Extended Team Orienteering Problem (ETOP)

• ETOP is a routing-with-profits framework that extends TOP by incorporating risk, service time, and resource constraints in multi-agent route planning.
• It includes formal variants like the canonical ETOP/TSO, ETOP-ST-MIN, and Dubins-based formulations, each addressing specific operational challenges.
• Solution methods range from exact MIP and cutting-plane techniques to greedy and metaheuristic algorithms, ensuring scalability and high-quality routing in complex scenarios.

The Extended Team Orienteering Problem (ETOP) is a family of routing-with-profits problems that extends the classical Team Orienteering Problem (TOP) through the incorporation of path-level survivability constraints, complex motion or resource budgets, service times, mandatory or incompatible nodes, and multi-agent or vehicle coordination. Originally motivated by applications in hazardous robotics, unmanned aerial vehicle (UAV) tasking, and real-world logistics, ETOP provides a unified framework for modeling routing scenarios where agents must maximize collected reward under nontrivial risk or feasibility constraints.

1. Formal Definitions and Principal Variants

At its core, ETOP generalizes the TOP by supplementing its constraints and objective function to reflect real-world operational uncertainties, additional combinatorial requirements, or problem-specific features.

Canonical ETOP/TSO (Team Surviving Orienteers) Formulation:

Given a graph $G = (V, E)$ with rewards $d_j > 0$ at nodes, $K$ robots, a fixed start node $v_s$, terminal node $v_t$, and survival probabilities $p_e \in (0, 1]$ on edges, the goal is:

• For each robot $k = 1,\dots,K$, choose a $v_s$-$v_t$ path $\rho_k$ such that the product of survival probabilities along $\rho_k$ is at least $p_s$.
• Maximize the total expected reward, $$J(\rho_1,\dots,\rho_K) = \sum_{j \in V} d_j \mathbb{E}[x_j]$$, where node $j$ is considered visited if any robot survives to visit $j$ (Jorgensen et al., 2016).

ETOP-ST-MIN Variant:

Incorporates service times $s_k$, mandatory customer sets $M$, physical incompatibilities $I$ (forbidden arcs), and logical incompatibilities $C$ (node pairs that cannot be visited by the same route). The model admits multiple mathematical formulations, including mixed route-based and compact flow-based models with additional variables to enforce service logic (Guastalla et al., 2024).

Dubins-based ETOP:

For fixed-wing UAVs (with minimum turn radius—Dubins constraint), each node is replaced by multiple "heading-nodes"; paths are feasible only if the total Dubins path length does not exceed a per-vehicle limit $L_{max}$. The problem then seeks up to $m$ such Dubins-feasible routes to cover as many targets as possible (Sundar et al., 2019).

Further variants:

Other instantiations allow for resource/time budgets, heterogeneous agent capabilities, revisitation costs or rewards (informative path planning), and online or uncertain environments. Each model retains the profit-maximizing, resource-constrained, multi-route selection structure but extends feasibility and objective logic.

2. Complexity and Mathematical Modeling

Computational Complexity:

All ETOP variants are at least NP-hard. For the canonical TSO, when $p_s = 1$ the model collapses to TOP, already NP-hard. The addition of service times and (mandatory, incompatible) node constraints in ETOP-ST-MIN strengthens the infeasibility threshold: even verifying the existence of a feasible solution is NP-complete, via reduction from Hamiltonian Path (Guastalla et al., 2024).

Mixed-Integer Programming (MIP) Schemas:

Key modeling ingredients, illustrated here for ETOP-ST-MIN:

Variable	Type	Role
$x_{ij}^r$	Binary	Route $r$ uses arc $(i, j)$
$y_k^r$	Binary	Route $r$ visits customer $k$
$z_{ij}^r$	Cont.	Time flow for arrival/service along $(i,j)$

Coupled by constraints enforcing route start/end, flow conservation, at-most-once visitation, mandatory coverage, arc/node incompatibility, and duration budgets. Logical incompatibilities (e.g., "no pair $(i, j)$ in $C$ visited together") are modeled by auxiliary index variables or combinatorial linearizations. Compact formulations aggregate variables across all vehicles, adding additional structural constraints (Guastalla et al., 2024).

Dubins-based ETOP formulations embed heading-discretization and a combinatorial "set-packing" approach, with arc costs arising from precomputed Dubins path lengths and resource-constrained path enumeration (Sundar et al., 2019).

3. Exact and Approximate Algorithms

Cutting-Plane and Branch-and-Price:

• The CPA for ETOP-ST-MIN operates on the LP relaxation of the compact model, dynamically introducing five classes of valid inequalities: route, set, sub-path, subtour elimination, and logic-based cuts. This approach results in substantial reductions in node counts and CPU time over plain commercial solvers (Guastalla et al., 2024).
• For Dubins-based ETOP, branch-and-price leverages column generation to manage the vast space of feasible Dubins tours, with a resource-constrained elementary shortest path pricing subproblem solved via label-setting DP and Decremental State-Space Relaxation. Specialized branching on targets and connections restores integrality (Sundar et al., 2019).

Greedy and Submodular Approximation:

• The TSO admits a constant-factor greedy algorithm utilizing submodularity: at each step, paths are selected to greedily maximize the marginal expected reward, with feasibility enforced via log-domain linearization of the survival constraint. If an orienteering subproblem oracle attains 1/$\lambda$ of the optimum, the overall procedure achieves $1-e^{-p_s/\lambda}$ approximation (Jorgensen et al., 2016).
• In practice, the greedy approach attains $>98\%$ of the best bound in simulations and scales polynomially in graph size and linearly in team size.

4. Heuristic and Metaheuristic Methods

Metaheuristic Benchmarks:

For multi-UAV ETOP, three prominent metaheuristics—Genetic Algorithm (GA), Ant Colony Optimization (ACO), and Particle Swarm Optimization (PSO)—are evaluated:

Algorithm	Strengths	Performance trends
GA	Fast, robust	Fastest, slightly lower reward on large
ACO	Explorative	Best reward on large, slowest
PSO	Balanced, tunable	Intermediate reward and speed

GA offers best speed–quality tradeoff for time-critical situations, while ACO is preferable for off-line, value-maximizing planning. No algorithm falls more than $5\%$ short of the best on the tested benchmarks (Xiao et al., 2020).

Neighborhood Descent and Matheuristics:

Recent work introduces Variable Neighbourhood Descent (VND) and Cuts Separation Heuristic (CSH), with specialized routines to generate initial feasible solutions (via mandatory node insertion), exploit both profit- and cost-based neighborhoods, and efficiently move between feasible and (controlled) infeasible regions with tabu mechanisms. CSH further embeds route set-packing over pools of locally generated routes, bridging local and global search (Guastalla et al., 8 Jul 2025).

Parallel implementations of both VND and CSH yield linear speedup on multicore hardware up to $8$–$12$ threads.

5. Computational Results and Empirical Insights

Benchmark Datasets and Scaling:

• ETOP and its variants have been evaluated on modified Chao–Golden–Wasil datasets (21–102 nodes), randomized large-scale complete graphs (up to $900$ nodes), and topologically complex instances with up to $66$ Dubins-constrained nodes for UAV routing (Jorgensen et al., 2016, Guastalla et al., 2024, Sundar et al., 2019).
• CPA, VND, and CSH outperform default MIP solvers, with CPA solving up to $99\%$ of small to medium instances to optimality, and both VND and CSH reaching within $1$–$5\%$ of best-known solutions on large or heavily constrained variants (Guastalla et al., 2024, Guastalla et al., 8 Jul 2025).

Empirical Trade-offs:

• CSH achieves best objective values but incurs $1.3\times$ (physical only) to $1.7\times$ (logical + physical) higher runtime than VND. Both outperform state-of-the-art TOP heuristics on extended benchmarks.
• Increasing sophistication of constraints (mandatory/incompatible nodes, Dubins kinematics) increases solution time, but specialized algorithms accommodate scaling with high solution quality (Sundar et al., 2019, Guastalla et al., 8 Jul 2025).
• Discretizing heading in Dubins ETOP enlarges solution space, monotonically improves scores, but increases runtime (Sundar et al., 2019).

6. Applications, Extensions, and Generalizations

ETOP's modeling and solution paradigms extend directly to time windows, vehicle capacity, precedence relations, multi-period planning, and stochastic service/travel times. Specific extensions include:

• Time Windows: Linear inequalities enforce service and traversal within allowed intervals.
• Resource (Capacity) Constraints: Additional flows and knapsack/packing cuts.
• Precedence/Delivery: Order constraints within/between routes.
• Heterogeneous Teams: Per-agent budgets or survival thresholds $p_s^k$ or speed/profit profiles (Jorgensen et al., 2016).
• Real-time Replanning: On-line correction under agent failure or dynamically arriving requests (Jorgensen et al., 2016, Guastalla et al., 8 Jul 2025).

These extensions are realized by augmenting the core MILP or compact models and expanding CPA or cutting-plane logic with new cut structures. A plausible implication is that the flexibility of the CPA/VND/CSH frameworks supports rapid adaptation to industry-specific constraints and operational requirements.

7. Relationship to Classical TOP and Related Problems

When risk constraints are deactivated ($p_s=1$), the TSO/ETOP reduces to the classical TOP, and all associated machinery/cuts reduce to familiar orienteering or vehicle routing forms. The submodular perspective unites ETOP with submodular relaxation-based algorithms for informative path planning and patrolling (Jorgensen et al., 2016). Many ETOP techniques, particularly in separation and rounding, draw from developments in the VRP, RCESPP, and set-packing literatures (Guastalla et al., 2024, Sundar et al., 2019).

The ETOP framework continues to see active adaptation in fields such as UAV fleet routing, healthcare logistics, stochastic patrolling, and resilience-focused robotic operations. It serves as a canonical testbed for algorithmic innovations balancing reward optimality, constraint complexity, and runtime scalability.