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
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
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
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
Example S R S R H B B R H S B B B R R H H H B H S R B Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 5
Example S R S R H B B R H S B Aussois B B R R H H H B H S R B Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 5
Previous work Theorem [Fleischer et al.,2010] The decision version of MDCS is NP-hard for L p metric with 1 < p < ∞ , in two or 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 L 2 metric → exponential in the number of dimensions √ • Kazemi et al.[2018] introduced a 2 + ǫ approximation algorithm for MDCS with L 2 metric suitable for high dimensions Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 6
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 (2 k n ), 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
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 v i we create an n ( n − 1) -dimensional point p i with the same 2 colors • every coordinate p i , jh with 1 ≤ j < h ≤ n of p i corresponds to a pair of vertices 1 , if i = j and v i , v h not adjacent • p i , jh := − 1 , if i = h and v i , v j 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
Parameterized Complexity Example 12 0 0 0 0 13 0 0 0 0 v 4 v 3 14 1 0 0 − 1 p 1 = p 2 = p 3 = p 4 = 23 0 0 0 0 24 0 1 0 − 1 34 0 0 0 0 v 1 v 2 { p 1 , p 2 , p 3 } is a Rainbow Set with diameter 1 ⇒ { v 1 , v 2 , v 3 } is a clique containing all colors Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 9
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 L p -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 L p -metric with 1 < p < ∞ MDCS can not be approximated within an � 2 n − 2+2 p p < 2 in polynomial time if the number of dimensions m is part of α < 2 n the input, unless P = NP. Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 10
Exact solution methods Theorem One IQP formulation for MDCS is: min z s.t. z ≥ x i x j d ( p i , p j ) ∀ 1 ≤ i < j ≤ n � x i ≥ 1 ∀ 1 ≤ c ≤ k c ∈ f ( p i ) x i ∈ { 0 , 1 } ∀ 1 ≤ i ≤ n z ∈ R • advantage : suitable for all metrics • advantage : number of dimensions does not influence running time significantly in L p case • disadvantage : quadratic constraints Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 11
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 L p Output: Subset of P with minimum diameter with respect to L P containing all colors 1: Compute all pairs of points p i , p j with f ( p i ) �⊂ f ( p j ) and f ( p j ) �⊂ f ( p i ), sort them upward by their distance into the queue Q and save their distances. 2: for all set of points P ij ∈ Q do Do the recursive step with P ij . 3: 4: end for Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 12
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 P ij ⊆ P ; fixed L p 1: Add all points from the interior of its convex hull to P ij . 2: if P ij contains all colors then return P ij and GLOBAL STOP 3: 4: end if 5: Build M ( P ij ) = { p ′ ∈ P | d ( p , p ′ ) ≤ diam ( P ij ) ∀ p ∈ P ij } . 6: if M ( P ij ) does not contain all colors then return 7: 8: end if 9: Find under all colors which are not covered by the solution the color c with the least occurrence in M ( P ij ) \ P ij . 10: for all Point p ∈ M ( P ij ) \ P ij with color c do Do the recursive step with P ij ∪ { p } . 11: 12: end for Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 13
Exact solution methods Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14
Exact solution methods Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14
Exact solution methods Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14
Exact solution methods Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14
Exact solution methods Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14
Exact solution methods Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14
Exact solution methods Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14
Exact solution methods Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14
Exact solution methods Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14
Exact solution methods Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14
Exact solution methods Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14
Exact solution methods Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14
Exact solution methods Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14
Exact solution methods Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14
Exact solution methods Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14
Exact solution methods Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14
Exact solution methods Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 14
Exact solution methods • advantage : suitable for all metrics • advantage : number of dimensions does not influence running time significantly in L p case • disadvantage : no good theoretical bound on running time Jonas Pr¨ unte Minimum Diameter Color-Spanning Sets Aussois 2019 15
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