# Analysis of Baseline Evolutionary Algorithms for the Packing While Travelling Problem

## TL;DR

This paper analyzes the performance of various evolutionary algorithms on the non-linear Packing While Traveling problem, providing theoretical runtime bounds and experimental insights into their efficiency.

## Contribution

It offers the first theoretical analysis of EAs on the non-linear PWT problem, including new algorithms and expected runtime bounds.

## Key findings

- RLS_swap finds optimal solutions in O(n^3) expected time.
- Enhanced GSEMO finds the Pareto front in the same asymptotic time.
- (1+1) EA finds optimal solutions in O(n^2 log(max{n,p_max})) expected time.

## Abstract

The performance of base-line Evolutionary Algorithms (EAs) on combinatorial problems has been studied rigorously. From the theoretical viewpoint, the literature extensively investigates the linear problems, while the theoretical analysis of the non-linear problems is still far behind. In this paper, variations of the Packing While Travelling (PWT) -- also known as the non-linear knapsack problem -- are studied as an attempt to analyse the behaviour of EAs on non-linear problems from theoretical perspective. We investigate PWT for two cities and $n$ items with correlated weights and profits, using single-objective and multi-objective algorithms. Our results show that RLS\_swap, which differs from the classical RLS by having the ability to swap two bits in one iteration, finds the optimal solution in $O(n^3)$ expected time. We also study an enhanced version of GSEMO, which a specific selection operator to deal with exponential population size, and prove that it finds the Pareto front in the same asymptotic expected time. In the case of uniform weights, (1+1)~EA is able to find the optimal solution in expected time $O(n^2\log{(\max\{n,p_{\max}\})})$, where $p_{\max}$ is the largest profit of the given items. We also perform an experimental analysis to complement our theoretical investigations and provide additional insights into the runtime behavior.