Leveraging Symbolic Regression for Heuristic Design in the Traveling Thief Problem

TL;DR

This work tackles the Traveling Thief Problem (TTP), an NP-hard fusion of TSP and KP, by learning packing heuristics through Symbolic Regression (SR) to initialize fast, interpretable metaheuristic GAs. It characterizes near-optimal packing plans via a nonlinear boundary between packed and unpacked items in terms of standardized $IPR$ and distance-to-end $rDist$, derives multiple SR-derived feature sets, and uses SR to predict GA genotype parameters, enabling a family of effective packing initializations. Extensive experiments on a large TTP benchmark set show that SR-guided initializations, particularly the 6T variant, achieve higher objective values with competitive or lower objective-value evaluations compared to state-of-the-art baselines. The approach improves interpretability and speed of packing initialization while providing a path toward generalization across knapsack types and instance distributions, with potential to integrate multiple SR-derived heuristics in practical solvers.

Abstract

The Traveling Thief Problem is an NP-hard combination of the well known traveling salesman and knapsack packing problems. In this paper, we use symbolic regression to learn useful features of near-optimal packing plans, which we then use to design efficient metaheuristic genetic algorithms for the traveling thief algorithm. By using symbolic regression again to initialize the metaheuristic GA with near-optimal individuals, we are able to design a fast, interpretable, and effective packing initialization scheme. Comparisons against previous initialization schemes validates our algorithm design.

Figures (9)

• Figure 1: Plot of the packed and unpacked items in near-optimal tours generated by the (1+1) EA against the normalized item profitability ratio and distance to the end of the tour. A smooth nonlinear boundary can be seen separating packed items from unpacked items. While the specific boundary varies from instance to instance, it keeps the same overall curved shape.
• Figure 2: Plot of the optimal parameter values found by our metaheuristic GA in each training instance against the capacity factor, along with a line of best fit. Parameter values were normalized such that $w_0^2+w_1^2+w_2^2=1$.
• Figure 3: Ranking results for all heuristics on all 450 instances. The 6T heuristic is has the best objective most frequently, but the 4T algorithm generally requires fewer objective value computations.
• Figure 4: Ranking results for all heuristics on all 450 instances. 6T is often the best in terms of objective value, and generally requires fewer objective value computations than packIterative, but is slower than Insertion.
• Figure 5: Plot of the optimal parameter values against the capacity factor for our 2T metaheuristic algorithm. The solution evolved by SR is plotted in black.