Feasible Vehicle-to-Request Assignments

• Feasible vehicle-to-request assignments are mappings that satisfy operational constraints like capacity, time windows, and sequencing, ensuring viable service in mobility systems.
• They employ algorithmic techniques such as LP relaxations, dynamic trees, and shareability hypergraphs to efficiently prune infeasible assignments in real time.
• Optimization strategies in these frameworks balance cost, fairness, and service rate while scaling to large dynamic networks and ensuring prompt decision-making.

A feasible vehicle-to-request assignment is a mapping of vehicles to travel requests that satisfies all operational and quality-of-service constraints imposed by the system or application context. This fundamental construct underpins dynamic ridesharing, mobility-on-demand, and general vehicle routing systems. Assigning vehicles to requests in a manner that ensures feasibility—i.e., no violation of capacity, time-window, or sequencing constraints—is computationally challenging, especially under real-time and large-scale conditions. The technical landscape has evolved from single-ride bipartite matchings to batch trip-vehicle assignment, combinatorial shareability hypergraphs, and sophisticated optimization or heuristic frameworks, addressing objectives such as cost, service fairness, user delay, and fleet sustainability.

1. Mathematical Formulations of Feasibility

The canonical feasibility structure is represented by a set of vehicles $V$ and a set of requests $R$, with requests having specific parameters (pickup location, desired time, number of passengers, etc.), and vehicles characterized by capacity and current/planned route state. The core requirement is to determine an assignment $x_{rv}$ (or more generally $x_{tv}$ for assigning trips/bundles of requests $t$ to vehicle $v$) such that:

• All decision variables are binary: $x_{rv} \in \{0,1\}$, indicating assignment of request $r$ to vehicle $v$ (Vakayil et al., 2023, Mori et al., 2020).
• Each request is assigned at most once: $\sum_v x_{rv} \le 1$ for all $r \in R$.
• Vehicle capacity is respected: $\sum_r Q_r x_{rv} \leq \bar Q_v$ for all $v \in V$, where $Q_r$ is the number of passengers for $r$ (Vakayil et al., 2023).
• Time-window, pickup-delay, and travel-delay constraints: assignment is pruned if $\max(\text{pickup delay}) > D^P$ or any passenger's route exceeds their $D^T$ (Vakayil et al., 2023).
• Sequencing/Precedence: pickup must precede drop-off for any request (Yao et al., 2020).

Assignments are further constrained via strong feasibility pruning, eliminating edges (or trip-vehicle pairs) that would result in infeasible service due to a violation of any of the above.

Integer programs or mixed-integer linear programs are the standard exact representation, with the feasible set defined by linear constraints and, potentially, subtour elimination constraints for multi-target coverage (Kumar et al., 30 Oct 2024, Kumar et al., 2023).

2. Algorithmic Approaches for Feasible Assignments

Several algorithmic paradigms are prominent:

• Iterative LP-based Peeling: The iterative assignment (IA) algorithm solves a sequence of LP relaxations, peeling off assignments whenever integral $x_{rv}$ solutions are found, pruning infeasible assignments at each iteration; guaranteed progress and termination in at most $|E|$ steps (Vakayil et al., 2023).
• Dynamic Tree Insertion: For per-vehicle dial-a-ride formulations, dynamic trees maintain feasible sequences, inserting request pickup and drop-off in all positions and pruning branches violating time or capacity constraints, achieving near real-time feasibility checking (Yao et al., 2020).
• Shareability Hypergraph: Hyperedges encode feasible trip-vehicle (bundles of requests routed by one vehicle), and only those satisfying all constraints (capacity, order, time windows) are considered. Solution methods include greedy rounding or combinatorial local search with provable approximation ratios, and capacity/time-window feasibility is strictly enforced during enumeration (Luo et al., 2021).
• Zone Path Construction: The ZAC approach abstracts feasible request combinations via paths through clustered geographic "zones," greatly reducing combinatorial explosion while encoding all pickup/delivery sequencing and timing constraints (Lowalekar et al., 2020).
• MILP Heuristics in Min-Max Problems: In applications like heterogeneous min–max multi-vehicle multi-depot TSP, feasibility is enforced by explicit assignment and routing constraints, including functional/structural vehicle heterogeneity. Local search and perturbation preserve feasibility throughout (Kumar et al., 30 Oct 2024, Kumar et al., 2023).

Feasibility can also be tested explicitly in single-vehicle subtour routing problems using polynomial-time greedy matchings under handover and deadline constraints, exploiting strong structural decompositions (Held et al., 2018).

3. Feasibility-Pruning and Preprocessing Techniques

To enable tractable solution of large-scale problems, aggressive pruning or preprocessing is essential:

• Batch-wise pruned edge sets: Candidate request-vehicle assignments are prefiltered based on spatial proximity, available capacity, and violation of any timing bounds (Vakayil et al., 2023).
• Geometric region exclusion: In dynamic trees, accessible vehicle regions (ellipses) and request waiting regions (circles) are checked for intersection; non-intersecting pairs are immediately pruned, reducing assignment attempts by up to 30% (Yao et al., 2020).
• Trip/Hyperedge Enumeration with Local Checks: Only trip-vehicle combinations passing quick capacity, delay, and sequencing checks are enumerated (Luo et al., 2021, Lowalekar et al., 2020).
• Automata-based Feasibility: For requests with temporal constraints (e.g., LTL formulas), feasibility is determined by building a product-automaton of vehicle route and request automata, using shortest-path algorithms to check for feasible satisfaction within all time/delay bounds (Liang et al., 2022, Liang et al., 7 Feb 2024).

These strategies ensure that only feasible assignments are considered during the optimization or matching stage, which is necessary for computational scalability in real-time deployments.

4. Optimization Objectives and Feasibility-Driven Tradeoffs

Assignment feasibility is necessary but not sufficient; the ultimate goal is to optimize one or more objectives:

• Operator Cost vs. Passenger Service: Objective functions typically blend total fleet travel time (or vehicle hours traveled, VHT) and worst-case passenger delay, via a parameter $\Lambda$ tuning operator- vs. customer-centric performance. Empirical results demonstrate sensitive tradeoffs: increasing $\Lambda$ yields heavier pooling and higher wait/ride times, but lower VHT (Vakayil et al., 2023).
• Fairness Criteria: Some frameworks enforce approximate envy-freeness or max-min utility, introducing assignment cost corrections or auction-based bid weighting to equalize service across vehicles (Liang et al., 2022, Liang et al., 7 Feb 2024).
• Coverage/Service Rate Maximization: Many formulations directly maximize the number of requests fulfilled while keeping all assignments feasible (Lowalekar et al., 2020, Yao et al., 2020).
• Privacy-Preserving Feasibility: Assignments are feasible under obfuscated locations with privacy guarantees; redundant vehicle dispatch is used to restore service levels lost to privacy-induced assignment noise (Prorok et al., 2017).
• Stochastic/Demand-Aware Assignment: Future demand distributions are incorporated into feasibility calculations via probabilistic or two-stage models, only generating assignments that are feasible in expectation or across sampled scenarios (Lin et al., 2019, Luo et al., 2021).

Policy parameters (e.g., $\Lambda$, fairness weights, redundancy, zone granularity) structure the feasible set and therefore system-level tradeoffs.

5. Implementation and Computational Considerations

The primary challenge is scaling feasible assignment algorithms to large fleets and dense request arrival patterns in real-time:

• LP Relaxations and Incremental Updates: Modern LP solvers efficiently handle constraints for thousands of vehicles/requests; iterative and batch frameworks exploit high fractional integrality and "peeling" to incrementally build feasible assignments (Vakayil et al., 2023, Mori et al., 2020).
• Greedy and Heuristic Algorithms: On-arrival greedy assignment, dynamic tree best-leaf extension, and zone-based clustering allow near-instantaneous feasible assignment, at the cost of suboptimality (Yao et al., 2020, Vakayil et al., 2023).
• Parallelization and Decomposition: Shareability hypergraph methods, scenario-based stochastic assignments, and Benders decomposition for non-myopic approaches are highly parallelizable, facilitating batch assignment across large urban networks (Luo et al., 2021, Lowalekar et al., 2020).
• Feasibility Certification: For pure feasibility (ignoring cost), polynomial algorithms exist in some domains (vehicle subtours, batch handover VRP) using matching or greedy assignment based on key structural theorems (Held et al., 2018).

The table below summarizes typical feasibility constraints enforced in recent research systems:

Constraint	Common Notation	Enforcement Mechanism
Capacity	$\sum_{r} Q_r x_{rv} \leq \bar Q_v$	Pruning/preprocessing, explicit constraint in (M)ILP (Vakayil et al., 2023, Yao et al., 2020)
Waiting-time window	$\max(\text{pickup delay}) \leq D^P$	Feasibility check in edge/trip enumeration (Vakayil et al., 2023)
Travel delay	$\text{delay}_i \leq D^T_i$	Shortest-path/automata search (Liang et al., 2022)
Sequencing/precedence	Pickup before drop-off	Route expansion in dynamic trees, automata, or zone-path (Yao et al., 2020, Lowalekar et al., 2020)
Functional/structural heterogeneity	Assignment variables $y_{it}$ restricted; per-vehicle costs $c^i_{mn}$	Assignment matrix construction, MILP constraints (Kumar et al., 30 Oct 2024)

6. Extensions: Fairness, Privacy, and Anticipation

Recent advances extend the feasibility paradigm beyond pure operational constraints:

• Approximate Envy-Freeness: Via ILP constraints or auction-based weighting, service parity is enforced, and feasibility must now include fairness constraints in addition to core operational limits (Liang et al., 2022, Liang et al., 7 Feb 2024).
• Privacy Guarantees: ε-geo-indistinguishability requires vehicle locations to be randomized, with batch assignment algorithms adapted to work with expected cost matrices and redundant vehicle assignment to offset information loss; all assignments continue to strictly enforce capacity and time window feasibility (Prorok et al., 2017).
• Anticipatory Routing: Artificial future requests ("ghosts") or reward-shifting terms in the offline cost function draw vehicles proactively into regions likely to generate future feasible assignments, still ensuring all constraints are satisfied for real and virtual requests (Fielbaum et al., 2021).

These extensions preserve core feasibility while broadening the assignment framework's objectives and robustness.

7. Empirical Performance and Sensitivity

Empirical studies indicate:

• Tuning assignment cost parameters ($\Lambda$ for VHT-delay tradeoff, fairness/utility corrections) has the largest impact on key system-level metrics (wait time, VHT, occupancy) (Vakayil et al., 2023, Liang et al., 2022).
• Aggressive pruning and zone-path strategies can reduce assignment computation from tens of seconds to near real time, while increasing request coverage by 3–10% over prior batching/trip-formation approaches (Lowalekar et al., 2020).
• Under wide operational settings (vehicle capacities, network sizes), feasible assignment frameworks consistently achieve >95% coverage at within 2–5% of optimal travel cost, with solution times scaling linearly with batch size due to pruning and decomposition (Vakayil et al., 2023, Luo et al., 2021, Yao et al., 2020).
• Sensitivity to operational constraints (e.g., tighter time windows, higher minimum capacity) can invert expected pooling/consolidation gains, as excessive detours render no feasible assignments, highlighting the importance of parameter selection in system design (Yao et al., 2020).

Feasible vehicle-to-request assignment frameworks thus balance expressive modeling of practical constraints with algorithmic scalability, real-time deployability, and multi-objective operational flexibility across a spectrum of modern mobility systems.