kRental-Variable: Online Reusable Allocation

• kRental-Variable is an online problem defining resource allocation for k identical, reusable units subject to variable-duration rental requests under adversarial conditions.
• It employs a price-based fractional allocation and limited-correlation rounding scheme to achieve an order-optimal competitive ratio that depends logarithmically on the ratio d_max/d_min.
• The framework integrates dual formulation and pseudo-utility maximization to manage temporal constraints, ensuring proper marginal probability preservation in online decisions.

kRental-Variable is an online resource-allocation problem with reusability in an adversarial setting: a decision-maker manages $k$ identical reusable units and faces a sequence of rental requests over time, each request $n$ being specified by an arrival time $a_n$, a rental duration $d_n$, and a valuation $v_n=d_n$. If an item is assigned to request $n$ at time $a_n$, it becomes unavailable until time $a_n+d_n$, then returns. The decision-maker must accept or reject each request irrevocably upon arrival, without knowledge of future $(a_j,d_j)$, and seeks to maximize total accepted valuation $\sum_n d_n x_n$ subject to at most $n$0 simultaneous rentals. In (Nekouyan et al., 25 Jul 2025), the variable-duration setting is treated by a relax-and-round framework consisting of a price-based fractional allocation and a limited-correlation online rounding scheme, yielding an order-optimal competitive ratio.

1. Formal problem specification

The model assumes $n$1 identical, reusable units and a sequence of $n$2 requests arriving online. Request $n$3 is specified by arrival time $n$4, rental duration $n$5, and valuation $n$6. If a unit is assigned to $n$7, that unit is unavailable on the interval $n$8 and returns afterward. The adversary chooses all $n$9, but is subject to bounded durations $a_n$0 and arbitrary arrival order, with no stochastic assumptions (Nekouyan et al., 25 Jul 2025).

The objective is to maximize total accepted valuation $a_n$1 subject to at most $a_n$2 simultaneous rentals. In this formulation, feasibility is temporal rather than purely cardinal: a unit may be reused, but only after the previously assigned request has expired. This makes duration heterogeneity central to the model. A plausible implication is that the difficulty of the problem is governed not only by capacity $a_n$3, but also by the spread $a_n$4, which later appears explicitly in the competitive ratio.

The model is online and irrevocable. Each request must be accepted or rejected at its arrival time, and future arrivals are unknown. This places kRental-Variable within adversarial online allocation, but with reusable resources and interval-based occupancy rather than one-shot consumption.

2. Fractional relaxation and price-based allocation

The analysis in (Nekouyan et al., 25 Jul 2025) begins with a fractional relaxation in which $a_n$5 is relaxed to $a_n$6, while capacity is enforced at each arrival. The resulting linear program maximizes $a_n$7; its optimum is denoted $a_n$8.

The online fractional algorithm is price-based. It maintains a nondecreasing price function

$a_n$9

where the argument represents the normalized utilization of a particular unit. At each arrival $d_n$0, the algorithm tracks for each unit $d_n$1 a fractional utilization $d_n$2, selects

$d_n$3

and then solves a one-dimensional pseudo-utility problem: $d_n$4 It then updates the chosen unit’s utilization by

$d_n$5

This rule is interpreted in the source as charging a marginal cost $d_n$6 for each infinitesimal fraction of the unit allocated over its current utilization. The pricing function is therefore not an ex post accounting device, but the mechanism by which the online fractional acceptance level is determined. Because $d_n$7 is nondecreasing, additional load on a more-utilized unit is penalized more heavily, which biases the algorithm toward lower-utilization units.

3. Limited-correlation rounding

The integral algorithm converts each fractional $d_n$8 into a binary decision $d_n$9 by a limited-correlation rounding scheme (Nekouyan et al., 25 Jul 2025). The paper treats each unit independently but preserves the correct marginal probabilities. For each arrival $v_n=d_n$0, after computing the fractional quantities and choosing $v_n=d_n$1, the algorithm samples $v_n=d_n$2 and accepts iff the designated unit is currently available and

$v_n=d_n$3

The scheme has two stated structural properties. First, by construction,

$v_n=d_n$4

Second, different requests use independent random seeds $v_n=d_n$5, but correlation across decisions involving the same unit arises through availability, specifically through the denominators $v_n=d_n$6; different units’ randomizations remain independent.

The paper emphasizes that this is not lossless online rounding. For kRental-Variable, lossless online rounding is impossible, and the integral analysis therefore requires controlled dependence rather than full decoupling. This point distinguishes the variable-duration setting from kRental-Fixed in the same work, where an optimal randomized algorithm is obtained by first computing an optimal fractional allocation via a price-based approach and then applying a novel lossless online rounding scheme (Nekouyan et al., 25 Jul 2025).

4. Competitive analysis and order-optimality

The main guarantee is given as Theorem 6 in (Nekouyan et al., 25 Jul 2025). Suppose $v_n=d_n$7 is nondecreasing and, for all $v_n=d_n$8, with

$v_n=d_n$9

two families of inequalities hold for all $n$0 and $n$1. Under these conditions, the relax-and-round algorithm is $n$2-competitive.

The proof sketch is LP-free and certificate-based. The paper constructs dual variables $n$3 and $n$4 whose value telescopes to

$n$5

The two families of inequalities guarantee that each dual constraint is satisfied exactly at equality. The rounding stage incurs a loss factor of $n$6, reflected in the $n$7 splits in the updates, but no further loss beyond that.

A closed-form feasible choice is

$n$8

The paper states that this $n$9 satisfies the required inequalities, so the algorithm attains the competitive ratio

$a_n$0

The assumptions attached to this result are explicit. Durations must lie in $a_n$1; $a_n$2; and $a_n$3. The factor of $a_n$4 arises from limited-correlation rounding, and the resulting bound is order-optimal: up to constant factors,

$a_n$5

matches the lower bound of any online algorithm under the same adversarial assumptions (Nekouyan et al., 25 Jul 2025).

5. Relation to multislope ski rental and older k-rental formulations

The term “k-rental” also appears in older online-optimization literature summarized from “Rent, Lease or Buy: Randomized Algorithms for Multislope Ski Rental” (0802.2832). There, the setting is the Multislope Ski Rental problem, a generalization of classical Ski Rental, with $a_n$6 slopes indexed by $a_n$7. Slope $a_n$8 has a one-time buy cost $a_n$9 and a per-unit-time rent rate $a_n+d_n$0, with

$a_n+d_n$1

and the adversary chooses an unknown usage duration $a_n+d_n$2. The offline optimum is

$a_n+d_n$3

For additive instances, the main result summarized from (0802.2832) is that there exists an instance-specific best $a_n+d_n$4 and a corresponding tight profile $a_n+d_n$5 achieving

$a_n+d_n$6

with $a_n+d_n$7 and $a_n+d_n$8 computable in $a_n+d_n$9 time to within additive $(a_j,d_j)$0. The analysis relies on prudent profiles, tightness, and a piecewise-exponential ODE characterization. For non-additive instances, the same summary gives a simple randomized doubling algorithm with expected competitive ratio at most $(a_j,d_j)$1.

The reusable-unit model of kRental-Variable in (Nekouyan et al., 25 Jul 2025) is structurally different. It is driven by overlapping requests with explicit arrival times and durations, and its feasibility constraint is “at most $(a_j,d_j)$2 simultaneous rentals.” By contrast, the multislope model is organized around switching among cost states over an unknown horizon $(a_j,d_j)$3. This suggests a shared conceptual lineage—online rental decisions under adversarial uncertainty—but not a shared state representation or proof template.

6. Nomenclature, adjacent topics, and common confusions

A common source of confusion is the phrase “kRental-Variable” itself. In the available material, it refers unambiguously to the variable-duration reusable-unit problem of (Nekouyan et al., 25 Jul 2025). In a separate summary of (0802.2832), “k-rental-variable” is used as an alias for the k-slope ski-rental problem. This suggests that the nomenclature is not uniform across summaries, even when the underlying online-allocation structures differ.

Another potential confusion arises with “Peer to Peer Sharing of Distributed Energy Resources” (Patel et al., 2018). That paper models a peer-to-peer rental market for rooftop solar and energy storage in the residential sector, using a rental price $(a_j,d_j)$4 and its market equilibrium $(a_j,d_j)$5, but it does not introduce or calibrate any parameter called “kRental,” “$(a_j,d_j)$6,” or anything similar. The source states that there are no equations involving kRental, no definition of kRental, no units, and no normalization; kRental plays no role in shaping equilibrium rental price, quantity, or participation rate; no sensitivity analysis or policy implication hinges on such a parameter (Patel et al., 2018).

Within the variable-duration online-allocation setting itself, a further misconception would be to expect lossless rounding from the fractional to the integral solution. The paper explicitly states that lossless online rounding is impossible for kRental-Variable, and that the limited-correlation construction is the mechanism by which the integral algorithm preserves correct marginals while controlling the dependence induced by reusable capacity (Nekouyan et al., 25 Jul 2025). This impossibility result is central to the form of the final guarantee: the competitive ratio is logarithmic in $(a_j,d_j)$7 up to a constant factor, and that constant factor is tied to the rounding stage rather than to the fractional pricing rule alone.