scholarly journals W[1]-hardness of the k-center problem parameterized by the skeleton dimension

2021 ◽  
Author(s):  
Johannes Blum
Keyword(s):  
Urban Planning ◽  
Planar Graphs ◽  
Transportation Networks ◽  
Exact Algorithm ◽  
Computable Function ◽  
Maximum Distance ◽  
Center Problem ◽  
Exponential Time ◽  
Positive Edge ◽  
Doubling Dimension

AbstractWe study the k-Center problem, where the input is a graph $$G=(V,E)$$ G = ( V , E ) with positive edge weights and an integer k, and the goal is to select k center vertices $$C \subseteq V$$ C ⊆ V such that the maximum distance from any vertex to the closest center vertex is minimized. In general, this problem is $$\mathsf {NP}$$ NP -hard and cannot be approximated within a factor less than 2. Typical applications of the k-Center problem can be found in logistics or urban planning and hence, it is natural to study the problem on transportation networks. Common characterizations of such networks are graphs that are (almost) planar or have low doubling dimension, highway dimension or skeleton dimension. It was shown by Feldmann and Marx that k-Center is $$\mathsf {W[1]}$$ W [ 1 ] -hard on planar graphs of constant doubling dimension when parameterized by the number of centers k, the highway dimension $$hd$$ hd and the pathwidth $$pw$$ pw (Feldmann and Marx 2020). We extend their result and show that even if we additionally parameterize by the skeleton dimension $$\kappa $$ κ , the k-Center problem remains $$\mathsf {W[1]}$$ W [ 1 ] -hard. Moreover, we prove that under the Exponential Time Hypothesis there is no exact algorithm for k-Center that has runtime $$f(k,hd,pw,\kappa ) \cdot \vert V \vert ^{o(pw+ \kappa + \sqrt{k+hd})}$$ f ( k , h d , p w , κ ) · | V | o ( p w + κ + k + h d ) for any computable function f.

APPROXIMATING ASYMMETRIC TSP IN EXPONENTIAL TIME

2014 ◽  
Vol 25 (01) ◽  
pp. 89-99 ◽  
Author(s):  
ALEXANDER GOLOVNEV
Keyword(s):  
Traveling Salesman Problem ◽  
Directed Graph ◽  
Short Note ◽  
Simple Algorithm ◽  
Computable Function ◽  
Traveling Salesman ◽  
Polynomial Space ◽  
Exponential Time ◽  
Running Time ◽  
Polynomial Factor

Let G be a complete directed graph with n vertices and integer edge weights in range [0,M]. It is well known that an optimal Traveling Salesman Problem (TSP) in G can be solved in 2n time and space (all bounds are given within a polynomial factor of the input length, i.e., poly(n, log M)) and this is still the fastest known algorithm. If we allow a polynomial space only, then the best known algorithm has running time 4nnlog n. For TSP with bounded weights there is an algorithm with 1.657n · M running time. It is a big challenge to develop an algorithm with 2n time and polynomial space. Also, it is well-known that TSP cannot be approximated within any polynomial time computable function unless P=NP. In this short note we propose a very simple algorithm that, for any 0 < ε < 1, finds (1+ε)-approximation to asymmetric TSP in 2nε−1 time and ε−1 · poly(n, log M) space. Thereby, for any fixed ε, the algorithm needs 2n steps and polynomial space to compute (1 + ε)-approximation.

scholarly journals The Parameterized Hardness of the k-Center Problem in Transportation Networks

Algorithmica ◽  
2020 ◽  
Vol 82 (7) ◽  
pp. 1989-2005 ◽  
Author(s):  
Andreas Emil Feldmann ◽  
Dániel Marx
Keyword(s):  
Transportation Networks ◽  
Center Problem

scholarly journals A scalable exact algorithm for the vertex p-center problem

2019 ◽  
Vol 103 ◽  
pp. 211-220 ◽  
Author(s):  
Claudio Contardo ◽  
Manuel Iori ◽  
Raphael Kramer
Keyword(s):  
Exact Algorithm ◽  
Center Problem

scholarly journals The capacitated single-source p-center problem in the presence of fixed cost and multilevel capacities using VNS and aggregation technique

2018 ◽  
Vol 28 (4) ◽  
pp. 453-473
Author(s):  
Chandra Irawan ◽  
Kusmaningrum Soemadi
Keyword(s):  
Mathematical Model ◽  
Exact Method ◽  
Maximum Distance ◽  
Variable Neighbourhood Search ◽  
Fixed Cost ◽  
Single Source ◽  
Center Problem ◽  
Limited Budget ◽  
Neighbourhood Search ◽  
Aggregation Technique

In this study, the discrete p-center problem with the presence of multilevel capacities and fixed (opening) cost of a facility under a limited budget is investigated. A mathematical model of the problem is produced, where we seek the location of open facilities, their corresponding capacities, and the allocation of the customers to the open facilities in order to minimise the maximum distance between customers and their assigned facilities. Two matheuristic approaches are also proposed to deal with larger instances. The first approach is a hybridisation of a clustering-based technique, an exact method, while the second one is based on Variable Neighbourhood Search (VNS). Computational experiments show that the proposed methods produce interesting and competitive results on newly and randomly generated datasets.

Adjustable Preference Path Strategies for Use in Multicriteria, Stochastic, and Time-Varying Transportation Networks

2005 ◽  
Vol 1923 (1) ◽  
pp. 137-143 ◽  
Author(s):  
Sathaporn Opasanon ◽  
Elise Miller-Hooks
Keyword(s):  
Real Time ◽  
Transportation Networks ◽  
Exact Algorithm ◽  
Random Variable ◽  
Expected Value ◽  
Multiple Criteria ◽  
Time Varying ◽  
Travel Times ◽  
Solution Strategies ◽  
Time Path

In this paper, an exact algorithm is proposed for determining adjustable preference path strategies in multicriteria, stochastic, and time-varying (MSTV) networks. In MSTV networks, multiple arc attributes are associated with each arc, each being a time-varying random variable. Solution paths that seek to minimize the expected value of each of multiple criteria are sought from all origins to a specified destination for all departure times in a period of interest. These solution strategies allow a traveler to update a preference for which criterion of multiple criteria is of greatest importance and then adaptively select the best path for the selected criterion at each node in response to knowledge of experienced travel times on the arcs. Such adjustable preference path strategies are particularly useful for providing real-time path-finding assistance.

scholarly journals A technique for obtaining true approximations for k-center with covering constraints

2021 ◽  
Author(s):  
Georg Anegg ◽  
Haris Angelidakis ◽  
Adam Kurpisz ◽  
Rico Zenklusen
Keyword(s):  
Approximation Algorithm ◽  
Constant Factor ◽  
Center Problem ◽  
New Approach ◽  
Exponential Time ◽  
Hard Constraints ◽  
Finite Approximation ◽  
Approximation Guarantee ◽  
Open Question ◽  
Clustering Problems

AbstractThere has been a recent surge of interest in incorporating fairness aspects into classical clustering problems. Two recently introduced variants of the k-Center problem in this spirit are Colorful k-Center, introduced by Bandyapadhyay, Inamdar, Pai, and Varadarajan, and lottery models, such as the Fair Robust k-Center problem introduced by Harris, Pensyl, Srinivasan, and Trinh. To address fairness aspects, these models, compared to traditional k-Center, include additional covering constraints. Prior approximation results for these models require to relax some of the normally hard constraints, like the number of centers to be opened or the involved covering constraints, and therefore, only obtain constant-factor pseudo-approximations. In this paper, we introduce a new approach to deal with such covering constraints that leads to (true) approximations, including a 4-approximation for Colorful k-Center with constantly many colors—settling an open question raised by Bandyapadhyay, Inamdar, Pai, and Varadarajan—and a 4-approximation for Fair Robust k-Center, for which the existence of a (true) constant-factor approximation was also open. We complement our results by showing that if one allows an unbounded number of colors, then Colorful k-Center admits no approximation algorithm with finite approximation guarantee, assuming that $$\mathtt {P}\ne \mathtt {NP}$$ P ≠ NP . Moreover, under the Exponential Time Hypothesis, the problem is inapproximable if the number of colors grows faster than logarithmic in the size of the ground set.

scholarly journals New Algorithms for Mixed Dominating Set

2021 ◽  
Vol vol. 23 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Louis Dublois ◽  
Michael Lampis ◽  
Vangelis Th. Paschos
Keyword(s):  
Lower Bound ◽  
Dominating Set ◽  
Exact Algorithm ◽  
Polynomial Space ◽  
Parameterized Algorithms ◽  
Exponential Time ◽  
Open Questions ◽  
Fpt Algorithm ◽  
Exponential Space ◽  
New Algorithms

A mixed dominating set is a collection of vertices and edges that dominates all vertices and edges of a graph. We study the complexity of exact and parameterized algorithms for \textsc{Mixed Dominating Set}, resolving some open questions. In particular, we settle the problem's complexity parameterized by treewidth and pathwidth by giving an algorithm running in time $O^*(5^{tw})$ (improving the current best $O^*(6^{tw})$), as well as a lower bound showing that our algorithm cannot be improved under the Strong Exponential Time Hypothesis (SETH), even if parameterized by pathwidth (improving a lower bound of $O^*((2 - \varepsilon)^{pw})$). Furthermore, by using a simple but so far overlooked observation on the structure of minimal solutions, we obtain branching algorithms which improve both the best known FPT algorithm for this problem, from $O^*(4.172^k)$ to $O^*(3.510^k)$, and the best known exponential-time exact algorithm, from $O^*(2^n)$ and exponential space, to $O^*(1.912^n)$ and polynomial space. Comment: This paper has been accepted to IPEC 2020

scholarly journals An analysis between exact and approximate algorithms for the k-center problem in graphs

2022 ◽  
Vol 12 (2) ◽  
pp. 2058
Author(s):  
Velin Kralev ◽  
Radoslava Kraleva ◽  
Viktor Ankov ◽  
Dimitar Chakalov
Keyword(s):  
Operating System ◽  
Execution Time ◽  
Source Code ◽  
Exact Algorithm ◽  
Approximate Solutions ◽  
Approximate Algorithm ◽  
Approximate Algorithms ◽  
Optimal Solutions ◽  
Center Problem ◽  
Computation Complexity

<span lang="EN-US">This research focuses on the k-center problem and its applications. Different methods for solving this problem are analyzed. The implementations of an exact algorithm and of an approximate algorithm are presented. The source code and the computation complexity of these algorithms are presented and analyzed. The multitasking mode of the operating system is taken into account considering the execution time of the algorithms. The results show that the approximate algorithm finds solutions that are not worse than two times optimal. In some case these solutions are very close to the optimal solutions, but this is true only for graphs with a smaller number of nodes. As the number of nodes in the graph increases (respectively the number of edges increases), the approximate solutions deviate from the optimal ones, but remain acceptable. These results give reason to conclude that for graphs with a small number of nodes the approximate algorithm finds comparable solutions with those founds by the exact algorithm.</span>

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