Parameterized Maximum Path Coloring Michael Lampis September 9, - - PowerPoint PPT Presentation

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Parameterized Maximum Path Coloring Michael Lampis September 9, 2011 1 / 17 Path Coloring Definition Path Coloring Path Coloring Example Known results Input : A graph G and a multi-set of paths on that graph Edge slicing

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Parameterized Maximum Path Coloring

Michael Lampis

September 9, 2011

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Path Coloring Definition

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 2 / 17

Path Coloring

Input: A graph G and a multi-set of paths on that graph Constraint: Assign colors from {1, . . . , W} to the paths so that paths that share an edge receive different colors. Objective: min W

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Path Coloring Definition

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 2 / 17

Path Coloring

Input: A graph G and a multi-set of paths on that graph Constraint: Assign colors from {1, . . . , W} to the paths so that paths that share an edge receive different colors. Objective: min W

  • Graph could be undirected or bi-directed
  • Instead of paths we could be given endpoints (Routing

and Path Coloring)

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Path Coloring Definition

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 2 / 17

Path Coloring

Input: A graph G and a multi-set of paths on that graph Constraint: Assign colors from {1, . . . , W} to the paths so that paths that share an edge receive different colors. Objective: min W

  • Graph could be undirected or bi-directed
  • Instead of paths we could be given endpoints (Routing

and Path Coloring)

  • We’ll mostly talk about trees (→ unique routing)
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Example

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 3 / 17

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Example

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 3 / 17

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Example

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 3 / 17

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Example

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 3 / 17

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Example

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 3 / 17

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Known results

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 4 / 17

  • PC is very hard!

✦ NP-hard on stars [Erlebach, Jansen 2001] ✦ NP-hard on rings [Garey, Johnson, Miller,

Papadimitriou 1980]

✦ NP-hard on bi-directed binary trees [Kumar,

Panigrahy, Russel, Sundaram 1997]

  • Good news: Thanks to a simple trick undirected trees

are no harder than stars.

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Edge slicing

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 5 / 17

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Edge slicing

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 5 / 17

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Edge slicing

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 5 / 17

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Edge slicing

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 5 / 17

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Edge slicing

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 5 / 17

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Edge slicing

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 5 / 17

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Edge slicing

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 5 / 17

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Edge slicing

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 5 / 17

  • Repeated

edge slicing can break down any undirected tree to a star

  • If we could solve PC on stars → poly-

time algorithm (we can’t!)

✦ But FPT algorithm when parame-

terized by ∆. [Erlebach, Jansen 2001]

  • Ironically, this doesn’t work for bi-

directed trees, where stars are easy.

✦ But FPT algorithm when param-

eterized by ∆ + W. [Erlebach, Jansen 2001]

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Max PC

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 6 / 17

Max Path Coloring

Input: A graph G and a multi-set of paths on that graph, color buget W Constraint: Assign colors from {1, . . . , W} to B of the paths so that paths that share an edge receive different colors. Objective: max B

  • Strict generalization of PC as a decision problem

✦ → At least as hard to solve exactly

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Max PC

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 6 / 17

Max Path Coloring

Input: A graph G and a multi-set of paths on that graph, color buget W Constraint: Assign colors from {1, . . . , W} to B of the paths so that paths that share an edge receive different colors. Objective: max B

  • Strict generalization of PC as a decision problem

✦ → At least as hard to solve exactly

  • Max PC is solvable in n∆W on trees. [Erlebach, Jansen

1998]

  • Can we do this in FPT time for either parameter?
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Max PC hardness results

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 7 / 17

  • An n∆W algorithm is known to solve Max PC exactly on
  • trees. Can we do better?
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Max PC hardness results

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 7 / 17

  • An n∆Wt algorithm is known to solve Max PC exactly on
  • trees. (t = treewidth)
  • We show that:

✦ Max PC is W-hard parameterized by W, even for

trees with ∆ = 3.

✦ Max PC is W-hard parameterized by ∆, even for

trees with W = 4.

✦ Max PC is W-hard parameterized by t, even for

∆ = W = 4.

  • → No no(

√ ∆Wt) algorithm exists (assuming ETH).

  • Strategy: Ind Set ≤ DNP ≤ Cap Max PC ≤ Max PC
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Disjoint Neighborhood Packing

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 8 / 17

In this problem we want to select a maximum-size set of vertices such that all pairs have distance > 2

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Disjoint Neighborhood Packing

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 8 / 17

Selecting a vertex will disqualify . . .

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Disjoint Neighborhood Packing

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 8 / 17

Selecting a vertex will disqualify its neighbors . . .

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Disjoint Neighborhood Packing

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 8 / 17

Selecting a vertex will disqualify its neighbors and their neighbors from future selection

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Disjoint Neighborhood Packing

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 8 / 17

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Disjoint Neighborhood Packing

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 8 / 17

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Disjoint Neighborhood Packing

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 8 / 17

Problem is similar to Independent Set (and similarly W-hard)

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Reduction

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 9 / 17

  • Our basic gadget is a path on

n vertices, to be attached to a “backbone” through an edge of capacity 2.

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Reduction

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 9 / 17

  • To each vertex of a gadget we

attach a short path with a local demand

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Reduction

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 9 / 17

  • The local demand will overlap

with two global demands so that either the local demand is se- lected or both global demands

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Reduction

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 10 / 17

  • We put n copies of the gadget on the backbone, one for

each vertex.

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Reduction

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 10 / 17

  • For each vertex of the original graph we will make a set
  • f global demands.
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Reduction

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 10 / 17

  • The global demands use up all the neighbors’ branches

and are enough to increase the solution by one.

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Reduction

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 10 / 17

  • If we can go from n2 to n2 + k satisfied demands then
  • riginal graph has DNP of size k.
  • ∆ = 3 and W = 2k.
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Reduction

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 10 / 17

  • Reduction for treewidth: Replace backbone with a k × n

grid.

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Reduction

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 10 / 17

  • Reduction for ∆: Replace backbone with a vertex of

degree k2. Use

k

2

copies of this gadget to check

compatibility between all pairs.

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Complexity jump

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 11 / 17

  • What makes Max PC harder than PC?

✦ Intuitively, we first have to decide which requests to

drop, then color the rest. The first part appears to be harder.

  • What if we only want to drop a small number of requests

T?

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Complexity jump

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 11 / 17

  • What makes Max PC harder than PC?

✦ Intuitively, we first have to decide which requests to

drop, then color the rest. The first part appears to be harder.

  • What if we only want to drop a small number of requests

T? Another parameter is born. . .

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(p∆, pW, pT)-MaxPC

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 12 / 17

  • Recall that (p∆, pW)-PC is FPT

✦ Bottom-up dynamic programming algorithm

  • So, the problem is (essentially) to pick the dropped

requests

  • Observation: if more than 2∆W + T + 1 requests touch

a vertex reject immediately

✦ Even if we drop T requests one of its incident edges

will have > W requests going through it.

  • Otherwise, do bottom-up dynamic programming again.
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(pT)-MaxPC binary trees

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 13 / 17

MaxPC on undirected binary trees

  • Slice edges until we are left with stars, solve PC on each

star

  • Some stars are “good”, other “bad” (if all are good

accept)

  • If a sub-tree contains only good stars cut it off the tree

✦ We can color everything there even if we don’t drop

any requests

  • All leaf-stars are now bad
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(pT)-MaxPC binary trees

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 14 / 17

  • All leaf-stars are now bad
  • All of them must be touched by a dropped request
  • No dropped request can touch more than two leaves

✦ If more than 2T leaf-stars reject

  • Now graph has O(T) leaf-stars and internal-stars

(∆ = 3)

  • Easy to pick one endpoint of a dropped request. If we

have O(T) choices for the other endpoint → recursive T O(T) algorithm.

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SLIDE 44

Algoritm cont’d

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 15 / 17

  • Now graph has O(T) leaf-stars and internal-stars

(∆ = 3)

  • Easy to pick one endpoint of a dropped request. If we

have O(T) choices for the other endpoint → recursive T O(T) algorithm.

  • Possible candidates are the O(T) special stars and a

(possible large) number of degree two stars.

✦ But we can be greedy with degree two stars! ✦ Pick the one that is furthest away

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SLIDE 45

Algoritm cont’d

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 15 / 17

  • Now graph has O(T) leaf-stars and internal-stars

(∆ = 3)

  • Easy to pick one endpoint of a dropped request. If we

have O(T) choices for the other endpoint → recursive T O(T) algorithm.

  • Possible candidates are the O(T) special stars and a

(possible large) number of degree two stars.

✦ But we can be greedy with degree two stars! ✦ Pick the one that is furthest away

  • The greedy part is the only part that requires ∆ = 3
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Open problems

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 16 / 17

Conclusions:

  • PC is easy parameterized by ∆, W, but MaxPC is hard!
  • In our reductions we have to drop many requests. If we

parameterize also by T things get better. What next?

  • (p∆, pT)-MaxPC on undirected trees
  • Other parameters?
  • (pW)-MaxPC on rings?
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The End

❖ Path Coloring ❖ Example ❖ Known results ❖ Edge slicing ❖ Max PC ❖ Max PC hardness results ❖ DNP ❖ Reduction ❖ Reduction ❖ Complexity jump ❖ (p∆, pW, pT )- MaxPC ❖ (pT )-MaxPC binary trees ❖ (pT )-MaxPC binary trees ❖ Algoritm cont’d ❖ Open problems 17 / 17

Thank you! Questions?