Multiple-Depot Split Delivery VRP

• Multiple-Depot Split Delivery VRP is a complex routing problem where vehicles from several depots deliver portions of customer demand to minimize travel and operational costs.
• It generalizes the classical CVRP by allowing split deliveries and depot-based routing, thereby offering practical flexibility for diverse logistics applications.
• Recent advances include refined approximation algorithms and efficient heuristics that reduce cost factors significantly, enhancing solutions for large-scale distribution scenarios.

The Multiple-Depot Split Delivery Vehicle Routing Problem (MD-SDVRP) is a generalization of the classical capacitated vehicle routing problem (CVRP) where a fleet of vehicles, located at multiple depots, must serve the demand of a set of customers. Each customer demand may be split among multiple deliveries, and the objective is to minimize the total travel or operational cost. The MD-SDVRP incorporates both the logistical complexity of multiple depots and the operational flexibility of split deliveries, and arises in a variety of logistics, distribution, and collection scenarios.

1. Formal Problem Definition

MD-SDVRP is defined on a complete undirected graph $G = (V,E)$, with vertex set $V = J \cup D$, where $J$ is the set of customers and $D$ is the set of depots. Each customer $v \in J$ has integer demand $q_v > 0$. Each depot $u \in D$ has a set of physically identical vehicles (capacity $Q$), and possibly limited total vehicle count $r_u$, with total fleet size $m = \sum_{u \in D} r_u$. Edge costs $c(e)$ form a metric on $E$.

A feasible solution consists of, for each vehicle, a closed route starting and ending at its depot and delivering total load at most $Q$. Customer demand may be fractionally split across different vehicles (split delivery). Every demand must be exactly fulfilled: $$\sum_{\tau \in L} \lambda_{v, \tau} = q_v \quad \forall v \in J, \qquad \sum_{v \in J: v \in T_\tau} \lambda_{v,\tau} \leq Q \quad \forall \tau \in L.$$ The objective is to minimize total route cost $c(T) = \sum_\tau c(T_\tau)$. For model completeness, the graph is often assumed metric, symmetric, and to satisfy the triangle inequality (Zhao et al., 5 Jan 2026).

A standard integer-programming formulation uses edge variables $x_e$ and flow variables $f_{v,e}$ to represent route selection and demand assignment, together with degree, connectivity, and capacity constraints, ensuring each route is an elementary cycle passing through a single depot, and delivering split customer demand subject to capacity limits (Zhao et al., 5 Jan 2026, Zhao et al., 2023).

2. Complexity and Structural Properties

MD-SDVRP is NP-hard. Even in the single-depot, unit-demand, capacity $k \ge 3$ case, it is APX-hard and admits no polynomial-time approximation scheme (PTAS) unless $\text{P} = \text{NP}$ (Zhao et al., 2023). No constant-factor lower bound better than 2 is known under standard complexity assumptions. The split delivery property increases flexibility but does not, in general, render the problem tractable.

A key structural property is that the optimal number of vehicles required admits a tightly characterized analytical upper bound under standard conditions (including single-depot route assignment, homogeneous vehicle capacities, and integer data). Under split delivery, the bound is achieved by recursively maximizing, via a dynamic program, the sum over depots of maximal possible single-depot vehicle use, given overall demand and capacity parameters (Eshragh et al., 2019).

3. Approximation Algorithms

Classical Methods

Before recent breakthroughs, the best-known approximation ratios for MD-SDVRP (metric graphs) were as follows:

Variant	Approximation Ratio	Reference
Unit-demand/splittable MCVRP	$4 - \Theta(1/k)$	Tree partition (Zhao et al., 2023)
Unsplittable MCVRP	$4$	(Zhao et al., 2023)
Split-delivery, fixed depots	$6-4/k$, i.e., 6	(Zhao et al., 5 Jan 2026)

These ratios are achieved by two central frameworks:

• Cycle-Partition (ITT/UITP): Computes a Hamiltonian cycle in a merged-depot "supergraph", partitions it into capacity-constrained segments, and replicates depot connections. Total cost is bounded by a function of $$\Delta = \sum_{v} d(v) \cdot \text{dist}(v, \text{nearest depot})$$ and the Hamiltonian cycle weight, yielding a factor close to 4 (Zhao et al., 2023).
• Tree-Partition: Constructs a minimum spanning forest covering customers and depots and recursively splits overloaded trees, producing tours via shortcutting. This yields ratio $4-\Theta(1/k)$ (Zhao et al., 2023).

Algorithmic Advances

Zhao and Xiao (Zhao et al., 2023) provided an improved $(4-1/1500)$-approximation for splittable MD-SDVRP valid for all $k$, by refining the tree-partition scheme and employing a trade-off with an enhanced cycle-partition using near-optimal TSP tours. The refined tree-partition algorithm, outlined below, guarantees the improved bound independently of $k$:

• High-demand customers ($d(v) > k/2$) are served by dedicated tours.
• Remaining customers are connected via a minimum spanning forest, with further splits made to avoid violating vehicle capacity.
• Subtree and final leftover demands are processed by efficient shortcutting to construct tours.

If $(2/k)\Delta \leq (1-\epsilon) \text{OPT}'$, the tree partition is used; otherwise, cycle partitioning with a near-optimal TSP tour is performed, yielding the $4-1/1500$ guarantee (Zhao et al., 2023).

4. New PTAS-Like and Parameterized Approximations

A recent milestone was achieved by Chen, Dinitz, Hwang, and Xiao (Zhao et al., 5 Jan 2026), who established the first improvement over the previously unbroken "6-approximation barrier" for constant-depot MD-SDVRP. Their algorithm achieves a $6 - 2 \cdot 10^{-36}$ approximation for constant $k$, utilizing:

• Cycle Cover Construction: Solve the multiple-depot TSP relaxation, augment to ensure Eulerian components, and transform cycles into paths respecting capacity.
• Feasible Assignment via Flow: Decompose cycles into capacity-limited paths and allocate path load to available vehicles via a minimum-cost flow.

Extensions include:

• FPT $(6+\varepsilon)$-, and deterministic $(7-2/k)$-approximation for arbitrary $k$ using moving partitions of TSP cycles and FPT "Eulerian-extension".
• A $5$-approximation arises by parameterizing by vehicle capacity and offering an XP or FPT solution depending on $Q$, $k$, and fleet slackness ($mQ-\sum q_v$).

This unified "good-component + flow-assignment" framework is robust to bi-factor relaxation, allowing $(\rho, \gamma)$-approximations in which both cost and route capacity are multiplicatively controlled. For instance, a polytime $(6+2/\varepsilon+2\varepsilon, 1+\varepsilon)$-bi-factor approximation is established.

5. Fleet Size Bounds and Theoretical Implications

An explicit analytical upper bound on the fleet size required to serve any MD-SDVRP instance is given by (Eshragh et al., 2019): $$V_n^*(\Delta) = \max\left\{ \sum_{j=1}^n \pi(x_j) : \sum_{j=1}^n x_j \leq \Delta,\, 0 \leq x_j \leq c_j,\, x_j \in \mathbb{Z} \right\}$$ where $\pi(\cdot)$ computes the maximal number of vehicles needed at a depot given assigned load, considering the pairwise-merge condition (vehicles loads not mergeable into a single vehicle under triangle inequality).

A greedy, nearly closed-form optimal allocation/recursion solves this integer program efficiently. The bound holds for heterogeneous fleets (by typewise disaggregation), for both split and unsplit delivery, and can be computed in $O(n\log n)$ preprocessing (Eshragh et al., 2019).

6. Heuristic and Practical Algorithms

Efficient heuristics for MD-SDVRP are critical for large-scale practical instances. The HILS-RVRP metaheuristic (Penna et al., 2018) extends to MD-SDVRP via:

• Multi-level local search (RVND) over customer, depot, and split assignment neighborhoods.
• Specialized moves: ShiftDepot and SwapDepot for depot reassignment; split-specific neighborhoods $k$-Split and quantity-adjusting Swap.
• Solution improvement by hybridization with set partitioning, where the SP step picks among a pool of promising routes with associated split and depot assignments.

Empirical results on benchmark instances (up to 150 customers, 5 depots) indicate an average optimality gap under 0.12%, with computation times averaging 24s on modern workstations. Solution quality does not improve significantly with more complex combined neighborhoods, suggesting that computational resources are better allocated to intensive intra-route search rather than frequent split or depot reassignments (Penna et al., 2018).

Recommendations for scaling include limiting complex moves to rare perturbations, restricting SP size, and calibrating penalties to avoid excessive customer demand fragmentation.

7. Comparative Summary and Open Directions

The evolution of MD-SDVRP approximation algorithms is summarized below.

Algorithm/Framework	Approximation Ratio	Complexity	Reference
Tree/Cycle Partition (prior best)	$4-\Theta(1/k)$	Poly. in $n$, $k$	(Zhao et al., 2023)
Refined Tree/Cycle Part. (Zhao, X)	$4-\frac{1}{1500}$	Poly. in $n$	(Zhao et al., 2023)
XP via MD-TSP (Chen et al.)	$6-2\cdot 10^{-36}$ (const. $k$)	$n^{O(k/\varepsilon)}$	(Zhao et al., 5 Jan 2026)
FPT (randomized/derm.)	$6+\varepsilon,\;7-2/k$	FPT in $k$	(Zhao et al., 5 Jan 2026)
Parameterized by capacity	$5$	XP/FPT/poly. (see paper)	(Zhao et al., 5 Jan 2026)
Bi-factor poly ($\varepsilon$)	$(6+2/\varepsilon+2\varepsilon,1+\varepsilon)$	Poly. in $n$	(Zhao et al., 5 Jan 2026)

The constant-factor barrier for MD-SDVRP, once thought to be 6, has been broken for the constant-depot case. Bi-factor relaxations and parameterized schemes provide a spectrum of trade-offs among cost, capacity violation, and computational complexity.

Open questions include closing the gap to the single-depot theoretical minimum ($\alpha+1 \approx 2.5$), extending improvements to arbitrary or unbounded $k$, and further refining the dependence on parameters $k$, $Q$, and approximation slack $\varepsilon$ in the FPT/XP regimes (Zhao et al., 5 Jan 2026).

• (Zhao et al., 5 Jan 2026) Improved Approximation Algorithms for the Multiple-Depot Split Delivery Vehicle Routing Problem (Chen, Dinitz, Hwang, Xiao, 2026)
• (Zhao et al., 2023) Multidepot Capacitated Vehicle Routing with Improved Approximation Guarantees (Zhao, Xiao, 2023)
• (Eshragh et al., 2019) An analytical bound on the fleet size in vehicle routing problems: a dynamic programming approach
• (Penna et al., 2018) A Hybrid Heuristic for a Broad Class of Vehicle Routing Problems with Heterogeneous Fleet