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

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Minimum Diameter Color-Spanning Sets

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

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Overview

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

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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

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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

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Example

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

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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

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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

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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].

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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

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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

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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.

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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

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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

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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}.