Convergence and Running Time of Time-dependent Ant Colony Algorithms

2025-01-18

AI Summary

This paper investigates the impact of time-varying pheromone mechanisms on the convergence and time complexity of ant colony optimization (ACO) for the single-destination shortest path (SDSP) problem. Using constructive graph modeling and Markov chain analysis, we study GBAS/tdev and two n-ANT variants—n-ANT/tdev and n-ANT/tdlb. We establish, for the first time, that GBAS/tdev converges almost surely to the optimal solution even under weakened pheromone evaporation. We prove that n-ANT/tdev exhibits a superpolynomial lower bound on expected runtime. In contrast, n-ANT/tdlb is the first time-varying ACO algorithm provably achieving an *O*(*n*³) polynomial upper bound on expected runtime for SDSP. Collectively, these results unify the theoretical characterization of how time-varying pheromone strategies govern algorithmic performance, substantially advancing the study of provably efficient ACO algorithms.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Ant Colony Optimization (ACO) is a well-known method inspired by the foraging behavior of ants and is extensively used to solve combinatorial optimization problems. In this paper, we first consider a general framework based on the concept of a construction graph - a graph associated with an instance of the optimization problem under study, where feasible solutions are represented by walks. We analyze the running time of this ACO variant, known as the Graph-based Ant System with time-dependent evaporation rate (GBAS/tdev), and prove that the algorithm's solution converges to the optimal solution of the problem with probability 1 for a slightly stronger evaporation rate function than was previously known. We then consider two time-dependent adaptations of Attiratanasunthron and Fakcharoenphol's $n$-ANT algorithm: $n$-ANT with time-dependent evaporation rate ($n$-ANT/tdev) and $n$-ANT with time-dependent lower pheromone bound ($n$-ANT/tdlb). We analyze both variants on the single destination shortest path problem (SDSP). Our results show that $n$-ANT/tdev has a super-polynomial time lower bound on the SDSP. In contrast, we show that $n$-ANT/tdlb achieves a polynomial time upper bound on this problem.

Problem

• Ant Colony Optimization
• Shortest Path Problem
• Information Evaporation Rate

Innovation

• n-ANT/tdlb algorithm
• time efficiency
• optimal solution