kRental-Fixed: Online Rental Allocation Model

• kRental-Fixed is an adversarial online resource allocation model where a decision-maker manages k reusable units, accepting or rejecting fixed-duration rental requests.
• The methodology employs a relax-and-round approach with a lossless online rounding (loRounding) scheme that preserves allocation probabilities under fixed duration.
• It integrates a primal–dual pricing algorithm, dop-$, to optimize rental decisions and achieve a competitive ratio of 1+ln(v_max/v_min) relative to the offline optimum.

Searching arXiv for the cited kRental-Fixed and related rental papers to ground the article. arXiv Search Query: id:(Nekouyan et al., 25 Jul 2025) arXiv Search Query: id:(Khare et al., 2012) arXiv Search Query: id:(Kim et al., 31 Mar 2026) kRental-Fixed is an adversarial online resource-allocation problem with reusability in which a decision-maker manages $k$ identical reusable units, faces a sequence of online rental requests, and must accept or reject each request irrevocably upon arrival. Each accepted request occupies one unit for a fixed, common duration $d$, after which the unit becomes reusable again; the objective is to maximize the total value of accepted requests. In the terminology of "Online Rounding Schemes for $k$-Rental Problems" (Nekouyan et al., 25 Jul 2025), kRental-Fixed is the fixed-duration member of a two-problem family, and its algorithmic analysis is built around a relax-and-round architecture with a lossless online rounding procedure.

1. Formal model and optimization objective

The kRental-Fixed instance is

$I = \{(v_n, a_n)\}_{n \in [N]},$

where request $n$ arrives at time $a_n$ and has value $v_n \in [v_{\min}, v_{\max}]$. Each request asks to rent exactly one unit for the same fixed duration $d$. If accepted, request $n$ occupies one unit on the interval from $a_n$ to $d$0; if rejected, it contributes no value. The algorithm learns $d$1 only at time $d$2 and must choose $d$3 irreversibly.

The offline clairvoyant benchmark is the integer program

$d$4

subject to

$d$5

and

$d$6

The overlap constraint states that, at each arrival time $d$7, the number of accepted intervals still active is at most $d$8. Because the worst load happens at arrivals, it suffices to enforce capacity at each $d$9 (Nekouyan et al., 25 Jul 2025).

The online performance criterion is competitive analysis. An algorithm ALG is $k$0-competitive if

$k$1

for every instance $k$2. Equivalently, ALG achieves at least a $k$3 fraction of the offline optimum on every adversarial input. The bounded-value assumption

$k$4

is part of the formal model and rules out trivial impossibility.

2. Fractional relaxation and the ocr abstraction

The 2025 treatment isolates a general online rounding primitive for reusable resources with fixed duration, called Online Correlated $k$5-Rental (ocr) (Nekouyan et al., 25 Jul 2025). In ocr, player $k$6 arrives with a pair $k$7, where $k$8 is a target assignment probability. The fractional inputs must satisfy

$k$9

This is the fractional capacity condition at arrival times: the new fractional allocation cannot exceed either one full unit or the remaining expected capacity.

For $I = \{(v_n, a_n)\}_{n \in [N]},$0, a $I = \{(v_n, a_n)\}_{n \in [N]},$1-ocr is an online randomized rounding scheme that guarantees allocation probability at least $I = \{(v_n, a_n)\}_{n \in [N]},$2 for every valid input request $I = \{(v_n, a_n)\}_{n \in [N]},$3. The distinction between lossless and lossy rounding is central. A lossless scheme has $I = \{(v_n, a_n)\}_{n \in [N]},$4, so it preserves the target probabilities exactly; generic independent rounding does not achieve this uniformly for all $I = \{(v_n, a_n)\}_{n \in [N]},$5, whereas the fixed-duration setting admits a specially constructed dependent scheme.

This abstraction separates two tasks. First, a fractional algorithm computes feasible targets $I = \{(v_n, a_n)\}_{n \in [N]},$6. Second, ocr converts those targets into an integral schedule without violating reuse constraints. The significance of this factorization is structural: it permits the competitive analysis to proceed through a primal–dual argument for the fractional part while treating the rounding stage as probability-preserving.

3. loRounding and lossless dependent online rounding

The lossless rounding scheme for kRental-Fixed is loRounding, identified as Algorithm 2 in the 2025 paper (Nekouyan et al., 25 Jul 2025). It samples a single random seed

$I = \{(v_n, a_n)\}_{n \in [N]},$7

once at the beginning, initializes $I = \{(v_n, a_n)\}_{n \in [N]},$8 and $I = \{(v_n, a_n)\}_{n \in [N]},$9, and then processes requests online using two pointers $n$0. The algorithm interprets $n$1 as a cyclic probability space and assigns to each request a subinterval of total length exactly $n$2.

If $n$3, the request is associated with the interval $n$4 on ball $n$5; if $n$6 lies in that interval, the request is accepted and assigned ball $n$7. The state then updates to

$n$8

If $n$9, the interval wraps around $a_n$0 and is split into

$a_n$1

with the first part assigned to ball $a_n$2 and the second to ball $a_n$3 modulo $a_n$4. The state becomes

$a_n$5

The crucial invariant is that the intervals currently corresponding to overlapping real-time rentals on the same ball are disjoint. The paper formalizes this through sets $a_n$6 of triples $a_n$7 representing active probability subintervals for ball $a_n$8. Disjointness of these sets implies that, for any realization of $a_n$9, no ball is assigned to two overlapping requests. Consequently, when $v_n \in [v_{\min}, v_{\max}]$0 falls in the newly assigned subinterval for request $v_n \in [v_{\min}, v_{\max}]$1, the corresponding ball is guaranteed to be available.

The correctness theorem states that Algorithm 2 is a $v_n \in [v_{\min}, v_{\max}]$2-ocr: for every valid input sequence satisfying the fractional feasibility condition, player $v_n \in [v_{\min}, v_{\max}]$3 is allocated a ball with probability exactly $v_n \in [v_{\min}, v_{\max}]$4. The argument is immediate once the invariants are established: in either case, the total measure of the acceptance set for request $v_n \in [v_{\min}, v_{\max}]$5 is precisely $v_n \in [v_{\min}, v_{\max}]$6, and $v_n \in [v_{\min}, v_{\max}]$7 is uniform on $v_n \in [v_{\min}, v_{\max}]$8. This losslessness is the technical fulcrum of kRental-Fixed. It is also where the fixed-duration assumption is used most sharply, because the monotone ordering of release times is what permits the cyclic pointer construction.

4. The dop-\$ algorithm and primal–dual pricing

The complete online algorithm for kRental-Fixed is dop-\$, a duration-oblivious price-based method coupled with loRounding (Nekouyan et al., 25 Jul 2025). At arrival $v_n \in [v_{\min}, v_{\max}]$9, it computes the current utilization

$d$0

and, provided $d$1, chooses a fractional allocation by solving

$d$2

The first term is the immediate value of serving a fraction $d$3 of the request; the second is a pseudo-cost for increasing normalized utilization from $d$4 to $d$5, where $d$6 is a marginal price function.

For the optimal guarantee, the pricing function is

$d$7

With this $d$8, the induced rule has the closed-form interpretation

$d$9

so the algorithm allocates up to the point where marginal price equals request value, capped by residual capacity.

The analysis relaxes the offline integer program to $n$0 and studies the dual

$n$1

subject to

$n$2

and

$n$3

The proof constructs a feasible dual using deferred updates. Writing

$n$4

and letting

$n$5

the dual increments are

$n$6

and

$n$7

The resulting comparison shows

$n$8

where $n$9 is the fractional primal value and $a_n$0 is the dual objective. Because loRounding is lossless, the expected integral value equals $a_n$1. The main theorem is therefore

$a_n$2

5. Optimality and the role of fixed duration

The competitive ratio achieved by dop-\$ is optimal. The lower bound states that no online algorithm, deterministic or randomized, can achieve a competitive ratio better than

$a_n$3

for kRental-Fixed under the bounded-value assumption (Nekouyan et al., 25 Jul 2025).

The hard family uses batches $a_n$4 of $a_n$5 identical requests of value $a_n$6, with values drawn from a grid

$a_n$7

where

$a_n$8

The instances are

$a_n$9

and all batches arrive inside a tiny interval $d$00 with $d$01. Because no accepted request can finish before the final batch arrives, no reuse is possible within the window. The problem therefore reduces to an online tradeoff between accepting early low-value requests and reserving capacity for potentially higher-value later requests, and the representative-function analysis yields the logarithmic lower bound.

The same paper also shows why the fixed-duration model is qualitatively special. For the variable-duration analogue, denoted ocr-v, lossless online rounding is impossible. The underlying reason is that variable durations destroy the monotone ordering of return times: a later request may release its unit earlier than an earlier request. loRounding depends on the opposite property, namely that under a common duration $d$02, later allocation implies later release. A common misconception is that the fixed-duration and variable-duration models should differ only quantitatively; the 2025 results show that the distinction is structural, because exact probability preservation is possible in one model and impossible in the other.

6. Relations to adjacent “rental” models and terminological boundaries

kRental-Fixed belongs to a broader online-algorithmic family of rental-style problems, but it is not the same as rental caching. In rental caching, the system maintains a cache of size $d$03, each file $d$04 has size $d$05 and retrieval cost $d$06, and a rental cost $d$07 is charged for each file in the cache at each time step; the objective is to minimize retrieval plus rental cost (Khare et al., 2012). That model is analyzed through online covering LPs with eviction variables $d$08 and rental variables $d$09, leading in the fault model to deterministic competitive ratios of $d$10, $d$11, or $d$12 depending on the regime of $d$13. By contrast, kRental-Fixed is a value-maximization problem over reusable units with fixed service duration. The shared word “rental” therefore marks a broad economic motif rather than a shared mathematical formulation.

The ski-rental literature is methodologically closer. In the distributional-advice framework for ski rental, the unknown horizon $d$14, buy time $d$15, expected competitive ratio, Clamp Policy, and randomized Water-Filling Algorithm are all formulated for single rent-or-buy decisions (Kim et al., 31 Mar 2026). The extracted analysis of that work describes how its threshold and water-filling machinery can be adapted per item in a kRental-Fixed-type generalization. This suggests a learning-augmented direction for reusable-resource models in which distributional advice informs per-unit or per-phase admission decisions, although such a formulation is distinct from the adversarial fixed-duration model of dop-\$.

The term should also be distinguished from the fair-division literature on “rental harmony,” where the problem is to divide rent among rooms and roommates, possibly with one secretive roommate, using Sperner-type arguments on the simplex of price vectors (Frick et al., 2017). There, “rental” concerns envy-free rent division in an $d$16-bedroom apartment, not reusable online resources. The distinction matters because the phrase “kRental-Fixed” may otherwise invite a misleading association with fixed-$d$17 room-allocation problems rather than with fixed-duration reusable-capacity allocation.

Taken together, these comparisons place kRental-Fixed at a specific intersection: it inherits the online adversarial perspective of classical resource-allocation theory, uses relax-and-round and primal–dual methods rather than covering-LP eviction dynamics, and derives its sharpest technical advantage from the combinatorial regularity induced by a common rental duration.