Joint Routing-Assignment Optimization

• Joint Routing-Assignment optimization is a complex problem that integrates alternating Hamiltonian cycles with one-to-one item-placeholder assignments to minimize total travel cost.
• It enforces strict alternation constraints by forbidding transitions within the same node set, significantly extending the classical TSP formulation.
• Heuristic methods such as cycle merging, shaking, and partial path reconstruction enable near-optimal solutions for large-scale instances that traditional MIP cannot efficiently solve.

The Joint Routing-Assignment (JRA) optimization problem seeks a minimum-cost Hamiltonian cycle on a bipartite set of items and placeholders, simultaneously determining the assignment of each item to a placeholder and the alternating route that visits every node exactly once. This integrated combinatorial structure arises in robotics, logistics, and paired pickup-delivery systems where both pairing and sequencing must be optimized under strict alternation constraints. Recent algorithmic advances have enabled near-optimal solutions for problem sizes previously intractable to exact mixed-integer programming.

1. Formal Statement and Mathematical Structure

The JRA problem is defined on two disjoint node sets, items $I = \{1,\ldots,n\}$ and placeholders $P = \{n\!+\!1,\ldots,2n\}$, forming the full set $V = I \cup P$. The directed edge set $E = \{(i,j) \mid i,j \in V,\,i\ne j\}$ is weighted by a cost function $c_{ij}$, typically representing Euclidean distance. The decision variable $x_{ij}\in \{0,1\}$ encodes whether edge $(i,j)$ belongs to the cycle. The combinatorial constraints are:

• Alternation & Non-connectable Constraints: All transitions $I\to I$, $P\to P$, or $P\to I$ are forbidden: $x_{ij}=0$ if $$(i,j) \in I\times I \cup P\times P \cup (P\times I)$$.
• Degree Constraints: Each $i\in I$ must have exactly two outgoing ($\sum_{p\in P}x_{ip}=2$), and each $p\in P$ two incoming edges ($\sum_{i\in I}x_{ip}=2$). The alternation requirement thus enforces the tour structure $I \to P \to I \to P$ strictly.
• Subtour Elimination: For every $S\subset V$, $2\le |S|<2n$, $\sum_{i,j\in S}x_{ij} \le |S|-1$ prevents disconnected cycles.
• (Optional) Fixed Start/Goal: $x_{n,2n}=1$ pins the initial/terminal transition.

The objective is:

$\min_{x} \sum_{(i,j)\in E} c_{ij}x_{ij}$

This models a Hamiltonian cycle alternating between assigned pairs, minimizing the total travel cost while ensuring a one-to-one assignment and feasible cyclic order.

2. Classical MIP Formulation and Scalability Limits

The JRA MIP can be regarded as a generalization of TSP with added assignment constraints and enforced alternation. Introducing $a_{ip}\in\{0,1\}$ for explicit item–placeholder assignments, a standard formulation is:

• $x_{ip}\le a_{ip}$, $x_{pi}\le a_{ip}$ for all $i\in I, p\in P$
• $\sum_{p\in P} a_{ip}=1$ for all $i\in I$; $\sum_{i\in I}a_{ip}=1$ for all $p\in P$

Subtour constraints are introduced dynamically during branch-and-cut (via lazy callback) due to their exponential number. The overall model has $O(n^2)$ binaries and $O(n^2)$ explicit constraints but an exponential separation phase.

Benchmark studies (Yuan, 18 Oct 2025) report that for $n\leq200$ this approach is practical, with runtimes (e.g., $n=32$ in 0.36s, $n=100$ in 11.5s, $n=200$ around 108s), but for $n\gtrsim 300$ the number of cuts and solution time become prohibitive, often resulting in timeouts or failure to reach proven optimality. The number of generated subtour cuts grows exponentially with $n$, confirming the inherent combinatorial hardness: joint JRA scales as $O(n! n^3)$, dominated by the assignment plus cycle search complexity.

3. Scalable Heuristics: Cycle Merging and Shaking Methods

To enable high-quality solutions for $n>300$, heuristic strategies have been proposed. Cycle-merging initializes with two-way Hungarian assignment—once for $I\to P$ and once for $P\to I$—to create a union of small cycles, which are greedily merged pairwise to form an initial feasible tour. The “shaking” improvement alternates between Hungarian assignment for fixed tour order and reordering for fixed assignment, converging to a local minimum. This process can be enhanced using simulated annealing over partial sliding windows to escape local minima (Qilong et al., 24 Oct 2025).

Empirical evidence demonstrates:

• Cycle merging alone yields tours within $\sim$1.1% of MIP-opt (Qilong et al., 24 Oct 2025).
• Shaking and annealing halve this gap, achieving $0.68$–$0.87\%$ (n=100–300).
• Solutions for $n=1000$ are computable within 1 minute; MIP fails to deliver.

These heuristics are polynomial-time per iteration ($O(T n^3)$ for $T$ alternations, $O(n^4)$ worst-case for initial merging), robust for large-scale instances, and extendable to multiple item/placeholder types.

4. Partial JRA and Large-α Optimization Framework

The “Partial JRA and Large-α Optimization” approach combines local reoptimization, localized MIP, and edge-retention constraints for further advances in solution accuracy and scalability (Yuan, 7 Nov 2025). This framework comprises several key algorithmic ingredients:

Partial Path Reconstruction (PPR):

• Partition the current tour by selecting a subset $V_S$ of nodes for reoptimization, removing incident edges and decomposing into chains $Q_1,\ldots,Q_m$.
• Construct a reduced JRA subproblem on $V_S$ and boundary nodes, with reserved edges fixed, and solve via Gurobi.
• Reintegrate the optimized local solution into the global tour.

Spatially-Localized PPR (SLPPR):

• Slide a window or circle of radius $r_r$ along the tour, repeatedly applying PPR for local refinements.

Large-α Constraint:

• Enforce a global edge-retention constraint on a high-quality incumbent tour $L_s$: $\sum_{(i,j)\in L_s}x_{ij} \ge 2n(1-\alpha)$ with small $\alpha$ (e.g., 0.05–0.15), forcing the final MIP to only consider solutions within a large but tractable neighborhood of the incumbent (a “κ-opt envelope” significantly larger than classical $k$-opt, yet still solvable).

Pipeline Structure:

1. Two-way assignment and cycle-merge.
2. PPR-based merging at merge-affected nodes.
3. SLPPR (1–2 passes).
4. Final Large-α MIP.

Algorithmic Flowchart

Initial Assignment → Cycle Merging → PPR-Merge → SLPPR Polishing → Large-α MIP

5. Computational Performance and Scaling

Extensive computational results on random Euclidean instances (Yuan, 7 Nov 2025):

Method	n=300	n=500	n=1000	Relative Gap	Runtime
Cycle-merging	+1.12%	+0.98%	—	vs. proven optimal (MIP)	17–53 s
PPR post-merge	+0.69%	+0.63%	—		17–53 s
SLPPR (1 pass)	+0.094%	+0.092%	—		17–53 s
SLPPR (2 passes)	+0.071%	+0.070%	—
Large-α (α=0.15)	+0.063%	+0.054%	—		up to 73 s
n=1000, L_Large - prior heuristic	—	—	-0.62 to -0.93	over prior methods	—

All experiments used a 22-core Intel Core Ultra 7 155H. For $n\geq 1000$, the proposed approach delivers feasible, nearly optimal solutions while exact MIP fails to converge.

6. Theoretical and Algorithmic Insights

The PPR/SLPPR pipeline leverages the locality of subproblem updates and the strong combinatorial structure arising from alternation and assignment constraints. The Large-α constraint quickly contracts the feasible polytope after multiple heuristic or local steps, efficiently capturing large neighborhoods that would correspond to prohibitively many $k$-opt moves if attempted directly. Classical $k$-opt local search is ineffective for JRA, as meaningful improvements often require overturning hundreds of edges—exponentially expensive in $k$. By contrast, the global constraint restricts changes to just $2n\alpha$ edges, allowing solution spaces large enough for refinements, but small enough for tractable MIP search.

7. Generalizations and Applications

The Partial JRA and Large-α framework is noted as broadly applicable (Yuan, 7 Nov 2025):

• Classical TSP: SLPPR and Large-α can be directly adapted by dropping alternation constraints.
• Vehicle Routing with Pickup/Delivery: The approach localizes pickup–delivery reassignments for dynamic routes.
• Robotics, Warehouse Order-Picking: Subproblem reoptimization and edge-retention are well-suited for path planning and order sequencing under dynamic assignment and precedence.
• Hamiltonian Cycle Problems with Local Structure: Any domain where local tour subdomains can be identified for partial reoptimization, and global edge-retention constraints can be meaningfully imposed, stands to benefit.

This modular, pipeline-oriented methodology allows for incremental improvement from fast heuristics to near-optimal global solutions, providing a rigorous and scalable alternative to both classical local search and infeasible exact global MIP approaches for real-world, large-scale JRA instances.