Convergence and Running Time of Time-dependent Ant Colony Algorithms

TL;DR

This paper analyzes the convergence and running time of time-dependent ant colony algorithms, proving convergence to optimal solutions and establishing polynomial or super-polynomial bounds for specific variants on shortest path problems.

Contribution

It introduces a general framework for time-dependent ACO algorithms, proves convergence properties, and compares two variants' running times on the SDSP.

Findings

GBAS/tdev converges to optimal solutions with probability 1.

n-ANT/tdlb achieves polynomial time bounds on SDSP.

n-ANT/tdev has a super-polynomial lower bound on SDSP.

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.