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Solving the Probabilistic Profitable Tour Problem on a Tree

Enrico Angelelli, Renata Mansini, Romeo Rizzi

TL;DR

The paper investigates the Probabilistic Profitable Tour Problem on a tree (PPTP-T), where a priori a subset of customers is selected to guarantee service and daily profits depend on probabilistic requests. It introduces a tree-structured dynamic programming framework based on the sub-tree characteristic function $f^{({\\mathcal{A};I})}(x)$ and a matryoshka representation of optimal sets, enabling efficient combination of sub-tree solutions. A key technical contribution is showing that $f^{({\\mathcal{A};I})}(x)$ is monotone, convex, and piecewise linear with a nested optimal-set structure, which allows computing the global solution in $O(n^2)$ time. The resulting PPTP-Tree algorithm builds and merges sub-tree descriptions through Merge and SolveTree steps to produce the optimal a priori set and the corresponding expected profit, with broad potential applications in logistics and networked service delivery under uncertainty.

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.

Solving the Probabilistic Profitable Tour Problem on a Tree

TL;DR

The paper investigates the Probabilistic Profitable Tour Problem on a tree (PPTP-T), where a priori a subset of customers is selected to guarantee service and daily profits depend on probabilistic requests. It introduces a tree-structured dynamic programming framework based on the sub-tree characteristic function and a matryoshka representation of optimal sets, enabling efficient combination of sub-tree solutions. A key technical contribution is showing that is monotone, convex, and piecewise linear with a nested optimal-set structure, which allows computing the global solution in time. The resulting PPTP-Tree algorithm builds and merges sub-tree descriptions through Merge and SolveTree steps to produce the optimal a priori set and the corresponding expected profit, with broad potential applications in logistics and networked service delivery under uncertainty.

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 , where is the number of nodes in the tree.
Paper Structure (19 sections, 12 theorems, 63 equations, 1 table, 3 algorithms)

This paper contains 19 sections, 12 theorems, 63 equations, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notation and problem definition
  3. Notation
  4. Problem definition
  5. Revenue
  6. Discounted cost
  7. Profit
  8. Properties
  9. Properties of the sub-trees characteristic function
  10. Maximal and minimal optimal sets.
  11. Matryoshka property.
  12. Computing $f^{({\mathcal{A}})}({x})$
  13. Characteristic function representation
  14. The basic case ($\mathcal{A}$ is a leaf)
  15. The general case
  16. ...and 4 more sections

Key Result

Proposition 1

Let $\mathcal{A}$ and $\mathcal{B}$ be two trees such that $S\subseteq\mathbf{A}$ and $S\subseteq\mathbf{B}$ and $x\leq\min(d_{\textbf{a}},d_{\textbf{b}})$. Then, for each outcome $\omega$, it holds that

Theorems & Definitions (14)

  • Proposition 1
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • Proposition 9
  • Definition 10
  • Definition 11
  • ...and 4 more