Parameterized Rural Postman Problem
Gregory Gutin, Magnus Wahlstrom, Anders Yeo

TL;DR
This paper presents a fixed-parameter tractable randomized algorithm for the Directed Rural Postman Problem, parameterized by the number of connected components, with implications for both directed and undirected cases.
Contribution
It introduces an algebraic, randomized algorithm with runtime $O^*(2^k)$ for DRPP when the total weight is polynomially bounded, resolving a long-standing open question.
Findings
Algorithm runs in $O^*(2^k)$ time for directed case
Algorithm also applies to undirected case
Addresses a 30-year open problem in parameterized complexity
Abstract
The Directed Rural Postman Problem (DRPP) can be formulated as follows: given a strongly connected directed multigraph with nonnegative integral weights on the arcs, a subset of and a nonnegative integer , decide whether has a closed directed walk containing every arc of and of total weight at most . Let be the number of weakly connected components in the the subgraph of induced by . Sorge et al. (2012) ask whether the DRPP is fixed-parameter tractable (FPT) when parameterized by , i.e., whether there is an algorithm of running time where is a function of only and the notation suppresses polynomial factors. Sorge et al. (2012) note that this question is of significant practical relevance and has been open for more than thirty years. Using an algebraic approach, we prove that DRPP has a randomized algorithm of…
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