Solving the Probabilistic Profitable Tour Problem on a Tree

TL;DR

This paper introduces a polynomial-time algorithm for solving the probabilistic profitable tour problem on trees, optimizing expected profit by selecting customer subsets with known service probabilities.

Contribution

It presents the first polynomial-time solution for the probabilistic PTP on trees, extending the understanding of this NP-hard problem in special graph structures.

Findings

The algorithm runs in O(n^2) time for trees.

Optimal subset selection maximizes expected profit.

The approach generalizes deterministic solutions to probabilistic scenarios.

Abstract

The profitable tour problem (PTP) is a well-known NP-hard routing problem searching for a tour visiting a subset of customers while maximizing profit as the difference between total revenue collected and traveling costs. PTP is known to be solvable in polynomial time when special structures of the underlying graph are considered. However, the computational complexity of the corresponding probabilistic generalizations is still an open issue in many cases. In this paper, we analyze the probabilistic PTP where customers are located on a tree and need, with a known probability, for a service provision at a predefined prize. The problem objective is to select a priori a subset of customers with whom to commit the service so to maximize the expected profit. We provide a polynomial time algorithm computing the optimal solution in $O(n^2)$, where $n$ is the number of nodes in the tree.