Parameterized Power Vertex Cover Eric Angel, Evripidis Bampis, Bruno - - PowerPoint PPT Presentation

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Parameterized Power Vertex Cover Eric Angel, Evripidis Bampis, Bruno - - PowerPoint PPT Presentation

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Parameterized Power Vertex Cover Eric Angel, Evripidis Bampis, Bruno Escoffier, Michael Lampis Universities in Paris WG 2016 Overview Parameterized Power Vertex Cover Parameterized Power Vertex Cover 2 / 17 Overview Parameterized Power

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SLIDE 1

Parameterized Power Vertex Cover

Eric Angel, Evripidis Bampis, Bruno Escoffier, Michael Lampis Universities in Paris

WG 2016

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SLIDE 2

Overview

Parameterized Power Vertex Cover 2 / 17

Parameterized Power Vertex Cover

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SLIDE 3

Overview

Parameterized Power Vertex Cover 2 / 17

Parameterized Power Vertex Cover

  • Parameterized
  • Dealing with NP-hard problem
  • Goal: Algorithm exponential in some parameter FPT
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SLIDE 4

Overview

Parameterized Power Vertex Cover 2 / 17

Parameterized Power Vertex Cover

  • Parameterized
  • Dealing with NP-hard problem
  • Goal: Algorithm exponential in some parameter FPT
  • Vertex Cover
  • Given graph G, find minimum set of vertices that hit all edges
  • Standard NP-hard problem
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SLIDE 5

Overview

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Parameterized Power Vertex Cover

  • Parameterized
  • Dealing with NP-hard problem
  • Goal: Algorithm exponential in some parameter FPT
  • Vertex Cover
  • Given graph G, find minimum set of vertices that hit all edges
  • Standard NP-hard problem
  • Power?
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SLIDE 6

Power Vertex Cover

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Vertex Cover: Select vertices that touch all edges

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Power Vertex Cover

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Vertex Cover: Select vertices that touch all edges

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Power Vertex Cover

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Power: Some edges demand more power to be covered

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SLIDE 9

Power Vertex Cover

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Power: Some edges demand more power to be covered

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Power Vertex Cover

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Power: Some edges demand more power to be covered

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Power Vertex Cover

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Power Vertex Cover: Must decide which vertices get power . . . and how much

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Power Vertex Cover

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Power Vertex Cover: Must decide which vertices get power . . . and how much

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Power Vertex Cover

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Formal Definition: min

  • p(v)

max{p(u), p(v)} ≥ d((u, v)) ∀(u, v) ∈ E

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Motivation

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  • Applications to communication networks
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Motivation

Parameterized Power Vertex Cover 4 / 17

  • Applications to communication networks ??
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SLIDE 16

Motivation

Parameterized Power Vertex Cover 4 / 17

  • Applications to communication networks ??
  • Interesting Generalization of Vertex Cover
  • Note: added non-linear constraint

max{p(u), p(v)} ≥ d((u, v)) ∀(u, v) ∈ E

  • Compare: p(u) + p(v) ≥ d((u, v))
  • Is this problem really different/harder from Vertex Cover?
  • Admits 2 approximation
  • In P for bipartite graphs [Angel et al. ISAAC ’15]
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SLIDE 17

Motivation

Parameterized Power Vertex Cover 4 / 17

  • Applications to communication networks ??
  • Interesting Generalization of Vertex Cover
  • Note: added non-linear constraint

max{p(u), p(v)} ≥ d((u, v)) ∀(u, v) ∈ E

  • Compare: p(u) + p(v) ≥ d((u, v))
  • Is this problem really different/harder from Vertex Cover?
  • Admits 2 approximation
  • In P for bipartite graphs [Angel et al. ISAAC ’15]
  • What about Parameterized algorithms?
  • Vertex Cover is flagship problem
  • Compare: Weighted VC, Capacitated VC, Connected VC, . . .
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SLIDE 18

Motivation

Parameterized Power Vertex Cover 4 / 17

  • Applications to communication networks ??
  • Interesting Generalization of Vertex Cover
  • Note: added non-linear constraint

max{p(u), p(v)} ≥ d((u, v)) ∀(u, v) ∈ E

  • Compare: p(u) + p(v) ≥ d((u, v))
  • Is this problem really different/harder from Vertex Cover?
  • Admits 2 approximation
  • In P for bipartite graphs [Angel et al. ISAAC ’15]
  • What about Parameterized algorithms?
  • Vertex Cover is flagship problem
  • Compare: Weighted VC, Capacitated VC, Connected VC, . . .

Bottom line: Natural and interesting generalization of VC

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Results

Parameterized Power Vertex Cover 5 / 17

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Results

Parameterized Power Vertex Cover 5 / 17

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Results

Parameterized Power Vertex Cover 5 / 17

  • Good
  • FPT parameterized by budget
  • Same complexity as VC!
  • FPT parameterized by used vertices
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SLIDE 22

Results

Parameterized Power Vertex Cover 5 / 17

  • Good
  • FPT parameterized by budget
  • Same complexity as VC!
  • FPT parameterized by used vertices
  • Bad
  • W-hard parameterized by treewidth!
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SLIDE 23

Results

Parameterized Power Vertex Cover 5 / 17

  • Good
  • FPT parameterized by budget
  • Same complexity as VC!
  • FPT parameterized by used vertices
  • FPT (1 + ǫ)-approximation for treewidth

time (log n/ǫ)tw

  • Bad
  • W-hard parameterized by treewidth!
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SLIDE 24

Results

Parameterized Power Vertex Cover 5 / 17

  • Good
  • FPT parameterized by budget
  • Same complexity as VC!
  • FPT parameterized by used vertices
  • FPT (1 + ǫ)-approximation for treewidth

time (log n/ǫ)tw

  • Bad
  • W-hard parameterized by treewidth!
  • Ugly
  • Quadratic (bi)-kernel
  • Linear kernel?
  • kk for asymmetric case
  • ck? cn?
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Things you (almost) already know

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Basic FPT Algorithm

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Basic Branching Algorithm for Vertex Cover

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Basic FPT Algorithm

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Basic Branching Algorithm for Vertex Cover – Pick an uncovered edge

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Basic FPT Algorithm

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Basic Branching Algorithm for Vertex Cover – Pick an uncovered edge – Pick one of its endpoints (Branch)

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Basic FPT Algorithm

Parameterized Power Vertex Cover 7 / 17

Basic Branching Algorithm for Vertex Cover – Pick an uncovered edge – Pick one of its endpoints (Branch)

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Basic FPT Algorithm

Parameterized Power Vertex Cover 7 / 17

Basic Branching Algorithm for Vertex Cover – Pick an uncovered edge – Pick one of its endpoints (Branch) – Remove endpoint, decrease budget by 1 Running time: 2k

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Basic FPT Algorithm

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Basic Branching Algorithm for Vertex Cover – Pick an uncovered edge – Pick one of its endpoints (Branch) – Remove endpoint, decrease budget by 1 Running time: 2k . . . Can be improved to 1.28k with smarter branching

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Basic FPT Algorithm

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Power Vertex Cover Parameter: Total Budget P

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Basic FPT Algorithm

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Power Vertex Cover Parameter: Total Budget P Basic Branching Algorithm – Pick The heaviest edge to branch on – If unweighted call VC algorithm

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Basic FPT Algorithm

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Power Vertex Cover Parameter: Total Budget P Basic Branching Algorithm – Pick The heaviest edge to branch on – If unweighted call VC algorithm Almost as good as best VC algorithm

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Basic FPT Algorithm

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Power Vertex Cover Parameter: Total Budget P Better Branching Algorithm – If two heaviest edges share vertex branch there

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Basic FPT Algorithm

Parameterized Power Vertex Cover 7 / 17

Power Vertex Cover Parameter: Total Budget P Better Branching Algorithm – If two heaviest edges share vertex branch there

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SLIDE 37

Basic FPT Algorithm

Parameterized Power Vertex Cover 7 / 17

Power Vertex Cover Parameter: Total Budget P Better Branching Algorithm – If two heaviest edges share vertex branch there

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Basic FPT Algorithm

Parameterized Power Vertex Cover 7 / 17

Power Vertex Cover Parameter: Total Budget P Better Branching Algorithm – If two heaviest edges share vertex branch there – If not decrease weight of heaviest edge and budget by 1

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Basic FPT Algorithm

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Power Vertex Cover Parameter: Total Budget P Better Branching Algorithm – If two heaviest edges share vertex branch there – If not decrease weight of heaviest edge and budget by 1

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Basic FPT Algorithm

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Power Vertex Cover Parameter: Total Budget P Better Branching Algorithm – If two heaviest edges share vertex branch there – If not decrease weight of heaviest edge and budget by 1 As fast as best VC algorithm! (1.28P)

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Basic FPT Algorithm

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Power Vertex Cover Parameter: Total Budget P Parameter 2: Number of selected vertices k

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Basic FPT Algorithm

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Power Vertex Cover Parameter: Total Budget P Parameter 2: Number of selected vertices k Same algorithm gives 1.41k Note: k < P so this is a harder problem Q: Can we do as fast as VC here?

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The Asymmetric Case

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This is too easy! Let’s make things more interesting!

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The Asymmetric Case

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The Asymmetric Case

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Asymmetric Power Vertex Cover: Each edge has a different demand for each endpoint

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The Asymmetric Case

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Asymmetric Power Vertex Cover: Each edge has a different demand for each endpoint

  • Problem: what is a “heaviest” edge?
  • Branching not guaranteed to be fast
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The Asymmetric Case

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Asymmetric Power Vertex Cover: Each edge has a different demand for each endpoint

  • Problem: what is a “heaviest” edge?
  • Branching not guaranteed to be fast
  • Result: 1.325P algorithm with case analysis
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The Asymmetric Case

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Asymmetric Power Vertex Cover: Each edge has a different demand for each endpoint

  • Problem: what is a “heaviest” edge?
  • Branching not guaranteed to be fast
  • Result: 1.325P algorithm with case analysis
  • What about parameter k?
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A simple kernel for the Asymmetric case

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A simple kernel for parameter k

  • Consider a vertex withe degree > k
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A simple kernel for the Asymmetric case

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A simple kernel for parameter k

  • Consider a vertex withe degree > k
  • Order its incident edges by demand
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A simple kernel for the Asymmetric case

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A simple kernel for parameter k

  • Consider a vertex withe degree > k
  • Order its incident edges by demand
  • If the vertex gets power lower than the k + 1-th cost. . .
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A simple kernel for the Asymmetric case

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A simple kernel for parameter k

  • Consider a vertex withe degree > k
  • Order its incident edges by demand
  • If the vertex gets power lower than the k + 1-th cost. . .
  • we need to use > k vertices
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SLIDE 53

A simple kernel for the Asymmetric case

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A simple kernel for parameter k

  • Consider a vertex withe degree > k
  • Order its incident edges by demand
  • If the vertex gets power lower than the k + 1-th cost. . .
  • we need to use > k vertices
  • We can therefore give it power Wk+1, which covers the lower cost

edges

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A simple kernel for the Asymmetric case

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A simple kernel for parameter k

  • In the end graph has O(k2) edges left.
  • Q: Running time of FPT algorithm?
  • Q: Kernel inherently asymmetric?
  • Q: Linear (order) kernel?
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Things which are different

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W-hard for treewidth

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Reminder:

  • Treewidth is most basic graph width
  • Vertex Cover solvable in 2twn time
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W-hard for treewidth

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Theorem: There is no no(t) algorithm for PVC (under ETH)

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W-hard for treewidth

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Theorem: There is no no(t) algorithm for PVC (under ETH) Proof: Reduction from Multi-Colored Clique

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SLIDE 59

W-hard for treewidth

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Theorem: There is no no(t) algorithm for PVC (under ETH) Proof: Reduction from Multi-Colored Clique Vertex Selection Gadget:

  • Thick edges have weight n
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SLIDE 60

W-hard for treewidth

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Theorem: There is no no(t) algorithm for PVC (under ETH) Proof: Reduction from Multi-Colored Clique Vertex Selection Gadget:

  • Thick edges have weight n
  • At least one internal vertex must get power n
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SLIDE 61

W-hard for treewidth

Parameterized Power Vertex Cover 11 / 17

Theorem: There is no no(t) algorithm for PVC (under ETH) Proof: Reduction from Multi-Colored Clique Vertex Selection Gadget:

  • Thick edges have weight n
  • At least one internal vertex must get power n
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SLIDE 62

W-hard for treewidth

Parameterized Power Vertex Cover 11 / 17

Theorem: There is no no(t) algorithm for PVC (under ETH) Proof: Reduction from Multi-Colored Clique Vertex Selection Gadget:

  • Thick edges have weight n
  • At least one internal vertex must get power n
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W-hard for treewidth

Parameterized Power Vertex Cover 11 / 17

Theorem: There is no no(t) algorithm for PVC (under ETH) Proof: Reduction from Multi-Colored Clique Vertex Selection Gadget:

  • Thick edges have weight n
  • At least one internal vertex must get power n
  • Main claim: Optimal power gives i to u and n − i to u′
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W-hard for treewidth

Parameterized Power Vertex Cover 11 / 17

Theorem: There is no no(t) algorithm for PVC (under ETH) Proof: Reduction from Multi-Colored Clique Vertex Selection Gadget:

  • Thick edges have weight n
  • At least one internal vertex must get power n
  • Main claim: Optimal power gives i to u and n − i to u′
  • Encode vertex selection by power level for u
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W-hard for treewidth

Parameterized Power Vertex Cover 11 / 17

Theorem: There is no no(t) algorithm for PVC (under ETH) Proof: Reduction from Multi-Colored Clique

  • Take k copies of previous gadget
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SLIDE 66

W-hard for treewidth

Parameterized Power Vertex Cover 11 / 17

Theorem: There is no no(t) algorithm for PVC (under ETH) Proof: Reduction from Multi-Colored Clique

  • Take k copies of previous gadget
  • Add a (small) check gadget for each non-edge of original graph
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SLIDE 67

W-hard for treewidth

Parameterized Power Vertex Cover 11 / 17

Theorem: There is no no(t) algorithm for PVC (under ETH) Proof: Reduction from Multi-Colored Clique

  • Take k copies of previous gadget
  • Add a (small) check gadget for each non-edge of original graph
  • Whole graph has treewidth O(k)
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SLIDE 68

W-hard for treewidth

Parameterized Power Vertex Cover 11 / 17

Theorem: There is no no(t) algorithm for PVC (under ETH) Proof: Reduction from Multi-Colored Clique Check gadget: Meaning: not (i and j)

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Treewidth Algorithms

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Treewidth doesn’t work!

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Treewidth Algorithms

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Treewidth doesn’t work! Actually it’s not so bad. . .

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Treewidth Algorithms

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Easy Exact Algorithms

  • (∆ + 1)twn time
  • (M + 1)twn time (M=maximum weight)

Main observation: Each vertex has limited number of reasonable power values. (These running times are optimal)

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SLIDE 72

Treewidth Algorithms

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Easy Exact Algorithms

  • (∆ + 1)twn time
  • (M + 1)twn time (M=maximum weight)

Main observation: Each vertex has limited number of reasonable power values. (These running times are optimal) Can we do better?

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Treewidth Algorithms

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FPT Approximation Scheme

  • (M + 1)twn time to solve exactly
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Treewidth Algorithms

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FPT Approximation Scheme

  • (M + 1)twn time to solve exactly
  • Main idea: Rounding
  • Instead of power value p for each vertex store ⌊log1+ǫ(p)⌋
  • At most log M/ log(1 + ǫ) possible values
  • At most a (1 + ǫ) factor from correct value
  • If M = nO(1) running time (log n/ǫ)tw
  • (If not, easy: think Knapsack)
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Treewidth Algorithms

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FPT Approximation Scheme

  • (M + 1)twn time to solve exactly
  • Main idea: Rounding
  • Instead of power value p for each vertex store ⌊log1+ǫ(p)⌋
  • At most log M/ log(1 + ǫ) possible values
  • At most a (1 + ǫ) factor from correct value
  • If M = nO(1) running time (log n/ǫ)tw
  • (If not, easy: think Knapsack)

Bottom line: Fast FPT algorithm for W-hard problem, only (1 + ǫ) error! (This is part of a more general technique [L. ICALP ’14])

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Things we don’t understand

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Linear (bi)-kernel?

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  • Recall: O(k2) kernel for (Asymmetric) PVC
  • Can we do better?
  • Using LP perhaps?
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Linear (bi)-kernel?

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  • Recall: O(k2) kernel for (Asymmetric) PVC
  • Can we do better?
  • Using LP perhaps?
  • Recall: for VC we have if LP says v(x) = 0, we should not take x
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SLIDE 79

Linear (bi)-kernel?

Parameterized Power Vertex Cover 14 / 17

  • Recall: O(k2) kernel for (Asymmetric) PVC
  • Can we do better?
  • Using LP perhaps?
  • Recall: for VC we have if LP says v(x) = 0, we should not take x
  • Theorem: Given an instance of PVC and an optimal fractional LP

solution that sets p(x) = 0 it is NP-hard to decide whether to take x.

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LPs don’t help

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Theorem: Given an instance of PVC and an optimal fractional LP solution that sets p(x) = 0 it is NP-hard to decide whether to take x.

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SLIDE 81

LPs don’t help

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Theorem: Given an instance of PVC and an optimal fractional LP solution that sets p(x) = 0 it is NP-hard to decide whether to take x. Reduction from VC

  • Left side contains vertices, right edges
  • Incidence encoded with weight 1 edges
  • Optimal fractional solution: weight 1 to all right vertices
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Conclusions

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  • Interesting generalization of Vertex Cover
  • W-hard for treewidth
  • But approximable!

Open questions:

  • Linear kernel?
  • ck for asymmetric?
  • FPT for feedback vertex set?
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Thank you!

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