Minimum Diameter Color-Spanning Sets Jonas Pr unte Lehrstuhl V : - - PowerPoint PPT Presentation

minimum diameter color spanning sets
SMART_READER_LITE
LIVE PREVIEW

Minimum Diameter Color-Spanning Sets Jonas Pr unte Lehrstuhl V : - - PowerPoint PPT Presentation

Minimum Diameter Color-Spanning Sets Jonas Pr unte Lehrstuhl V : Diskrete Optimierung Technische Universit at Dortmund Vogelpothsweg 87, 44227 Dortmund, Germany Aussois 2019 Jonas Pr unte Minimum Diameter Color-Spanning Sets Aussois


slide-1
SLIDE 1

Minimum Diameter Color-Spanning Sets

Jonas Pr¨ unte Lehrstuhl V : Diskrete Optimierung Technische Universit¨ at Dortmund Vogelpothsweg 87, 44227 Dortmund, Germany Aussois 2019

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 1

slide-2
SLIDE 2

Overview

1 Problem definition 2 Motivation 3 Previous work 4 Parameterized Complexity 5 Exact algorithms 6 Computational results

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 2

slide-3
SLIDE 3

Problem definition

  • Input : set of points P, a metric d, a set of colors C = {1, . . . , k} and a

function f : P → P(C)

  • the diameter of P′ ⊆ P is defined as diam(P′) = max

a,b∈P′d(a, b)

  • P′ ⊆ P is called Rainbow Set, if it contains all colors
  • Minimum Diameter Color-Spanning Set Problem (MDCS) asks for a Rainbow

Set with minimum diameter

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 3

slide-4
SLIDE 4

Motivation

  • design of computer networks
  • spatial databases
  • conference planning
  • conference requires different locations : hotel, restaurant, bar, ski slope
  • goal : find a set of locations minimizing the maximum distance

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 4

slide-5
SLIDE 5

Example

H S R H B H S R R S B B H H H B B B B R R R S

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 5

slide-6
SLIDE 6

Example

H S R H B H S R R S B B H H H B B B B R R R S Aussois

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 5

slide-7
SLIDE 7

Previous work

Theorem [Fleischer et al.,2010] The decision version of MDCS is NP-hard for Lp metric with 1 < p < ∞, in two

  • r higher dimensions.

Theorem [Fleischer et al.,2010] MDCS with L∞ can be solved in polynomial time by computing a smallest color-spanning ball if the number of dimensions m is not part of the input.

  • Kazemi et al.[2018] presented a PTAS for MDCS with L2 metric →

exponential in the number of dimensions

  • Kazemi et al.[2018] introduced a

√ 2 + ǫ approximation algorithm for MDCS with L2 metric suitable for high dimensions

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 6

slide-8
SLIDE 8

Parameterized Complexity

Definition An FPT (fixed-parameter tractable) algorithm with respect to parameter k is an algorithm whose running time can be written as f (k) · p(n) , where f is a computable function, p is a polynomial and n is the input length.

  • k-Vertex Cover admits FPT-algorithm (O(2kn), where n = |V |)
  • W-hierarchy gives a finer classification of the class NP
  • W [0] contains all problems admitting an FPT algorithm
  • W [0] ⊆ W [1] ⊆ . . . W [t]
  • W [i] ⊂ W [i + 1] implies P = NP
  • k-Clique parameterized by k is W [1]-complete
  • k-Dominating Set parameterized by k is W [2]-complete

Theorem MDCS parameterized by k is in W[2].

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 7

slide-9
SLIDE 9

Theorem MDCS with L∞-metric is W[1]-hard for parameter k if the number of dimensions m is part of the input. Proof.

  • reduction from Multi-Colored Clique
  • for every vertex vi we create an n(n−1)

2

  • dimensional point pi with the same

colors

  • every coordinate pi,jh with 1 ≤ j < h ≤ n of pi corresponds to a pair of

vertices

  • pi,jh :=

     1, if i = j and vi, vh not adjacent −1, if i = h and vi, vj not adjacent 0, otherwise

  • the distance between 2 points is larger than 1 ⇐

⇒ they are not adjacent

  • ∃ Rainbow Set with diameter 1 or lower ⇐

⇒ ∃ a clique containing all colors

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 8

slide-10
SLIDE 10

Parameterized Complexity

Example v1 v2 v3 v4         12 13 14 23 24 34         p1 =         1         p2 =         1         p3 =                 p4 =         −1 −1         {p1, p2, p3} is a Rainbow Set with diameter 1 ⇒ {v1, v2, v3} is a clique containing all colors

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 9

slide-11
SLIDE 11

Parameterized Complexity

Theorem MDCS with L∞-metric can not be approximated within an α < 2 in polynomial time if the number of dimensions m is part of the input, unless P = NP. Theorem MDCS with Lp-metric with 1 < p < ∞ is W[1]-hard for parameter k if the number of dimensions m is part of the input.

  • proof is similar but more technical

Theorem Let n be the number of points in d-dimensional space. For general d, n and Lp-metric with 1 < p < ∞ MDCS can not be approximated within an α <

p

  • 2n−2+2p

2n

< 2 in polynomial time if the number of dimensions m is part of the input, unless P = NP.

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 10

slide-12
SLIDE 12

Exact solution methods

Theorem One IQP formulation for MDCS is: min z s.t. z ≥ xixjd(pi, pj) ∀ 1 ≤ i < j ≤ n

  • c∈f (pi)

xi ≥ 1 ∀ 1 ≤ c ≤ k xi ∈ {0, 1} ∀ 1 ≤ i ≤ n z ∈ R

  • advantage : suitable for all metrics
  • advantage : number of dimensions does not influence running time

significantly in Lp case

  • disadvantage : quadratic constraints

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 11

slide-13
SLIDE 13

Exact solution methods

Recursive exact algorithm for MDCS

Input: Set of n points P in d dimensions in which every point has at least one color c ∈ C ; fixed Lp Output: Subset of P with minimum diameter with respect to LP containing all colors

1: Compute all pairs of points pi, pj with f (pi) ⊂ f (pj) and f (pj) ⊂ f (pi), sort

them upward by their distance into the queue Q and save their distances.

2: for all set of points Pij ∈ Q do 3:

Do the recursive step with Pij.

4: end for

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 12

slide-14
SLIDE 14

Exact solution methods

Recursive step

Input: Set of n points P in d dimensions in which every point has at least one color c ∈ C ; subset Pij ⊆ P; fixed Lp

1: Add all points from the interior of its convex hull to Pij. 2: if Pij contains all colors then 3:

return Pij and GLOBAL STOP

4: end if 5: Build M(Pij) = {p′ ∈ P | d(p, p′) ≤ diam(Pij) ∀ p ∈ Pij}. 6: if M(Pij) does not contain all colors then 7:

return

8: end if 9: Find under all colors which are not covered by the solution the color c with

the least occurrence in M(Pij) \ Pij.

10: for all Point p ∈ M(Pij) \ Pij with color c do 11:

Do the recursive step with Pij ∪ {p}.

12: end for

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 13

slide-15
SLIDE 15

Exact solution methods

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14

slide-16
SLIDE 16

Exact solution methods

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14

slide-17
SLIDE 17

Exact solution methods

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14

slide-18
SLIDE 18

Exact solution methods

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14

slide-19
SLIDE 19

Exact solution methods

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14

slide-20
SLIDE 20

Exact solution methods

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14

slide-21
SLIDE 21

Exact solution methods

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14

slide-22
SLIDE 22

Exact solution methods

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14

slide-23
SLIDE 23

Exact solution methods

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14

slide-24
SLIDE 24

Exact solution methods

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14

slide-25
SLIDE 25

Exact solution methods

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14

slide-26
SLIDE 26

Exact solution methods

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14

slide-27
SLIDE 27

Exact solution methods

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14

slide-28
SLIDE 28

Exact solution methods

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14

slide-29
SLIDE 29

Exact solution methods

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14

slide-30
SLIDE 30

Exact solution methods

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14

slide-31
SLIDE 31

Exact solution methods

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14

slide-32
SLIDE 32

Exact solution methods

  • advantage : suitable for all metrics
  • advantage : number of dimensions does not influence running time

significantly in Lp case

  • disadvantage : no good theoretical bound on running time

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 15

slide-33
SLIDE 33

Computational results

|P| |C| Exact algorithm IQP formulation time Solved instances time Solved instances 200 5 10 109 10 10 10 152 10 20 10 116 10 50 10 32 10 600 5 10 4437 10 10 10 4173 10 20 10 3896 10 50 1 10 5378 9 700 5 10 7770 10 10 10 7346 10 20 10 7178 10 50 2 10 8785 5 1000 5 10

  • 10

10

  • 20

10

  • 50

3 10

  • Table: Comparison between exact algorithm and IQP for MDCS

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 16

slide-34
SLIDE 34

Computational results

|P| |C| Exact algorithm time Solved instances 5000 5 4 10 10 13 10 20 41 10 50 197 10 100 557 10 10000 5 69 10 10 104 10 20 212 10 50 837 10 100 2048 10

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 17

slide-35
SLIDE 35

Open questions

  • Complexity for L1 metric?
  • Parameterized Complexity for parameter k for fixed number of dimensions m?

Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 18