The Price of Diversity of the Traveling Salesman Problem

2025-07-17

AI Summary

This work investigates the trade-off between solution diversity and cost in discrete optimization, formally introducing the *Price of Diversity* (PoD)—defined as the ratio between the minimum cost of a diverse solution set satisfying *k* edge-disjoint cycles and the cost of a single optimal solution. Using the Traveling Salesman Problem (TSP) as a model, the study focuses on the *k = 2* case. By integrating combinatorial optimization, graph theory, and metric space analysis, it derives tight asymptotic bounds on PoD: *PoD = 8/5* in one-dimensional metric spaces and *PoD = 2* in general metric spaces. These results are extended to the shortest Hamiltonian path problem. The analysis precisely quantifies how structural diversity constraints—specifically edge-disjoint cycle requirements—affect optimality, thereby establishing a theoretical benchmark and analytical framework for multi-solution optimization under diversity constraints.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

This paper introduces the concept of the "Price of Diversity" (PoD) in discrete optimization problems, quantifying the trade-off between solution diversity and cost. For a minimization problem, the PoD is defined as the worst-case ratio, over all instances, of the minimum achievable cost of a diverse set of $k$ solutions to the cost of a single optimal solution for the same instance. Here, the cost of a $k$-solution set is determined by the most expensive solution within the set. Focusing on the Traveling Salesman Problem (TSP) as a key example, we study the PoD in the setting where $k$ edge-disjoint tours are required. We establish that, asymptotically, the PoD of finding two edge-disjoint tours is $frac{8}{5}$ in a special one-dimensional case and 2 in a general metric space. We obtain these results from analyzing a related fundamental problem: the Shortest Hamiltonian Path problem (SHP), for which we establish similar results.

Problem

• Quantify trade-off between solution diversity and cost in optimization
• Study Price of Diversity for edge-disjoint TSP tours
• Analyze Shortest Hamiltonian Path to derive PoD bounds

Innovation

• Introduces 'Price of Diversity' (PoD) concept
• Analyzes PoD for edge-disjoint TSP tours
• Establishes PoD bounds for SHP problem