Lossy Kernels for Connected Dominating Set on Sparse Graphs

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

doi:10.4230/lipics.stacs.2018.29

Cited by 2 publications

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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: Our Contributions and Outline Of This Papersupporting

confidence: 53%

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

Bevern

Fluschnik

Tsidulko

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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