# Combinatorics Seminar

When: Sunday, April 6, 10am

Where: Schreiber 309

Speaker: Gregory Z. Gutin (Royal Holloway, University of London)

Title: Planning for Snow Plowing in Berlin: Parameterized Rural Postman Problem

## 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 $\ell$. DRPP is NP-complete. Let $k$ be the number of weakly connected components in the subgraph of $D$ induced by $R$. Sorge et al. (2012) asked 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) noted that this question is of significant practical relevance (in Snow Plowing in Berlin, k is between 3 and 5) and has been open for more than thirty years. Using an algebraic approach, we prove that DRPP has a randomized algorithm with 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. We obtain similar results for problems on constrained matchings in bipartite and general undirected graphs.

Joint work with Magnus Wahlstrom (RHUL) and Anders Yeo (Singapore Univ. Technology and Design)

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