Fixed-parameter approximations for k-Center problems in low highway dimension graphs

We consider the k-Center problem and some generalizations. For k-Center a set of kcenter vertices needs to be found in a graph G with edge lengths, such that the distance from any vertex of G to its nearest center is minimized. This problem naturally occurs in transportation networks, and therefore we model the inputs as graphs with bounded highway dimension, as proposed by Abraham et al. (SODA, pp 782–793, 2010). We show both approximation and fixed-parameter hardness results, and how to overcome them using fixed-parameter approximations, where the two paradigms are combined. In particular, we prove that for any ε> 0 computing a (2 - ε) -approximation is W[2]-hard for parameter k, and NP-hard for graphs with highway dimension O(log 2 n). The latter does not rule out fixed-parameter (2 - ε) -approximations for the highway dimension parameter h, but implies that such an algorithm must have at least doubly exponential running time in h if it exists, unless ETH fails. On the positive side, we show how to get below the approximation factor of 2 by combining the parameters k and h: we develop a fixed-parameter 3 / 2-approximation with running time 2 O(khlogh) · n O(1) . Additionally we prove that, unless P=NP, our techniques cannot be used to compute fixed-parameter (2 - ε) -approximations for only the parameter h. We also provide similar fixed-parameter approximations for the weightedk-Center and (k, F) -Partition problems, which generalize k-Center

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

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

Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

The Directed Steiner Network (DSN) problem takes as input a directed edge-weighted graph $G=(V,E)$ and a set $\mathcal{D}\subseteq V\times V$ of $k$ demand pairs. The aim is to compute the cheapest network $N\subseteq G$ for which there is an $s\to t$ path for each $(s,t)\in\mathcal{D}$. It is known that this problem is notoriously hard as there is no $k^{1/4-o(1)}$-approximation algorithm under Gap-ETH, even when parametrizing the runtime by $k$ [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter $k$. For the bi-DSN$_\text{Planar}$ problem, the aim is to compute a planar optimum solution $N\subseteq G$ in a bidirected graph $G$, i.e., for every edge $uv$ of $G$ the reverse edge $vu$ exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for $k$. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists for any generalization of bi-DSN$_\text{Planar}$, unless FPT=W[1]. One important special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, for which the solution network $N\subseteq G$ needs to strongly connect a given set of $k$ terminals. It has been observed before that for SCSS a parameterized $2$-approximation exists when parameterized by $k$ [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for $k$ no parameterized $(2-\varepsilon)$-approximation algorithm exists under Gap-ETH. Additionally we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for $k$

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions

Algoritmy pro grafy malé highway dimension

V této práci navrhneme algoritmy pro problém k-Supplier with Outliers. V síti dostaneme zadanou množinu dodavatelů a množinu klientů. Cílem je vybrat k doda- vatelů tak, aby vzdálenost mezi každým obslouženým klientem a nejbližším vybraným dodavatelem byla co nejmenší. Je dovoleno ponechat některé klienty neobsloužené. Max- imální počet klientů, které nemusíme obsloužit, je dán na vstupu. Jelikož k-Supplier with Outliers má mnoho využití v logistice, soustředíme se na parametry, které jsou vhodné pro dopravní sítě. Zabýváme se grafy s malou highway dimension, která byla zavedena Abrahamem et al. [SODA 2010] a grafy s malou doubling dimension. Je známo, že za předpokladu P ̸= NP nelze pro žádné kladné ε problém k-Sup- plier with Outliers (3 − ε)-aproximovat. Problém k-Supplier with Outliers je W[1]-těžký pro grafy s konstantní doubling dimension a highway dimension. Oba tyto těžkostní výsledky překonáme pomocí paradigmatu parametrizovaných aproximačních algoritmů. V případě highway dimension navrhneme (1 + ε)-aproximační algoritmus pro jakéko- liv kladné ε pracující v čase f(k, p, h, ε) · nO(1) , kde p je povolený počet klientů, které nemusíme obsloužit, h je highway dimension grafu na vstupu a f je nějaká vyčíslitelná funkce. V případě doubling dimension navrhneme (1 + ε)-aproximační algoritmus pro...In this work we develop algorithms for the k-Supplier with Outliers problem. In a network, we are given a set of suppliers and a set of clients. The goal is to choose k suppliers so that the distance between every served client and its nearest supplier is minimized. Clients that are not served are called outliers and the number of allowed outliers is given on input. As k-Supplier with Outliers has numerous applications in logistics, we focus on parameters which are suitable for transportation networks. We study graphs with low highway dimension, which was proposed by Abraham et al. [SODA 2010], and low doubling dimension. It is known that unless P = NP, k-Supplier with Outliers does not admit a (3 − ε)-approximation algorithm for any constant ε > 0. The k-Supplier with Outliers problem is W[1]-hard on graphs of constant doubling dimension for parame- ters k and highway dimension. We overcome both of these barriers through the paradigm of parameterized approximation algorithms. In the case of highway dimension, we develop a (1 + ε)-approximation algorithm for any ε > 0 with running time f(k, p, h, ε) · nO(1) where p is the number of allowed outliers, h is the highway dimension of the input graph, and f is some computable function. In the case of doubling dimension, we develop a (1 + ε)-approximation...Department of Applied MathematicsKatedra aplikované matematikyMatematicko-fyzikální fakultaFaculty of Mathematics and Physic