Quadratic Traveling Salesperson Problem

• QTSP is a combinatorial optimization problem that extends the classic TSP by incorporating quadratic cost terms from ordered triples of consecutive nodes.
• Four modeling approaches—MIQP, MILP, CP, and DIDP—offer different trade-offs in scalability, memory usage, and optimality proofs.
• Empirical studies indicate MILP performs best on small instances while DIDP remains robust for large-scale problems, guiding effective model selection.

The Quadratic Traveling Salesperson Problem (QTSP) is a generalization of the classical Traveling Salesperson Problem (TSP) in which the travel cost is determined not only by pairs of consecutive sites but by all ordered triples of consecutive sites along the tour. This triple-dependent cost structure introduces a quadratic term in the objective and reflects more complex operational or geometric constraints, such as those that penalize sharp turns (AngleTSP) or combined distance-angle terms (AngleDistanceTSP). Recent research provides a systematic comparison of four compact optimization models for QTSP: Mixed-Integer Quadratic Programming (MIQP), Mixed-Integer Linear Programming (MILP), Constraint Programming (CP), and Domain-Independent Dynamic Programming (DIDP). Theoretical formulations, empirical scalability, and solution quality of these models distinguish them in the landscape of combinatorial optimization (Chen et al., 2024).

1. Formal Definition of the QTSP

Let $N = \{0, 1, \dots, n-1\}$ denote the set of customers (nodes), with tours rooted at customer $0$. A tour is a permutation $\sigma : \{0,\dots,n-1\} \to N$ such that $\sigma(0)=0$, visits all customers exactly once, and cyclically returns to $0$. By extending $\sigma$ cyclically, i.e., $\sigma(-1)=\sigma(n-1)$ and $\sigma(n)=\sigma(0)$, the total tour cost is defined as: $$\sum_{t=0}^{n-1} c_{\sigma(t-1),\,\sigma(t),\,\sigma(t+1)}$$ where $c_{ijk}$ are cost coefficients for each ordered triple of consecutive nodes $0$0. The QTSP seeks a tour minimizing this cost.

2. Optimization Models for QTSP

Four optimization paradigms are proposed, each modeling the triple-interaction with different formulations and variable encodings.

2.1 MIQP Formulation

• Variables: $0$1 (edge $0$2 selected), $0$3 (MTZ indexing for subtour elimination).
• Objective: Quadratic, with bilinear terms $0$4 activating $0$5 if $0$6 are consecutive:

$0$7

• Constraints: Degree constraints ensuring valid tours, and subtour-elimination via MTZ formulation.

2.2 MILP Formulation

• Variables: $0$8 as above, plus $0$9 indicating whether $\sigma : \{0,\dots,n-1\} \to N$0 form a consecutive triple.
• Objective: Linearized, since triple costs act directly on $\sigma : \{0,\dots,n-1\} \to N$1:

$\sigma : \{0,\dots,n-1\} \to N$2

• Constraints: Degree and MTZ constraints (as in MIQP), plus linking constraints between $\sigma : \{0,\dots,n-1\} \to N$3 and $\sigma : \{0,\dots,n-1\} \to N$4 to model consecutive triples.

2.3 CP Formulation

• Variables: $\sigma : \{0,\dots,n-1\} \to N$5 give the tour order (integers in $\sigma : \{0,\dots,n-1\} \to N$6).
• Constraints: All-different constraint over the tour sequence, symmetry breaking (fix $\sigma : \{0,\dots,n-1\} \to N$7), with the objective directly accumulating $\sigma : \{0,\dots,n-1\} \to N$8 over the tour.
• Objective:

$\sigma : \{0,\dots,n-1\} \to N$9

2.4 DIDP Formulation

• State: $\sigma(0)=0$0, with $\sigma(0)=0$1 unvisited, $\sigma(0)=0$2 predecessor, $\sigma(0)=0$3 current node, $\sigma(0)=0$4 first after $\sigma(0)=0$5.
• Value Function: $\sigma(0)=0$6 gives the min-cost from this state, with recursive transitions modeling next-tour-step expansions and direct handling of triple costs.
• Initialization: $\sigma(0)=0$7 solves the entire QTSP.
• Complexity: State-space size is $\sigma(0)=0$8 in the worst case.

3. Computational Study and Benchmarks

Empirical evaluation employed benchmark instances from Staněk et al. (2019) with $\sigma(0)=0$9, generating 400 instances across two QTSP types (AngleTSP, AngleDistanceTSP). Test settings imposed 1800s and 8 GB RAM per instance on single-core hardware.

• Solvers: MIQP/MILP (Gurobi 11.0.3), CP (IBM CP-Optimizer 22.1.1), DIDP (DIDPPy 0.8.0).
• Metrics: Memory-out count, optimality gap, primal gap, primal integral (gap vs. time).

Performance Table

Model	Feasibility (large $0$0)	Optimality (small $0$1)
MILP	Memory-out $0$2100	Best optimality proofs ($0$3)
MIQP	Memory-out $0$4100	Fewer optima proved
CP	Not viable $0$5	Fewer optima proved
DIDP	No memory-out (except tiny cases)	Slightly fewer optima ($0$6), best large-scale

• MILP is preferred for $0$7.
• For $0$8, only DIDP yields feasible tours and meaningful optimality gaps.
• MILP excels in optimality and quality for $0$9 where memory allows, but for $\sigma$0, only DIDP provides nontrivial bound guarantees.

4. Scalability and Empirical Model Comparison

The scalability profiles differ markedly:

• MILP/MIQP: Strong dual bounds for small $\sigma$1 but exponential memory blowup for $\sigma$2.
• CP: Prone to memory and branching blowup beyond $\sigma$3.
• DIDP: Graceful scaling in memory, with feasibility even as $\sigma$4—dual-bound tightness is the main bottleneck to optimality proof.

For large-scale problems, DIDP not only avoids memory exhaustion but also produces the best feasible tours and the lowest primal integrals, indicating rapidly convergent high-quality solutions (Chen et al., 2024).

5. Recommendations for Model Selection

The computational study leads to clear model selection guidance:

• Use MILP for instances where $\sigma$5, or up to $\sigma$6 if memory is sufficient, to achieve the best optimality proofs and high-quality tours.
• For $\sigma$7, only the domain-independent dynamic programming (DIDP) approach is practical and effective.
• A plausible implication is that hybrid schemes utilizing DIDP for lower bounds or warm-starting MILP may further enhance performance, but these extensions remain prospective.

6. Open Challenges and Directions

Advancing the scalability and optimality of QTSP solvers centers on:

• Strengthening DIDP dual bounds, including problem-specific relaxations, to improve proofs of optimality on large instances.
• Designing hybrid frameworks to combine the optimality proof capabilities of MILP with the feasibility and memory scaling of DIDP.
• Integrating cutting-plane techniques into MILP to defer memory-out at higher $\sigma$8.
• Extending formulations to accommodate more complex cost structures, such as higher-order arc costs or additional routing constraints (e.g., time-windows).

Chen et al. provide the first systematic, off-the-shelf comparative analysis of compact QTSP solvers and demonstrate that domain-independent DP models can, under suitable algorithmic enhancements, match or surpass traditional MILP/MIQP/CP techniques for large-scale quadratic routing problems (Chen et al., 2024).