2 research outputs found
A Unified PTAS for Prize Collecting TSP and Steiner Tree Problem in Doubling Metrics
We present a unified (randomized) polynomial-time approximation scheme (PTAS) for the prize collecting traveling salesman problem (PCTSP) and the prize collecting Steiner tree problem (PCSTP) in doubling metrics. Given a metric space and a penalty function on a subset of points known as terminals, a solution is a subgraph on points in the metric space, whose cost is the weight of its edges plus the penalty due to terminals not covered by the subgraph. Under our unified framework, the solution subgraph needs to be Eulerian for PCTSP, while it needs to be a tree for PCSTP. Before our work, even a QPTAS for the problems in doubling metrics is not known.
Our unified PTAS is based on the previous dynamic programming frameworks proposed in [Talwar STOC 2004] and [Bartal, Gottlieb, Krauthgamer STOC 2012]. However, since it is unknown which part of the optimal cost is due to edge lengths and which part is due to penalties of uncovered terminals, we need to develop new techniques to apply previous divide-and-conquer strategies and sparse instance decompositions
Approximation Algorithm for Unrooted Prize-Collecting Forest with Multiple Components and Its Application on Prize-Collecting Sweep Coverage
In this paper, we introduce a polynomial-time 2-approximation algorithm for
the Unrooted Prize-Collecting Forest with Components (URPCF) problem.
URPCF aims to find a forest with exactly connected components while
minimizing both the forest's weight and the penalties incurred by unspanned
vertices. Unlike the rooted version RPCF, where a 2-approximation algorithm
exists, solving the unrooted version by guessing roots leads to exponential
time complexity for non-constant . To address this challenge, we propose a
rootless growing and rootless pruning algorithm. We also apply this algorithm
to improve the approximation ratio for the Prize-Collecting Min-Sensor Sweep
Cover problem (PCMinSSC) from 8 to 5.
Keywords: approximation algorithm, prize-collecting Steiner forest, sweep
cover