Cumulative Vehicle Routing Problem (Cu-VRP)

• Cu-VRP is defined on a complete metric graph where the cost depends both on travel distance and the vehicle's carried load, making load management as crucial as route geometry.
• Approximation algorithms for unsplittable and splittable variants leverage TSP structure and depot-distance demand coupling to enhance performance over traditional VRP models.
• The framework uses randomized parameter tuning and LP rounding techniques to balance recourse actions with optimal route planning, resulting in significantly improved approximation ratios.

Searching arXiv for the cited Cu-VRP paper and closely related routing work. arxiv_search(query="Approximation Algorithms for the Cumulative Vehicle Routing Problem with Stochastic Demands", max_results=5, sort_by="relevance") Looking for related arXiv entries on cumulative or stochastic vehicle routing to ground the article. arxiv_search(query="cumulative vehicle routing problem stochastic demands", max_results=10, sort_by="relevance") Searching for approximation results on VRPSD and Cu-VRP that may provide context. arxiv_search(query="VRPSD approximation algorithm arXiv stochastic demands vehicle routing", max_results=10, sort_by="relevance") The Cumulative Vehicle Routing Problem (Cu-VRP) is a capacitated routing problem on a complete undirected metric graph in which the objective is not merely to find a feasible itinerary from a depot to satisfy customer demands, but to minimize a cumulative cost determined by the vehicle’s load throughout the itinerary. In the deterministic setting, each customer demand is known in advance; when demands are unknown until the vehicle visits a customer, the problem is the Cu-VRP with Stochastic Demands (Cu-VRPSD). A 2025 approximation study formulates both models in a unified framework with edge cost $a\cdot w(u,v)+b\cdot x\cdot w(u,v)$, derives lower bounds that couple TSP structure and depot-distance demand mass, and gives improved approximation algorithms for unsplittable and splittable variants (Zhao et al., 11 Jul 2025).

1. Core formulation

The standard formulation uses a complete undirected graph $G=(V,E)$ with vertex set $V=\{v_0,v_1,\ldots,v_n\}$, where $v_0$ is the depot and the remaining vertices are customers. Edge lengths are given by a function $w:E\to\mathbb{R}_+$ satisfying the triangle inequality, and $w(u,v)$ denotes the length of edge $uv$. The vehicle has capacity $Q>0$, which is often normalized to $Q=1$ by scaling. Two nonnegative parameters, $a$ and $G=(V,E)$0, govern the traversal cost: traversing an edge $G=(V,E)$1 with load $G=(V,E)$2 incurs cost

$G=(V,E)$3

This objective distinguishes Cu-VRP from formulations in which routing cost depends only on traveled distance, because the load carried on each edge directly affects the objective (Zhao et al., 11 Jul 2025).

In deterministic Cu-VRP, customer $G=(V,E)$4 has known demand $G=(V,E)$5. An itinerary $G=(V,E)$6 is a multiset of directed simple cycles, each starting and ending at the depot, such that exactly $G=(V,E)$7 units are delivered at $G=(V,E)$8 over the whole itinerary. For each occurrence of an edge $G=(V,E)$9 in $V=\{v_0,v_1,\ldots,v_n\}$0, the formulation defines a load-on-edge value $V=\{v_0,v_1,\ldots,v_n\}$1 with $V=\{v_0,v_1,\ldots,v_n\}$2. The cumulative cost is

$V=\{v_0,v_1,\ldots,v_n\}$3

Feasibility requires that when a tour visits customers in the order $V=\{v_0,v_1,\ldots,v_n\}$4, the initial load is at most $V=\{v_0,v_1,\ldots,v_n\}$5, traversal never exceeds capacity, and delivery at a node reduces the load by exactly the delivered amount.

This formulation makes the “cumulative” qualifier precise: cost is aggregated edge by edge with explicit dependence on carried load rather than only on tour count or total length. A plausible implication is that load placement and replenishment timing become as central as route geometry.

2. Deterministic, stochastic, unsplittable, and splittable variants

In Cu-VRPSD, demands are independent random variables $V=\{v_0,v_1,\ldots,v_n\}$6 with known distributions. A policy $V=\{v_0,v_1,\ldots,v_n\}$7 adaptively chooses which tour to follow next based on observed demands so far, and its expected cumulative cost is

$V=\{v_0,v_1,\ldots,v_n\}$8

Performance is compared to the expected cost of the optimal adaptive policy, denoted $V=\{v_0,v_1,\ldots,v_n\}$9. Because the demands are revealed only upon visit, the stochastic model couples route planning with online recourse decisions (Zhao et al., 11 Jul 2025).

The same study treats unsplittable and splittable service regimes separately. In the deterministic definition, each customer’s entire demand must be delivered on some visit or visits, and later algorithmic sections distinguish whether a customer is allowed to be satisfied by using multiple tours. For Cu-VRPSD and Cu-VRP, allowing multiple tours yields stronger approximation guarantees than the unsplittable case. The work also states that VRP with Stochastic Demands (VRPSD) is a special case of Cu-VRPSD. This is significant because approximation results for the cumulative model immediately induce corollaries for the classical stochastic-demand setting.

A common misconception is to treat Cu-VRPSD as merely VRPSD with different notation. The formal cost model shows otherwise: VRPSD appears as a special case, but Cu-VRPSD explicitly weights distance by current load, so the optimization criterion is structurally richer.

3. Lower bounds and analytical scaffold

The analysis introduces two benchmark quantities for a realization $v_0$0: first,

$v_0$1

where $v_0$2 is the depot distance of customer $v_0$3; second, $v_0$4, the length of a minimum TSP tour through all nodes. The lower-bound lemma states that for any realization $v_0$5, any feasible policy satisfies

$v_0$6

Equivalently, the study writes

$v_0$7

By conditioning on $v_0$8, it suffices to show for each realization that the algorithmic cost is at most $v_0$9 (Zhao et al., 11 Jul 2025).

This lower bound has two distinct structural components. The $w:E\to\mathbb{R}_+$0 term captures the geometric backbone imposed by visiting all nodes in a metric graph, while $w:E\to\mathbb{R}_+$1 captures demand mass weighted by depot distance. This suggests that approximation design for Cu-VRP cannot rely solely on TSP structure or solely on radial demand arguments; both must be balanced.

The analysis of the main randomized algorithm decomposes cost into a TSP-pass component and recourse visits. On each edge of the approximate TSP tour, the expected load is $w:E\to\mathbb{R}_+$2, leading to

$w:E\to\mathbb{R}_+$3

plus empty-vehicle cost $w:E\to\mathbb{R}_+$4. Additional depot returns are then integrated over the random initial load, with piecewise expressions depending on whether customer demand falls in $w:E\to\mathbb{R}_+$5, $w:E\to\mathbb{R}_+$6, $w:E\to\mathbb{R}_+$7, or $w:E\to\mathbb{R}_+$8. The resulting normalized expression is

$w:E\to\mathbb{R}_+$9

where $w(u,v)$0 and $w(u,v)$1.

4. Randomized approximation for unsplittable Cu-VRPSD

For unsplittable Cu-VRPSD, the principal construction is the randomized algorithm denoted $w(u,v)$2. It begins by computing an $w(u,v)$3-approximate TSP tour $w(u,v)$4 and fixing an orientation $w(u,v)$5, where $w(u,v)$6 is treated as a black-box approximation ratio for metric TSP. After normalizing $w(u,v)$7, the algorithm chooses parameters $w(u,v)$8 and $w(u,v)$9, samples an initial load $uv$0, and carries backup $uv$1, so the total initial load is $uv$2 (Zhao et al., 11 Jul 2025).

The algorithm visits customers in TSP order and distinguishes cases according to the realized demand at $uv$3 relative to the current normal load and the backup. If $uv$4, the demand is delivered from normal goods without returning to the depot. If $uv$5, the vehicle delivers all normal goods, uses part of the backup, and then returns once to the depot to reload. If $uv$6, the next decision depends on whether $uv$7. For $uv$8, the algorithm makes two returns to the depot, one to pick up bulk for that customer and one to restore the backup. For $uv$9, the customer is skipped during the TSP pass, recorded for clean-up, and then served individually by a single round trip after the pass ends.

The approximation guarantee is obtained by tuning $Q>0$0 and $Q>0$1 through balancing integrals over demand distributions. The paper states three parameter regimes: for $Q>0$2, one choice of $Q>0$3 achieves factor $Q>0$4; for $Q>0$5, another choice is used; and for $Q>0$6, $Q>0$7 is combined with a second algorithm $Q>0$8 to cap the approximation ratio at $Q>0$9. Under the assumption that the approximation ratio of metric TSP is $Q=1$0, this yields a randomized $Q=1$1-approximation algorithm for Cu-VRPSD and, as a corollary, a randomized $Q=1$2-approximation algorithm for VRPSD.

A further analytical point is that the basic choice $Q=1$3 yields $Q=1$4, whereas mixtures over two $Q=1$5-values, obtained by randomly choosing between $Q=1$6 and $Q=1$7, cancel the troublesome quadratic term in $Q=1$8 and deliver the uniform $Q=1$9 guarantee. This indicates that randomization is used not only for initial offsetting but also for smoothing the worst-case contribution of recourse.

5. Randomized approximation for unsplittable Cu-VRP

For deterministic unsplittable Cu-VRP, the algorithm $a$0 combines weighted set-cover rounding on “large” customers with $a$1 on the remainder. It defines

$a$2

constructs all feasible subsets $a$3 with total demand at most $a$4, and assigns each subset a weight $a$5 equal to the cost of an optimal single-trip tour for $a$6. This becomes an instance of $a$7-set-cover with

$a$8

The LP relaxation enforces $a$9, after which each set $G=(V,E)$00 is selected independently with probability $G=(V,E)$01 (Zhao et al., 11 Jul 2025).

The selected collections are shortcut to a set of tours $G=(V,E)$02 covering a subset $G=(V,E)$03. The algorithm then runs $G=(V,E)$04 to cover $G=(V,E)$05 on $G=(V,E)$06 and finally “optimize[s]” each $G=(V,E)$07 tour to make load = delivered and choose the better direction. The resulting approximation ratio is obtained by tuning $G=(V,E)$08 and taking $G=(V,E)$09 small, which yields an overall ratio $G=(V,E)$10.

The analysis uses two ingredients. First, the set-cover rounding preserves $G=(V,E)$11. Second, by standard “$G=(V,E)$12-rounding” analysis, each customer is missed with probability at most $G=(V,E)$13, so the leftover set has size at most $G=(V,E)$14 in expectation. Covering leftovers by $G=(V,E)$15 therefore adds at most $G=(V,E)$16, and the paper summarizes the bound as

$G=(V,E)$17

Under the same metric-TSP assumption, this improves the best-known approximation ratio for Cu-VRP from $G=(V,E)$18 to $G=(V,E)$19.

6. Splittable service and approximation landscape

When each customer is allowed to be satisfied by using multiple tours, the paper gives a simpler procedure, $G=(V,E)$20. It uses an initial uniform load $G=(V,E)$21 and no backup. If, at customer $G=(V,E)$22, the current load is less than the required amount, the vehicle returns to the depot for $G=(V,E)$23 units and then resumes. Setting $G=(V,E)$24 yields a randomized $G=(V,E)$25-approximation. In the deterministic case of Cu-VRP, this derandomizes to a $G=(V,E)$26-approximation (Zhao et al., 11 Jul 2025).

The paper’s comparative summary is as follows:

Problem class	Previous best	New ratio
Unsplit Cu-VRPSD	$G=(V,E)$27	$G=(V,E)$28
Unsplit VRPSD	$G=(V,E)$29	$G=(V,E)$30
Unsplit Cu-VRP	$G=(V,E)$31	$G=(V,E)$32
Splittable Cu-VRPSD/VRP	matches $G=(V,E)$33	$G=(V,E)$34

The main assumptions are metric $G=(V,E)$35, independent demands, and normalized capacity $G=(V,E)$36. Within those assumptions, the approximation framework shows that cumulative-load routing admits substantial improvements over previously known guarantees. A plausible implication is that the lower bound based on $G=(V,E)$37 and the separation between a TSP pass and depot-recoursed correction may provide a reusable template for other load-sensitive vehicle-routing objectives.