C*: A New Bounding Approach for the Moving-Target Traveling Salesman Problem

TL;DR

We tackle the Moving-Target Traveling Salesman Problem ($MT\text{-}TSP$) where targets move on known piecewise-linear or Dubins trajectories with time-windows, and seek to minimize the agent’s travel time. The core contribution is Continuity* ($C^*$), a bounding framework that relaxes trajectory continuity by partitioning target paths into trajectory-intervals, building a graph whose nodes are intervals, and assigning edge costs via Shortest Feasible Travel ($SFT$); solving a Generalized Traveling Salesman Problem (GTSP) on this graph yields valid lower-bounds on the MT-TSP, with convergence as discretization becomes finer. Four $C^*$ variants—$C^*$-Lite, $C^*$-Geometric, $C^*$-Sampling, and $C^*$-Linear—trade off bound tightness and computation, with $C^*$-Linear achieving the strongest bounds by solving $SFT$ exactly for piecewise-linear trajectories. Numerical results on up to 15 targets show that feasible tours are typically within about $4.5\%$ of the lower-bounds on average, and that $C^*$ can outperform the SOCP baseline in larger instances while remaining substantially faster. These results establish tight, scalable lower-bounds for MT-TSP and motivate future branch-and-cut and approximation schemes leveraging the problem’s trajectory-continuity structure.

Abstract

We introduce a new bounding approach called Continuity* C*, which provides optimality guarantees for the Moving-Target Traveling Salesman Problem (MT-TSP). Our approach relaxes the continuity constraints on the agent's tour by partitioning the targets' trajectories into smaller segments. This allows the agent to arrive at any point within a segment and depart from any point in the same segment when visiting each target. This formulation enables us to pose the bounding problem as a Generalized Traveling Salesman Problem (GTSP) on a graph, where the cost of traveling along an edge requires solving a new problem called the Shortest Feasible Travel (SFT). We present various methods for computing bounds for the SFT problem, leading to several variants of C*. We first prove that the proposed algorithms provide valid lower-bounds for the MT-TSP. Additionally, we provide computational results to validate the performance of all C* variants on instances with up to 15 targets. For the special case where targets move along straight lines, we compare our C* variants with a mixed-integer Second Order Conic Program (SOCP) based method, the current state-of-the-art solver for the MT-TSP. While the SOCP-based method performs well on instances with 5 and 10 targets, C* outperforms it on instances with 15 targets. For the general case, on average, our approaches find feasible solutions within approximately 4.5% of the lower-bounds for the tested instances.

Figures (12)

• Figure 1: A feasible solution for an instance of the MT-TSP with four targets. The blue lines shows the path of the vehicle. Also, the colored solid segments for each target indicates the part of its trajectory corresponding to its time-windows when the vehicle can visit the target.
• Figure 2: Illustration of the key steps in the C$^*$ Algorithm.
• Figure 3: Given target trajectories for AlgoSFT. In the trajectory of target $i$, there are 3 line-segments and 2 corner points. In the trajectory of target $j$, there are 4 line-segments and 3 corner points. Also, $T_i:=(\underline{t}_i,t_i^1,t_i^2,\overline{t}_i)$ and $T_j:=(\underline{t}_j,t_j^1,t_j^2,t_j^3,\overline{t}_j)$.
• Figure 4: Target trajectories from Fig. \ref{['fig:AlgSFTinput']} showing the updated lists of times in $T_i$ and $T_j$. At the end of the step 1 of AlgoSFT, $|T_i|=|T_j|=7$.
• Figure 5: Lower-bounds from the C$^*$ variants as compared to the feasible costs, for the simple instances (left), and the complex instances (right). The SOCP costs are also included for the simple instances. In (c), square markers indicate the outlier cases for instances 1, and 5.

Theorems & Definitions (14)

• Theorem 1
• proof
• Remark 1
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5