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

Abstract

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...

Polynomial-Time Data Reduction for Weighted Problems Beyond Additive Goal Functions

“…Hence, we put forward the following: Can Theorem 2.1 be executed in quadratic, or even linear time? We remark that van Bevern et al [6] give a linear-time variant Theorem 2.1 for approximate kernelizations.…”

“…Another direction, seemingly not addressed so far, aims on the running time. Note that Frank and Tardos [15] state no explicit running time of their algorithm, and Lenstra et al [20,Proposition 1.26] state that their simultaneous Diophantine approximation algorithm, which forms a subroutine in Frank and Tardos' technique, runs in O(d 6 (log( w ∞ )) O (1) ) time. Hence, we put forward the following: Can Theorem 2.1 be executed in quadratic, or even linear time?…”

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

“…We think that the approach taken by Reduction Rule 5.8, namely reducing all vertices that do not belong to some inclusion‐maximal set B of mutually sufficiently distant vertices, might be applicable to other metric graph problems: it ensures that, for each deleted vertex, some nearby representative in B is retained. In preliminary research, for example, we also found it to be applicable to a metric variant of the Min‐Power Symmetric Connectivity problem, where it is required to connect c disconnected parts of a wireless sensor network [4], and to the Location Rural Postman Problem [12]. Notably, this approach does not generalize well to asymmetric distances, so that another vexing question besides proving Conjecture 5.14 is whether the scheme for the parameter b + c presented in this work can be generalized to the directed Rural Postman Problem.…”

“…Since our main goal is evaluating the effect of our data reduction rather than the running time of our algorithm, we sacrificed speed for simplicity and implemented the part of our PSAKS described in Section 5.4.1 in approximately 200 lines of Python (not counting the testing environment) using the NetworkX library for finding minimum-weight perfect matchings, (bi)connected components, cut vertices, and spanning trees. 5 These routines are also contained in highly optimized C++ libraries like LEMON 6 and we expect that one could achieve a speedup by orders of magnitude by implementing our PSAKS in C++. We did not implement the weight reduction step described in Section 5.4.2, since it is mainly of theoretical interest (to prove a polynomial-size of the kernel rather than just a polynomial number of vertices and edges).…”

On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

“…Pioneering work of Lokshtanov et al [LPRS17] on the approximate kernel is being followed by a series of papers generalizing/improving results mentioned in this work and establishing lossy kernels for various other problems. Lossy kernels for some variations of Connected Vertex Cover [EHR17, KMR18], Connected Feedback Vertex Set [Ram19], Steiner Tree [DFK + 18] and Dominating Set [EKM + 19, Sie17] have been established (also see [Man19,vBFT18]). Krithika et al [KMRT16] were first to study graph contraction problems from the lenses of lossy kernelization.…”