Cutting-Plane Subtour Elimination Techniques

• Cutting-plane subtour elimination is a method that enforces global connectivity by dynamically adding constraints to eliminate disconnected subtours in routing and assignment problems.
• It employs a lazy cutting-plane approach that integrates efficient search routines, such as DFS, to identify and remove infeasible cycles during the branch-and-bound process.
• Empirical evaluations show that this technique significantly reduces computation times and optimality gaps in MIP solvers for TSP, JRA, and other routing applications.

Cutting-plane subtour elimination refers to a class of algorithmic strategies and constraints for enforcing connectivity in combinatorial optimization problems, particularly in mixed-integer programming (MIP) formulations of routing-type problems such as the Joint Routing–Assignment (JRA) problem and the Traveling Salesman Problem (TSP). These methods exploit the separation between feasible tours (Hamiltonian cycles or paths) and infeasible solutions containing disconnected subtours by dynamically detecting and excluding invalid cycles during the solution process. In recent approaches to assignment-routing and constrained packaging scenarios, cutting-plane subtour elimination is performed as a key, on-demand component within high-performance solvers, enabling rapid and scalable enforcement of global connectivity even under various additional constraints (Qilong et al., 21 Dec 2025, Yuan, 7 Nov 2025).

1. Subtour Elimination in Assignment-Routing Problems

In combinatorial routing problems, a subtour is a cycle covering a strict subset $S$ of the nodes, disconnected from the rest of the solution. Valid solutions must comprise a single tour or path covering all nodes. In MIP formulations for joint assignment-routing problems, such as the JRA, binary variables $x_{ij}$ indicate whether arc $(i \to j)$ is used in the tour, and the constraints must ensure that $x$ encodes a single connected Hamiltonian cycle or path across the union of items and placeholders.

The canonical subtour elimination constraint is: $$\sum_{i \in S} \sum_{j \in S\setminus\{i\}} x_{ij} \leq |S| - 1 \quad \forall~ S \subset V,\ |S| \geq 2,$$ where $V$ is the set of all routing nodes (items and placeholders) (Qilong et al., 21 Dec 2025). There are exponentially many such subsets, making explicit enumeration infeasible in practical solvers for nontrivial problems.

2. Dynamic Cut Generation: Lazy Cutting-Plane Approach

Rather than statically including all subtour elimination constraints, modern solvers employ a cutting-plane strategy: subtour constraints are added incrementally, only as needed, using lazy constraint callbacks. At each integer-feasible solution $\bar{x}$ produced during branch and bound, a fast graph search procedure (e.g., DFS) identifies disconnected cycles $S$. If a subtour is detected, the corresponding cut is dynamically injected into the model to forbid such a solution in subsequent searches: $\sum_{i, j \in S} x_{ij} \leq |S| - 1.$ This on-demand separation sharply limits the number of active constraints, yielding a substantially more scalable and memory-efficient formulation (Qilong et al., 21 Dec 2025, Yuan, 7 Nov 2025).

The subtour elimination routine is often combined with other strategies, such as the MTZ (Miller–Tucker–Zemlin) constraint system, which can ensure tour connectivity in certain contexts and may obviate cuts in restricted subproblems (e.g., over auxiliary item-graphs with temporal precedence) (Qilong et al., 21 Dec 2025).

3. Integration with Mixed-Integer Programming Solvers

State-of-the-art implementations utilize advanced MIP solvers such as Gurobi, leveraging features including:

• LazyConstraint callbacks for real-time cut generation.
• Rapid graph connectivity checks during callback execution.
• Model parameterization (e.g., MIPFocus=2, Presolve=1, Cuts=2, Heuristics=0.05) for optimal performance in large, constrained domains (Qilong et al., 21 Dec 2025).
• Handling degree and type constraints (no same-type arcs, assignment coupling) which interact with subtour elimination for specialized assignment-routing cycles (Yuan, 7 Nov 2025).

A summary of solver configuration details as reported:

Parameter	Typical Setting	Function
MIPFocus	2	Emphasize proving optimality
Presolve	1	Aggressive presolving
Cuts	2	Solver-generated cuts on
Heuristics	0.05	Light primal heuristics
Threads	8–22	Maximize parallelism

Disabling solution polishing and focusing on efficient cut callbacks is reported to keep pure MIP iterations rapid and the branch-and-bound tree manageable. Symmetry-breaking constraints are also integrated to reduce combinatorially equivalent tours where appropriate (Qilong et al., 21 Dec 2025).

4. Empirical Performance and Comparisons

Empirical evaluation demonstrates the marked impact of cutting-plane subtour elimination:

• For moderately sized instances ($n = 16$), mean MIP solve times are under $4$ seconds; for $n = 20$, under $10$ seconds (Qilong et al., 21 Dec 2025).
• All 46 mobile-manipulation instances tested yielded exact global optima—matching the best-known solutions of alternative methods down to $10^{-4}$—while computation times were reduced by over an order of magnitude versus shaking-based exact solvers (Qilong et al., 21 Dec 2025).
• In larger-scale settings ($n = 300, 500$), the cutting-plane approach facilitated provable $0.00\%$ optimality gaps with high efficiency (e.g., $76$–$178$ seconds for full MIP solves) (Yuan, 7 Nov 2025).
• Greedy and heuristic approaches achieved faster but suboptimal results, with tour lengths deviating $8$–$16\%$; cutting-plane MIPs outperformed these in solution quality and stability, reinforcing the necessity of global subtour elimination (Qilong et al., 21 Dec 2025).

5. Extensions, Frameworks, and Related Methodologies

Cutting-plane subtour elimination is adaptable to domain-specific constraints, such as:

• Time-frame or staged processing, using auxiliary variables and MTZ-type subtour elimination on item-graphs supporting temporal precedence (Qilong et al., 21 Dec 2025).
• Multi-class and extra-placeholder scenarios, adjusting assignment, degree, and selection constraints as required, with subtour cuts still enforced over the expanded variable set.
• Heuristic-accelerated frameworks (such as Partial Path Reconstruction and iterated local cut addition), which further exploit cutting-plane subtour elimination within restricted subproblems to polish or refine tours for large-scale settings (Yuan, 7 Nov 2025).

The cutting-plane paradigm generalizes beyond JRA, underpinning robust solvers for TSP, vehicle routing, and related combinatorial designs where maintaining a global tour or path is essential.

6. Significance and Practical Implications

The cutting-plane subtour elimination approach enables the efficient, exact enforcement of connectivity in mixed-integer models for routing-assignment, packaging, and logistical problems with complex constraints. By dynamically injecting only the necessary subtour cuts, models remain tractable for realistic problem sizes experienced in robotic manipulation and logistics planning. This methodology is indispensable in domains requiring tight tolerance and global optimality guarantees, especially where the assignment or multi-stage nature of the problem poses additional combinatorial complexity (Qilong et al., 21 Dec 2025, Yuan, 7 Nov 2025).

A plausible implication is that continual advances in solver integration, separation heuristics, and domain-specific cut management will further expand the scale and scope of problems solvable via cutting-plane subtour elimination strategies.