On approximability of minimum color-spanning ball in high dimensions

Discrete Applied Mathematics xxx (xxxx) xxx

Contents lists available at ScienceDirect

Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam

Note

On approximability of minimum color-spanning ball in high dimensions ∗

Mohammad Reza Kazemi a , Ali Mohades a , , Payam Khanteimouri b a b

Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran Department of Computer Science, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran

article

info

Article history: Received 27 January 2019 Received in revised form 19 September 2019 Accepted 9 October 2019 Available online xxxx

a b s t r a c t This paper presents a lower bound on the running time of any approximation scheme for Minimum Color-Spanning Ball (MCSB) problem in high dimensional spaces. This bound is based on the Exponential Time Hypothesis (ETH). © 2019 Elsevier B.V. All rights reserved.

Keywords: Approximation algorithm Approximability Color-spanning set High dimensional spaces Exponential Time Hypothesis (ETH)

1. Introduction Suppose P is a given set of n colored points in Rd . A color-spanning ball is a ball that contains at least one point from each color. By B(P) we denote the collection of all color-spanning balls for P. The MCSB problem is to compute a color-spanning ball B∗ ∈ B(P) where radius(B∗ ) = min radius(B). B∈B(P )

One of the main motivations for computing an extent measure of colored points comes from spatial database [7]. Consider a spatial database where each tuple is associated with a keyword. The m-closest keywords (mCK) query is the problem of finding the ‘‘closest’’ tuples that together match all the keywords given by the user. Considering each keyword as a distinct color and also each tuple as a d-dimensional point, we model this problem as computing an extent measure of colored points. Now, if the diameter is the measure of closeness, Minimum Diameter Color-Spanning Set (MDCSS) describes the problem. This problem is NP-Hard even in two dimensions [2]. MDCSS can be approximated in low dimensions by some PTASs√[3,5]. Kazemi et al. [5] also showed a lower bound on error dependency of any PTAS in low dimensions and proved 2 − ϵ inapproximability in high dimensions. Considering the Minimum Enclosing Ball (MEB) as another measure of closeness, an m-closest query turns into MCSB problem. This problem can be approximated efficiently in low dimensions [6]. However, in high dimensions, the situation is different as we discuss in this paper. The P ̸ = NP assumption excludes the possibility of finding polynomial time algorithms for NP-Hard problems. However, it does not even rule out the existence of super-polynomial time algorithms according to the current knowledge. On the other hand, finding subexponential time algorithms for many NP-Hard problems remains a major challenge. ∗ Corresponding author. E-mail addresses: [email protected] (M.R. Kazemi), [email protected] (A. Mohades), [email protected] (P. Khanteimouri). https://doi.org/10.1016/j.dam.2019.10.016 0166-218X/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: M.R. Kazemi, A. Mohades and P. Khanteimouri, On approximability of minimum color-spanning ball in high dimensions, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.10.016.

2

M.R. Kazemi, A. Mohades and P. Khanteimouri / Discrete Applied Mathematics xxx (xxxx) xxx

Fig. 1. Reduction illustration for an unsatisfiable 3-SAT formula of 4 variables where m = 2. e1 , . . . , e8 stand for partial assignments.

Exponential Time Hypothesis (ETH) [4], as a stronger assumption, enables us to prove sharp lower bounds on complexity of computational problems. A Boolean formula is in Conjunctive Normal Form (CNF) if it is a conjunction of clauses. A clause is a disjunction of literals and, a literal is a Boolean variable or its negation. A 3-CNF formula is a CNF formula in which each clause has at most 3 literals. 3-Satisfiability (3-SAT) problem is the problem of deciding whether a 3-CNF formula is satisfiable or not. The Exponential Time Hypothesis (ETH) states that there is no algorithm with running time of 2o(n) for the 3-SAT problem where n is the number of variables. Let P be a set of n points in Rd partitioned into some colors. A color-spanning set is a subset that contains at least one point from each color. Also, let CS be the collection of all color-spanning subsets of P. The MCSB problem is min

max ∥c − x∥2 .

S ∈CS , c ∈Rd x∈S

In the following, we study the approximability of the MCSB problem and as a contribution, we prove the following theorem. Theorem 1. Assuming ETH, there is no (1 + ϵ )-approximation algorithm running in f ( 1ϵ )n the dimension is considered as an input parameter.

o( √1 ) ϵ

time for MCSB problem when

Note that ϵ is the error parameter and f (·) is any unbounded time constructible function. We say a function f is time constructible if it can be computed by an algorithm with running time of O(f (n)). 2. Proof of Theorem 1 1

Suppose there exists a (1 + ϵ )-approximation algorithm for the MCSB problem running in f ( 1ϵ )N α time where N is √ the size of input, α = ϵ s( 1ϵ ) and s(·) is any nondecreasing unbounded function. We show that 3-SAT can be solved in subexponential time by this algorithm. Let φ be an arbitrary 3-SAT formula with n variables and let m be the greatest integer smaller than √1 . We partition 2ϵ the variables into m parts of equal size in an arbitrary way. Assume that n is divisible by m, since otherwise we add at n most m − 1 dummy variables to the input formula, and any deciding algorithm for the satisfiability of the new formula in subexponential time works as the same for the input formula. For simplicity, we name each clause and each part by n their colors. A partial assignment is a true/false assignment for a specified subset of variables. Since each part includes m n distinct variables, there are 2 m possible partial assignments for variables of that part. Thus, by considering all the parts, n we have m2 m partial assignments. n Now, suppose that {e1 , . . . , e mn } is the standard basis of Euclidean space of dimension m2 m . Recall that for any vector m2 in this basis, one coordinate is 1 and the others are 0. We define an arbitrary one-by-one correspondence between the set of all partial assignments and the standard basis. For ease of presentation, we name each partial assignment by its corresponding basis vector. Then, if partial assignment ei satisfies clause c, we place a point with color c in position ei . Moreover, if partial assignment ei belongs to part p, we place a point with color p in position ei (see Fig. 1). Please cite this article as: M.R. Kazemi, A. Mohades and P. Khanteimouri, On approximability of minimum color-spanning ball in high dimensions, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.10.016.

M.R. Kazemi, A. Mohades and P. Khanteimouri / Discrete Applied Mathematics xxx (xxxx) xxx

3

Fig. 2. There are not two vertices that cover all the colors.

Lemma 1. φ is satisfiable if and only if there is a color-spanning ball that contains exactly m distinct points. Proof. Suppose that φ is satisfiable by assignment e. We split e to m partial assignments e′1 , . . . , e′m according to our partitioning method. Since e′1 , . . . , e′m belong to distinct parts, they cover all colors of parts. Now, we show that the regular simplex formed by e′1 , . . . , e′m spans the remaining colors. Since assignment e satisfies all clauses, at least one of the literals in any clause c, called x, is true by e. Assuming x belongs to part p, the partial assignment e′p satisfies the literal x and also the clause c. In this situation, we have already placed a point with color c in the position corresponding to e′p . Therefore, the smallest ball covering e′1 , . . . , e′m is color-spanning. Now, to prove the reverse implication, suppose that there exists a color-spanning ball containing e′1 , . . . , e′m . Since e′1 , . . . , e′m cover all colors of parts, they belong to distinct parts. As they cover the colors of clauses, they also satisfy all clauses. Therefore, e′1 , . . . , e′m form an assignment that satisfies φ (See Fig. 2). □ Let Bm be the minimum enclosing ball of a regular (m − 1)-simplex with unit side. Then, we have radius(Bm+1 ) radius(Bm )

√ = = √

Since m <

√1

2ϵ

1 m2

⇒ 1−

m

/√

m+1 m m2 − 1

m−1 m

.

, we have

> 2ϵ 1 m2 m

< 1 − 2ϵ 1

m2

> √

1 − 2ϵ

>

1

> 1 + ϵ. −1 Thus, any (1 + ϵ )-approximation algorithm for MCSB decides the satisfiability of 3-SAT formula. Therefore, the running ⇒ √

1−ϵ

time needed to solve the 3-SAT problem is 1 √ 1 √ n f ( )m ϵ s(1/ϵ ) 2 m ϵ s(1/ϵ )

ϵ

√ √ 1 nO(1) = f ( )(1/ 2ϵ ) ϵ s(1/ϵ ) 2 s(1/ϵ ) . ϵ Let f −1 (n) = sup{ν|f (ν ) ≤ n}. f −1 is a nondecreasing unbounded function because f is unbounded. We set ϵ such that = min{f −1 (n), n}. Then, s( 1ϵ ) = min{s(f −1 (n)), s(n)} is also a nondecreasing unbounded function of n. We have, 1

1

ϵ

nO(1)

2 s(1/ϵ ) = 2o(n) . Please cite this article as: M.R. Kazemi, A. Mohades and P. Khanteimouri, On approximability of minimum color-spanning ball in high dimensions, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.10.016.

4

Since

M.R. Kazemi, A. Mohades and P. Khanteimouri / Discrete Applied Mathematics xxx (xxxx) xxx 1

≤ n we have, √ √ 1 √ √n (1/ 2ϵ ) ϵ s(1/ϵ ) ≤ n = 2o(n) ϵ

and also, f ( 1ϵ ) is less than n. Note that min{f −1 (n), n} is computable in polynomial time because f is time constructible. Therefore, 3-SAT can be computed in time 2o(n) . This contradicts ETH and Theorem 1 holds. In the above reduction, if we set m to be a constant fraction of n, we also gain a polynomial time reduction from 3-SAT to MCSB problem and the following result can be obtained directly. Corollary 1. It is NP-Hard to compute MCSB problem when the dimension is considered as an input parameter. 3. Upper bound In this section, we give a simple upper bound for the running time of the MCSB problem. To approximate MCSB, we use the optimal core-set for balls presented by Bădoiu and Clarkson’s [1]. According to their result, for any point set P there exists a subset Q of size ⌈ 1ϵ ⌉ where MEB(P) ⊂ (1 + ϵ )MEB(Q ). Suppose B∗ is an optimal solution for MCSB. Let Q ∗ be an optimal core-set for the set of points contained in B∗ . Then, B∗ ⊂ (1 + ϵ )MEB(Q ∗ ). Since (1 + ϵ )MEB(Q ∗ ) is also color-spanning, it can be considered as a (1 + ϵ )-approximation for MCSB. Therefore, by checking all candidates to be color-spanning we achieve a (1 + ϵ )-approximation in O(dn⌈1/ϵ⌉+1 ) time. 4. Conclusion Ω ( √1 )

ϵ for approximability of MCSB problem. In addition, we showed an Assuming ETH, we have a lower bound of n upper bound of O(dn⌈1/ϵ⌉+1 ) time. Therefore, a step toward filling this gap remains as an interesting open challenge.

References [1] M. Bădoiu, K.L. Clarkson, Optimal core-sets for balls, Comput. Geom. 40 (1) (2008) 14–22. [2] R. Fleischer, X. Xu, Computing minimum diameter color-spanning sets is hard, Inform. Process. Lett. 111 (21) (2011) 1054–1056. [3] M. Ghodsi, H. Homapour, M. Seddighin, Approximate minimum diameter, in: Proceding of 23rd International Conference, COCOON, 2017, pp. 237–249. [4] R. Impagliazzo, R. Paturi, Complexity of k-sat, in: Computational Complexity, 1999. Proceedings. Fourteenth Annual IEEE Conference on, 1999, pp. 237–240. [5] M.R. Kazemi, A. Mohades, P. Khanteimouri, Approximation algorithms for color spanning diameter, Inform. Process. Lett. 135 (2018) 53–56. [6] P. Khanteimouri, A. Mohades, M.A. Abam, M.R. Kazemi, Efficiently approximating color-spanning balls, Theoret. Comput. Sci. 634 (2016) 120–126. [7] D. Zhang, Y.M. Chee, A. Mondal, A.K. Tung, M. Kitsuregawa, Keyword search in spatial databases: Towards searching by document, in: Data Engineering, 2009. ICDE’09. IEEE 25th International Conference on, IEEE, 2009, pp. 688–699.

Please cite this article as: M.R. Kazemi, A. Mohades and P. Khanteimouri, On approximability of minimum color-spanning ball in high dimensions, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.10.016.