Lossy Kernels for Connected Dominating Set on Sparse Graphs

Eiben, Eduard; Kumar, Mithilesh; Mouawad, Amer E.; Panolan, Fahad; Siebertz, Sebastian

doi:10.1137/18m1172508

“…On the other hand, more and more sophisticated kernelization algorithms for distance-r dominating set on nowhere dense classes, which are all using the notion of uniform quasi-wideness, were developed [20,29,34,45]. The concept was also applied in the context of lossy kernelization [33] and for efficient algorithms for the reconfiguration variants of the above problems [52,79].…”

Section: Weak Coloring Numbermentioning

confidence: 99%

Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-wideness

Nadara

¹

,

Pilipczuk

²

,

Rabinovich

³

et al. 2019

ACM J. Exp. Algorithmics

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The notions of bounded expansion and nowhere denseness not only offer robust and general definitions of uniform sparseness of graphs, they also describe the tractability boundary for several important algorithmic questions. In this paper we study two structural properties of these graph classes that are of particular importance in this context, namely the property of having bounded generalized coloring numbers and the property of being uniformly quasi-wide. We provide experimental evaluations of several algorithms that approximate these parameters on real-world graphs. On the theoretical side, we provide a new algorithm for uniform quasi-wideness with polynomial size guarantees in graph classes of bounded expansion and show a lower bound indicating that the guarantees of this algorithm are close to optimal in graph classes with fixed excluded minor.

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“…Our Theorems 1.2 and 1.3 exhibit that RPP is such a problem. Among the so far few known lossy kernels [20,21,39,40,42], our Theorem 1.3 stands out since it shows a time and size efficient PSAKS, which is a property previously observed only in results of Krithika et al [40]. Moreover, Theorem 1.3 is apparently the first lossy kernelization result for parameters above lower bounds, which previously got attention in exact kernelization.…”

Section: Related Workmentioning

confidence: 51%

On $$(1+\varepsilon )$$ -approximate Data Reduction for the Rural Postman Problem

Bevern

¹

,

Fluschnik

²

,

Tsidulko

³

2019

Mathematical Optimization Theory and Operations Research

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Given an undirected graph with edge weights and a subset R of its edges, the Rural Postman Problem (RPP) is to find a closed walk of minimum total weight containing all edges of R. We prove that RPP is WK[1]-complete parameterized by the number and cost d of edges traversed additionally to the required ones. Thus, in particular, RPP instances cannot be polynomial-time compressed to instances of size polynomial in d unless the polynomial-time hierarchy collapses. In contrast, denoting by b ≤ 2d the number of vertices incident to an odd number of edges of R and by c ≤ d the number of connected components formed by the edges in R, we show how to reduce any RPP instance I to an RPP instance I with 2b + O(c/ε) vertices in O(n 3 ) time so that any α-approximate solution for I gives an α(1 + ε)-approximate solution for I, for any α ≥ 1 and ε > 0. That is, we provide a polynomial-size approximate kernelization scheme (PSAKS). We experimentally evaluate it on wide-spread benchmark data sets as well as on two real snow plowing instances from Berlin. On instances with few connected components, the number of vertices and required edges is reduced to about 50 % at a 1 % solution quality loss. We also make first steps towards a PSAKS for the parameter c. Problem 1.1 (Rural Postman Problem, RPP).Input: An undirected graph G = (V, E) with n vertices, edge weights ω : E → N ∪ {0}, and a multiset R of required edges of G. Task: Find a closed walk W * in G containing each edge of R and minimizing the total weight ω(W * ) of the edges on W * .We call any closed walk containing each edge of R an RPP tour. We will also consider the decision variant k-RPP, where one additionally gets a non-negative integer k ∈ N in the input and the task is to decide whether there is an RPP tour W of cost ω(W) ≤ k. RPP has direct applications in snow plowing, street sweeping, meter reading [14,22], vehicle depot location [29], drilling, and plotting [28,31]. The undirected version occurs especially in rural areas, where service vehicles can operate in both directions even on one-way roads [19]. Moreover, RPP is a special case of the Capacitated Arc Routing Problem (CARP) [30] and used in all "route first, cluster second" algorithms for CARP [1,10,49], which are notably the only ones with proven approximation guarantees [6,36,50]. Improved approximations for RPP automatically lead to better approximations for CARP.There is a folklore polynomial-time 3/2-approximation for RPP based on the Christofides-Serdyukov algorithm for the metric Traveling Salesman Problem [11,46] (we refer to Eiselt et al. [22] or van Bevern et al. [7] for a detailed algorithm description). We aim for (1 + ε)-approximations for all ε > 0. Unfortunately, containing the metric Traveling Salesman Problem as a special case, RPP is APX-hard [37]. Thus, finding such approximations typically requires exponential time, we present data reduction rules for this task. Their effectivity depends on the desired approximation factor. Graph theory. We generally consider multigraphs G = (V, E) wit...

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“…Our Theorems 1.2 and 1.3 exhibit that RPP is such a problem. Among the so far few known lossy kernels [22,23,44,45,47], our Theorem 1.3 stands out since it shows a time and size efficient PSAKS, which is a property previously observed only in results of Krithika et al [44]. Moreover, Theorem 1.3 is apparently the first lossy kernelization result for parameters above lower bounds, which previously got attention in exact kernelization.…”

Section: Introductionmentioning

confidence: 52%

On approximate data reduction for the Rural Postman Problem: Theory and experiments

Bevern

¹

,

Fluschnik

²

,

Tsidulko

³

2020

View full text Add to dashboard Cite

Given an undirected graph with edge weights and a subset R of its edges, the Rural Postman Problem (RPP) is to find a closed walk of minimum total weight containing all edges of R. We prove that RPP is WK[1]-complete parameterized by the number and weight d of edges traversed additionally to the required ones. Thus RPP instances cannot be polynomial-time compressed to instances of size polynomial in d unless the polynomial-time hierarchy collapses. In contrast, denoting by b ≤ 2d the number of vertices incident to an odd number of edges of R and by c ≤ d the number of connected components formed by the edges in R, we show how to reduce any RPP instance I to an RPP instance I ′ with 2b + O(c/ε) vertices in O(n 3) time so that any α-approximate solution for I ′ gives an α(1 + ε)-approximate solution for I, for any α ≥ 1 and ε > 0. That is, we provide a polynomial-size approximate kernelization scheme (PSAKS). We experimentally evaluate it on widespread benchmark data sets as well as on two real snow plowing instances from Berlin. We also make first steps toward a PSAKS for the parameter c.

show abstract

Lossy Kernels for Connected Dominating Set on Sparse Graphs

Cited by 25 publications

References 45 publications

Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-wideness

Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-wideness

On $$(1+\varepsilon )$$ -approximate Data Reduction for the Rural Postman Problem

On approximate data reduction for the Rural Postman Problem: Theory and experiments

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