n, p ≤ k ⇒ k is a too large parameter Idea: substracting a lower bound to the objective function

m sets n wafers p dimensions

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

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SLIDE 45

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

We just proved that we can suppose n, p ≤ k ⇒ k is a too large parameter Idea: substracting a lower bound to the objective function

m sets n wafers p dimensions the cost of any solution must be at least 11

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

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SLIDE 46

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

We just proved that we can suppose n, p ≤ k ⇒ k is a too large parameter Idea: substracting a lower bound to the objective function Let B be the maximum number of bad dies in a set ⇒ parameter ζB = k − B

m sets n wafers p dimensions the cost of any solution must be at least 11

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

slide-47
SLIDE 47

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total
  • Find in polynomial time if there is a costless assignment between two sets

m sets n wafers p dimensions

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

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SLIDE 48

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total
  • Find in polynomial time if there is a costless assignment between two sets

the cost of any solution must be at least 11 m sets n wafers p dimensions

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

slide-49
SLIDE 49

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total
  • Find in polynomial time if there is a costless assignment between two sets

the cost of any solution must be at least 11 m sets n wafers p dimensions

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

slide-50
SLIDE 50

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total
  • Find in polynomial time if there is a costless assignment between two sets

the cost of any solution must be at least 11 m sets n wafers p dimensions

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

slide-51
SLIDE 51

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total
  • Find in polynomial time if there is a costless assignment between two sets

the cost of any solution must be at least 11 m sets n wafers p dimensions

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

slide-52
SLIDE 52

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total
  • Find in polynomial time if there is a costless assignment between two sets
  • If no such assignment can be found, branch:

the cost of any solution must be at least 11 m sets n wafers p dimensions

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

slide-53
SLIDE 53

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total
  • Find in polynomial time if there is a costless assignment between two sets
  • If no such assignment can be found, branch:
  • guess all couples of wafers which will induce new bad dies

the cost of any solution must be at least 11 m sets n wafers p dimensions

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

slide-54
SLIDE 54

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total
  • Find in polynomial time if there is a costless assignment between two sets
  • If no such assignment can be found, branch:
  • guess all couples of wafers which will induce new bad dies

the cost of any solution must be at least 11 m sets n wafers p dimensions

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

slide-55
SLIDE 55

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total
  • Find in polynomial time if there is a costless assignment between two sets
  • If no such assignment can be found, branch:
  • guess all couples of wafers which will induce new bad dies
  • find a costless assignment for the others

the cost of any solution must be at least 11 m sets n wafers p dimensions

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

slide-56
SLIDE 56

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total
  • Find in polynomial time if there is a costless assignment between two sets
  • If no such assignment can be found, branch:
  • guess all couples of wafers which will induce new bad dies
  • find a costless assignment for the others

the cost of any solution must be at least 11 m sets n wafers p dimensions

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

slide-57
SLIDE 57

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total
  • Find in polynomial time if there is a costless assignment between two sets
  • If no such assignment can be found, branch:
  • guess all couples of wafers which will induce new bad dies
  • find a costless assignment for the others
  • branching of width n2, applied at most ζB times

the cost of any solution must be at least 11 m sets n wafers p dimensions

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

slide-58
SLIDE 58

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total
  • Find in polynomial time if there is a costless assignment between two sets
  • If no such assignment can be found, branch:
  • guess all couples of wafers which will induce new bad dies
  • find a costless assignment for the others
  • branching of width n2, applied at most ζB times

Theorem

There is an exact algorithm solving the problem in O∗(4ζB log(n)) time.

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

slide-59
SLIDE 59

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total
  • Find in polynomial time if there is a costless assignment between two sets
  • If no such assignment can be found, branch:
  • guess all couples of wafers which will induce new bad dies
  • find a costless assignment for the others
  • branching of width n2, applied at most ζB times

Theorem

There is an exact algorithm solving the problem in O∗(4ζB log(n)) time. Simple, but somehow tight, because:

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

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SLIDE 60

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total
  • Find in polynomial time if there is a costless assignment between two sets
  • If no such assignment can be found, branch:
  • guess all couples of wafers which will induce new bad dies
  • find a costless assignment for the others
  • branching of width n2, applied at most ζB times

Theorem

There is an exact algorithm solving the problem in O∗(4ζB log(n)) time. Simple, but somehow tight, because:

  • W [2]-hard parameterized by ζB only
  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

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SLIDE 61

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total
  • Find in polynomial time if there is a costless assignment between two sets
  • If no such assignment can be found, branch:
  • guess all couples of wafers which will induce new bad dies
  • find a costless assignment for the others
  • branching of width n2, applied at most ζB times

Theorem

There is an exact algorithm solving the problem in O∗(4ζB log(n)) time. Simple, but somehow tight, because:

  • W [2]-hard parameterized by ζB only
  • no 2o(ζB) log(n) nor 2ζBo(log(n)) under ETH
  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

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SLIDE 62

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total
  • Find in polynomial time if there is a costless assignment between two sets
  • If no such assignment can be found, branch:
  • guess all couples of wafers which will induce new bad dies
  • find a costless assignment for the others
  • branching of width n2, applied at most ζB times

Theorem

There is an exact algorithm solving the problem in O∗(4ζB log(n)) time. Simple, but somehow tight, because:

  • W [2]-hard parameterized by ζB only
  • no 2o(ζB) log(n) nor 2ζBo(log(n)) under ETH
  • no 2o(k) (and thus no 2o(ζB)) for fixed n under ETH.
  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

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SLIDE 63

Above guarantee parameterization

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total
  • Find in polynomial time if there is a costless assignment between two sets
  • If no such assignment can be found, branch:
  • guess all couples of wafers which will induce new bad dies
  • find a costless assignment for the others
  • branching of width n2, applied at most ζB times

Theorem

There is an exact algorithm solving the problem in O∗(4ζB log(n)) time. Simple, but somehow tight, because:

  • W [2]-hard parameterized by ζB only
  • no 2o(ζB) log(n) nor 2ζBo(log(n)) under ETH
  • no 2o(k) (and thus no 2o(ζB)) for fixed n under ETH.
  • R. Watrigant

Multidimensional Binary Vector Assignment problem 7/10

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SLIDE 64

Lower bounds

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Reduction from Hitting Set: ground set = {1, 2, 3, 4, 5} instance = {1, 3, 4}, {2, 3, 4}, {2, 3, 5}, {1, 4, 5}

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 8/10

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SLIDE 65

Lower bounds

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Reduction from Hitting Set: ground set = {1, 2, 3, 4, 5} instance = {1, 3, 4}, {2, 3, 4}, {2, 3, 5}, {1, 4, 5}

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 8/10

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SLIDE 66

Lower bounds

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Reduction from Hitting Set: ground set = {1, 2, 3, 4, 5} instance = {1, 3, 4}, {2, 3, 4}, {2, 3, 5}, {1, 4, 5}

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 8/10

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SLIDE 67

Lower bounds

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Reduction from Hitting Set: ground set = {1, 2, 3, 4, 5} instance = {1, 3, 4}, {2, 3, 4}, {2, 3, 5}, {1, 4, 5}

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 8/10

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SLIDE 68

Lower bounds

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Reduction from Hitting Set: ground set = {1, 2, 3, 4, 5} instance = {1, 3, 4}, {2, 3, 4}, {2, 3, 5}, {1, 4, 5}

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 8/10

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SLIDE 69

Lower bounds

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Reduction from Hitting Set: ground set = {1, 2, 3, 4, 5} instance = {1, 3, 4}, {2, 3, 4}, {2, 3, 5}, {1, 4, 5}

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 8/10

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SLIDE 70

Lower bounds

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Reduction from Hitting Set: ground set = {1, 2, 3, 4, 5} instance = {1, 3, 4}, {2, 3, 4}, {2, 3, 5}, {1, 4, 5}

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 8/10

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SLIDE 71

Lower bounds

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Reduction from Hitting Set: ground set = {1, 2, 3, 4, 5} instance = {1, 3, 4}, {2, 3, 4}, {2, 3, 5}, {1, 4, 5}

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 8/10

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SLIDE 72

Lower bounds

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Reduction from Hitting Set: ground set = {1, 2, 3, 4, 5} instance = {1, 3, 4}, {2, 3, 4}, {2, 3, 5}, {1, 4, 5}

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 8/10

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SLIDE 73

Lower bounds

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Reduction from Hitting Set: ground set = {1, 2, 3, 4, 5} instance = {1, 3, 4}, {2, 3, 4}, {2, 3, 5}, {1, 4, 5}

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 8/10

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SLIDE 74

Lower bounds

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Reduction from Hitting Set: ζB = size of the hitting set ⇒ W [2]-hardness parameterized by ζB

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 8/10

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SLIDE 75

Lower bounds

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Reduction from Hitting Set: ζB = size of the hitting set ⇒ W [2]-hardness parameterized by ζB

Theorem [Lokshtanov, Marx, Saurabh, ’11]

Assuming ETH, no O∗(2o(k log(k))) algorithm for Hitting Set where the goal is to find a hitting set of size k in an instance where the ground set is of size k2

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 8/10

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SLIDE 76

Lower bounds

Multidimensional Binary Vector Assignment

  • Input: m sets of n wafers (p-dimensional binary vectors)
  • Output: n stacks (of m wafers each)
  • Goal: obtain at most k bad dies in total

Reduction from Hitting Set: ζB = size of the hitting set ⇒ W [2]-hardness parameterized by ζB

Theorem [Lokshtanov, Marx, Saurabh, ’11]

Assuming ETH, no O∗(2o(k log(k))) algorithm for Hitting Set where the goal is to find a hitting set of size k in an instance where the ground set is of size k2 ⇒ no O∗(2o(ζB) log(n)) nor O∗(2ζBo(log(n))) for our problem, under ETH.

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 8/10

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SLIDE 77

Summary of the results

Positive results Negative results O(k2m) kernel no pO(1) kernel unless NP ⊆ coNP/poly O∗(4ζB log(n)) algorithm W [2]-hard for ζB only no 2o(ζB) log(n) nor 2ζBo(log(n)) under ETH no 2o(k) for fixed n under ETH O∗(dζp) algorithm NP-hard for ζp = 0 and fixed n ≥ 3 for n = 2

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 9/10

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SLIDE 78

Summary of the results

Positive results Negative results O(k2m) kernel no pO(1) kernel unless NP ⊆ coNP/poly O∗(4ζB log(n)) algorithm W [2]-hard for ζB only no 2o(ζB) log(n) nor 2ζBo(log(n)) under ETH no 2o(k) for fixed n under ETH O∗(dζp) algorithm NP-hard for ζp = 0 and fixed n ≥ 3 for n = 2 Open questions: algorithm in O∗(2k) ? (n part of the input) polynomial kernel parameterized by k only ?

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 9/10

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SLIDE 79

Thank you for your attention !

  • R. Watrigant

Multidimensional Binary Vector Assignment problem 10/10