Parameterized Rural Postman Problem

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 $D=(V,A)$ with nonnegative integral weights on the arcs, a subset $R$ of $A$ and a nonnegative integer $\ell$, decide whether $D$ has a closed directed walk containing every arc of $R$ and of total weight at most $\ell$. Let $k$ be the number of weakly connected components in the the subgraph of $D$ induced by $R$. Sorge et al. (2012) ask whether the DRPP is fixed-parameter tractable (FPT) when parameterized by $k$, i.e., whether there is an algorithm of running time $O^*(f(k))$ where $f$ is a function of $k$ only and the $O^*$ 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 running time $O^*(2^k)$ when $\ell$ is bounded by a polynomial in the number of vertices in $D$. We also show that the same result holds for the undirected version of DRPP, where $D$ is a connected undirected multigraph.