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A Mixed-Integer Conic Program for the Multi-Agent Moving-Target Traveling Salesman Problem

Allen George Philip, Zhongqiang Ren, Sivakumar Rathinam, Howie Choset

TL;DR

This work tackles the Multi-Agent Moving-Target Traveling Salesman Problem by formulating a new Mixed-Integer Conic Program that handles targets moving along piecewise-linear trajectories with multiple time windows. The authors first recast the existing baseline MICP as a nonconvex MINLP and then derive a compact, single-set edge formulation that encodes multiple tours via an integer flow, avoiding per-agent edge variables. They prove the equivalence of the new MICP to the nonconvex formulation and demonstrate, through extensive experiments, that the proposed approach achieves up to two orders of magnitude faster runtimes and substantial reductions in optimality gap compared with the state-of-the-art baseline. The results indicate strong scalability and practical impact for exact MA-MT-TSP solutions, with future work extending to multiple depots and heterogeneous agents.

Abstract

The Moving-Target Traveling Salesman Problem (MT-TSP) seeks a shortest path for an agent that starts at a stationary depot, visits a set of moving targets exactly once, each within one of their respective time windows, and returns to the depot. In this paper, we introduce a new Mixed-Integer Conic Program (MICP) formulation for the Multi-Agent Moving-Target Traveling Salesman Problem (MA-MT-TSP), a generalization of the MT-TSP involving multiple agents. Our approach begins by restating the current state-of-the-art MICP formulation for MA-MT-TSP as a Nonconvex Mixed-Integer Nonlinear Program (MINLP), followed by a novel reformulation into a new MICP. We present computational results demonstrating that our formulation outperforms the state-of-the-art, achieving up to two orders of magnitude reduction in runtime, and over 90% improvement in optimality gap.

A Mixed-Integer Conic Program for the Multi-Agent Moving-Target Traveling Salesman Problem

TL;DR

This work tackles the Multi-Agent Moving-Target Traveling Salesman Problem by formulating a new Mixed-Integer Conic Program that handles targets moving along piecewise-linear trajectories with multiple time windows. The authors first recast the existing baseline MICP as a nonconvex MINLP and then derive a compact, single-set edge formulation that encodes multiple tours via an integer flow, avoiding per-agent edge variables. They prove the equivalence of the new MICP to the nonconvex formulation and demonstrate, through extensive experiments, that the proposed approach achieves up to two orders of magnitude faster runtimes and substantial reductions in optimality gap compared with the state-of-the-art baseline. The results indicate strong scalability and practical impact for exact MA-MT-TSP solutions, with future work extending to multiple depots and heterogeneous agents.

Abstract

The Moving-Target Traveling Salesman Problem (MT-TSP) seeks a shortest path for an agent that starts at a stationary depot, visits a set of moving targets exactly once, each within one of their respective time windows, and returns to the depot. In this paper, we introduce a new Mixed-Integer Conic Program (MICP) formulation for the Multi-Agent Moving-Target Traveling Salesman Problem (MA-MT-TSP), a generalization of the MT-TSP involving multiple agents. Our approach begins by restating the current state-of-the-art MICP formulation for MA-MT-TSP as a Nonconvex Mixed-Integer Nonlinear Program (MINLP), followed by a novel reformulation into a new MICP. We present computational results demonstrating that our formulation outperforms the state-of-the-art, achieving up to two orders of magnitude reduction in runtime, and over 90% improvement in optimality gap.
Paper Structure (12 sections, 1 theorem, 6 equations, 3 figures)

This paper contains 12 sections, 1 theorem, 6 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Baseline MICP for MA-MT-TSP
  4. A New MICP for the MA-MT-TSP
  5. Nonconvex MINLP Formulation for MA-MT-TSP
  6. A New MICP Formulation for MA-MT-TSP
  7. Proof of Validity
  8. Numerical Results
  9. Test Settings and Instance Generation
  10. Varying the Total Time Window Duration
  11. Varying the Number of Agents
  12. Conclusion And Future Work

Key Result

Theorem 1

The optimal value of the MICP formulation is equal to that of the nonconvex formulation for the MA-MT-TSP. Additionally, optimal agent tours for the nonconvex formulation can be recovered from an optimal solution to the MICP by following the procedure explained previously.

Figures (3)

  • Figure 1: A feasible solution for an example instance of the MA-MT-TSP. The solid, colored portions of the target trajectories correspond to their time windows. The agents begin and end their tour at the depot. Note how one of the three agents is not assigned any targets in this solution, and simply waits at the depot.
  • Figure 2: Numerical results comparing the % Gap and runtime for MICP-Baseline and MICP, when the number of agents is fixed at 1, and the total time window duration is varied to be 20 (a), 40 (b), and 60 (c) $secs$. The MICP scales significantly better than MICP-Baseline with larger time window durations and more targets. For 10 targets in (c) and 15 targets in (b), the MICP shows a runtime improvement of two orders of magnitude. It also shows a % Gap improvement of more than 15 for 15 targets in (b). Similarly, for 15 targets in (c) and 20 targets in (b) and (c), the MICP runs up to 1000 seconds faster, while also providing a % Gap improvement within a 35-40 range.
  • Figure 3: Numerical results comparing the % Gap and runtime for MICP-Baseline and MICP, when the total time window duration is fixed at 40, and the number of agents is varied to be 2 (a), 3 (b), 4 (c), and 5 (d). The MICP significantly outperforms MICP-Baseline for all number of targets and agents. Note how for 15 targets, MICP runs two orders of magnitude faster, while giving a % Gap improvement of more than 50 for 2 agents to more than 90 for 5 agents. Similarly, for 20 targets, the MICP runs one-order of magnitude faster, while giving a % Gap improvement of 60 for 2 agents to more than 90 for 5 agents.

Theorems & Definitions (2)

  • Theorem 1
  • proof