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On approximate data reduction for the Rural Postman Problem: Theory and experiments
Networks ( IF 1.6 ) Pub Date : 2020-10-06 , DOI: 10.1002/net.21985
René Bevern 1 , Till Fluschnik 2 , Oxana Yu. Tsidulko 1, 3
Affiliation  

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 $b\leq 2d$ the number of vertices incident to an odd number of edges of $R$ and by $c\leq d$ 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 $2b+O(c/\varepsilon)$ vertices in $O(n^3)$ time so that any $\alpha$-approximate solution for $I'$ gives an $\alpha(1+\varepsilon)$-approximate solution for $I$, for any $\alpha\geq 1$ and $\varepsilon>0$. 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 $c$.

中文翻译:

农村邮递员问题的近似数据约简:理论与实验

给定具有边权重的无向图及其边的子集 $R$,农村邮递员问题 (RPP) 是找到包含 $R$ 的所有边的最小总权重的封闭步行。我们证明 RPP 是 WK[1] 完全参数化的,由除所需边之外所遍历的边的数量和成本 $d$ 进行参数化。因此,特别是,除非多项式时间层次结构崩溃,否则 RPP 实例不能被多项式时间压缩为 $d$ 中大小多项式的实例。相比之下,用 $b\leq 2d$ 表示与 $R$ 的奇数条边相关的顶点数,用 $c\leq d$ 表示由 $R$ 中的边形成的连通分量的数量,我们证明如何在 $O(n^3)$ 时间内将任何 RPP 实例 $I$ 减少为具有 $2b+O(c/\varepsilon)$ 顶点的 RPP 实例 $I'$,以便任何 $\alpha$-近似解对于$I' $ 为任何 $\alpha\geq 1$ 和 $\varepsilon>0$ 给出了 $\alpha(1+\varepsilon)$ 的近似解。也就是说,我们提供了一个多项式大小的近似核化方案(PSAKS)。我们在广泛的基准数据集以及来自柏林的两个真实除雪实例上对其进行了实验评估。在连接组件很少的实例上,顶点和所需边的数量减少到大约 $50\,\%$,解决方案质量损失为 $1\,\%$。我们还朝着参数 $c$ 的 PSAKS 迈出了第一步。在连接组件很少的实例上,顶点和所需边的数量减少到大约 $50\,\%$,解决方案质量损失为 $1\,\%$。我们还朝着参数 $c$ 的 PSAKS 迈出了第一步。在连接组件很少的实例上,顶点和所需边的数量减少到大约 $50\,\%$,解决方案质量损失为 $1\,\%$。我们还朝着参数 $c$ 的 PSAKS 迈出了第一步。
更新日期:2020-10-06
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