[Paper Review] A Cooperative Coordination Solver for Travelling Thief Problems

Majid Namazi, Conrad Sanderson+2 more|arXiv (Cornell University)|Nov 8, 2019

Data Management and Algorithms29 references4 citations

TL;DR

This paper proposes a cooperative coordination solver (CoCo) for the Travelling Thief Problem (TTP) that enhances synergy between the TSP and KP components. By introducing a Profit Guided Coordination Heuristic (PGCH) to co-optimise tour reversals with item collection changes and a Boundary Bit-Flip search to focus on high-impact items, CoCo achieves significantly improved solution quality, setting new best gains on most benchmark instances and outperforming state-of-the-art solvers.

ABSTRACT

The travelling thief problem (TTP) is a representative of multi-component optimisation problems with interacting components. TTP combines the knapsack problem (KP) and the travelling salesman problem (TSP). A thief performs a cyclic tour through a set of cities, and pursuant to a collection plan, collects a subset of items into a rented knapsack with finite capacity. The aim is to maximise profit while minimising renting cost. Existing TTP solvers typically solve the KP and TSP components in an interleaved manner: the solution of one component is kept fixed while the solution of the other component is modified. This suggests low coordination between solving the two components, possibly leading to low quality TTP solutions. The 2-OPT heuristic is often used for solving the TSP component, which reverses a segment in the tour. Within TTP, 2-OPT does not take into account the collection plan, which can result in a lower objective value. This in turn can result in the tour modification to be rejected by a solver. We propose an expanded form of 2-OPT to change the collection plan in coordination with tour modification. Items regarded as less profitable and collected in cities located earlier in the reversed segment are substituted by items that tend to be more profitable and not collected in cities located later in the reversed segment. The collection plan is further changed through a modified form of the hill-climbing bit-flip search, where changes in the collection state are only permitted for boundary items, which are defined as lowest profitable collected items or highest profitable uncollected items. This restriction reduces the time spent on the KP component, allowing more tours to be evaluated by the TSP component within a time budget. The proposed approaches form the basis of a new cooperative coordination solver, which is shown to outperform several state-of-the-art TTP solvers.

Motivation & Objective

• To address the low coordination between TSP and KP components in existing TTP solvers, which often leads to suboptimal solutions.
• To improve the effectiveness of 2-OPT by integrating collection plan adjustments during tour modifications, avoiding rejected moves due to objective value degradation.
• To accelerate KP component search by restricting bit-flip operations to boundary items—lowest profitable collected or highest profitable uncollected—reducing computation time.
• To enable more efficient exploration of the TTP solution space by reallocating time from KP to TSP evaluation within a fixed time budget.
• To develop a holistic solver that jointly optimises tour and collection plan, improving overall TTP performance.

Proposed method

• Proposes a Profit Guided Coordination Heuristic (PGCH) that modifies the collection plan in coordination with 2-OPT segment reversals: less profitable collected items in earlier cities of the reversed segment are replaced with more profitable uncollected items in later cities.
• Extends PGCH to use both the sequence of lowest profitable collected items and highest profitable uncollected items to guide item substitutions, improving solution quality.
• Introduces a Boundary Bit-Flip search that restricts KP updates to only two types of items: the least profitable collected items and the most profitable uncollected items.
• Applies the Chained Lin-Kernighan heuristic for initial TSP tour generation to ensure diversity in search space exploration.
• Implements a cooperative coordination framework where TSP and KP components are iteratively improved in tandem, with coordinated state changes.
• Employs a time-budgeted iterative refinement process, prioritising TSP evaluation by reducing KP computation time through targeted search.

Experimental results

Research questions

• RQ1 Can coordinated modification of the TSP tour and KP collection plan during 2-OPT segment reversals lead to higher-quality TTP solutions?
• RQ2 Does restricting KP bit-flip operations to boundary items improve search efficiency and solution quality in TTP?
• RQ3 Can a cooperative coordination approach outperform existing interleaved solvers in terms of best gain values on standard TTP benchmarks?
• RQ4 How does the integration of profitability-aware item substitution in tour changes affect the acceptance rate of 2-OPT moves?
• RQ5 To what extent does the proposed method reduce wasted computation on unproductive KP updates?

Key findings

• The proposed CoCo solver sets new best gain values for the majority of benchmark TTP instances, demonstrating superior performance.
• PGCH significantly increases the acceptance rate of 2-OPT moves by co-optimising the collection plan, enabling longer and more effective segment reversals.
• The Boundary Bit-Flip search reduces KP computation time, allowing more TSP tours to be evaluated within the same time budget.
• CoCo outperforms several state-of-the-art solvers, including MATLS, S5, CS2SA*, and MEA2P, on most TTP instances.
• For small instances with limited initial tour diversity, MEA2P still performs better, indicating a need for improved initialisation in CoCo.
• The integration of profitability-guided coordination between TSP and KP components proves effective in navigating the complex interdependency of the TTP.