2,693 research outputs found
A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface
Given a graph cellularly embedded on a surface of genus , a
cut graph is a subgraph of such that cutting along yields a
topological disk. We provide a fixed parameter tractable approximation scheme
for the problem of computing the shortest cut graph, that is, for any
, we show how to compute a approximation of
the shortest cut graph in time .
Our techniques first rely on the computation of a spanner for the problem
using the technique of brick decompositions, to reduce the problem to the case
of bounded tree-width. Then, to solve the bounded tree-width case, we introduce
a variant of the surface-cut decomposition of Ru\'e, Sau and Thilikos, which
may be of independent interest
A PTAS for Bounded-Capacity Vehicle Routing in Planar Graphs
The Capacitated Vehicle Routing problem is to find a minimum-cost set of
tours that collectively cover clients in a graph, such that each tour starts
and ends at a specified depot and is subject to a capacity bound on the number
of clients it can serve. In this paper, we present a polynomial-time
approximation scheme (PTAS) for instances in which the input graph is planar
and the capacity is bounded. Previously, only a quasipolynomial-time
approximation scheme was known for these instances. To obtain this result, we
show how to embed planar graphs into bounded-treewidth graphs while preserving,
in expectation, the client-to-client distances up to a small additive error
proportional to client distances to the depot
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