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The performance of base-line Evolutionary Algorithms (EAs) on combinatorialproblems has been studied rigorously. From the theoretical viewpoint, theliterature extensively investigates the linear problems, while the theoreticalanalysis of the non-linear problems is still far behind. In this paper,variations of the Packing While Travelling (PWT) -- also known as thenon-linear knapsack problem -- are studied as an attempt to analyse thebehaviour of EAs on non-linear problems from theoretical perspective. Weinvestigate PWT for two cities and $n$ items with correlated weights andprofits, using single-objective and multi-objective algorithms. Our resultsshow that RLS\_swap, which differs from the classical RLS by having the abilityto 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 specificselection operator to deal with exponential population size, and prove that itfinds the Pareto front in the same asymptotic expected time. In the case ofuniform 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 ofthe given items. We also perform an experimental analysis to complement ourtheoretical investigations and provide additional insights into the runtimebehavior.

Analysis of Baseline Evolutionary Algorithms for the Packing While Travelling Problem