Sample-Based Online Contention Resolution Schemes

• Sample-based OCRSs are online algorithms that use limited sample access to product distributions, ensuring each active element is selected with constant probability under matroid constraints.
• They employ chain-based constructions and greedy rules to iteratively reduce the candidate set, achieving nearly tight sample complexity bounds with a selectability of about 1/4 - ε.
• This approach bridges online rounding and prophet inequalities by balancing estimation accuracy and adversarial robustness in sample-based settings for matroids.

Sample-based online contention resolution schemes (OCRSs) are online rounding procedures that replace full distributional knowledge with sample access. In the standard product-distribution model, one starts from an unknown vector $x$ in a feasibility polytope, draws an active set $R(x)$ by including each element independently with marginal $x_e$, and seeks an online algorithm that selects a feasible subset while preserving each active element’s chance of survival up to a constant factor. In the sample-based variant, the algorithm does not know $x$; it receives only a bounded number of independent samples from $\mathcal D(x)$ before the online process begins, and is then evaluated by selectability against an adversarial arrival order. For matroid constraints, this perspective has recently yielded nearly tight sample complexity bounds and, through the standard OCRS-to-prophet-inequality reduction, sample-based matroid prophet inequalities with the best known competitiveness against an almighty adversary (Feldman et al., 13 Jul 2025).

1. Formal setting and performance criteria

Let $M=(N,\mathcal I)$ be a matroid with rank

$\rho := \rank(M)=\max\{|I|: I\in\mathcal I\}.$

For $x\in[0,1]^N$ in the matroid polytope $P_M$, the associated product distribution $\mathcal D(x)$ is defined by drawing a random active set $R(x)$0 with

$R(x)$1

independently for all $R(x)$2. An OCRS observes the active elements one by one in an adversarial order and must irrevocably accept or reject each active element while maintaining feasibility, so the selected set $R(x)$3 always lies in $R(x)$4. Its standard guarantee is $R(x)$5-selectability: $R(x)$6 This probability is over both the random active set and the algorithm’s internal randomness, and in the online model the minimum is taken over worst-case arrival orders (Feldman et al., 13 Jul 2025).

The sample-based model keeps the same online objective but removes knowledge of $R(x)$7. The algorithm is given only sample access to $R(x)$8: before the online phase it may draw a bounded number $R(x)$9 of independent samples $x_e$0, possibly adaptively, but it never sees $x_e$1 itself. In the matroid setting studied in recent work, the algorithm knows $x_e$2, an upper bound $x_e$3 on the rank, and parameters such as $x_e$4, and must remain $x_e$5-selectable against an almighty adversary that sees both the realization $x_e$6 and the algorithm’s random bits before choosing the arrival order (Feldman et al., 13 Jul 2025).

The central optimization criterion is sample complexity: the smallest number of samples per random variable sufficient to obtain constant selectability. This criterion is distinct from computational complexity. It measures how much prior distributional information must be extracted from data before the online contention-resolution step can reproduce the structural guarantees traditionally available only in the full-information regime (Feldman et al., 13 Jul 2025).

2. Chain-based structure for matroid OCRSs

A central combinatorial device for matroid OCRSs is the spanning chain

$x_e$7

whose links partition the ground set into layers $x_e$8. Given such a chain, the Feldman–Svensson–Zenklusen greedy rule processes an arriving active element $x_e$9 inside the matroid obtained by restricting to $x$0 and contracting $x$1; $x$2 is accepted iff adding it preserves independence in that contracted matroid. This always yields an independent set, because the layers are handled in mutually compatible contracted subproblems (Feldman et al., 13 Jul 2025).

Analysis is organized through the $x$3-freeness of an element. If $x$4, its freeness is

$x$5

Operationally, this is the probability that even if every other active element in the same or lower-priority links arrived before $x$6, the greedy rule could still accept $x$7. A chain is $x$8-balanced for $x$9 if every element has freeness at least $\mathcal D(x)$0; a distribution over chains is $\mathcal D(x)$1-balanced if the expected freeness of every element is at least $\mathcal D(x)$2 (Feldman et al., 13 Jul 2025).

This chain viewpoint is the bridge between combinatorial structure and online rounding. In the full-information setting, Feldman–Svensson–Zenklusen construct balanced chains by iteratively removing elements that are likely to be spanned. In the sample-based setting, the object of interest is a sample-based chain oracle: an algorithm that, using only sample access to $\mathcal D(x)$3, outputs a random spanning chain whose distribution is balanced for an appropriate scaling of $\mathcal D(x)$4 (Feldman et al., 13 Jul 2025).

The structural reduction from balanced chains to OCRSs is explicit. If there exists a distribution over chains that is $\mathcal D(x)$5-balanced for $\mathcal D(x)$6, then one can implement a $\mathcal D(x)$7-selectable OCRS for $\mathcal D(x)$8 without additional access to $\mathcal D(x)$9. This reduction is what turns the sample-based chain oracle into a full sample-based matroid OCRS (Feldman et al., 13 Jul 2025).

3. Nearly tight sample complexity for matroids

The current benchmark result for matroid constraints states that for every $M=(N,\mathcal I)$0 there is a polynomial-time sample-based randomized OCRS for any matroid $M=(N,\mathcal I)$1 with selectability $M=(N,\mathcal I)$2 against the almighty adversary, using

$M=(N,\mathcal I)$3

samples from $M=(N,\mathcal I)$4 (Feldman et al., 13 Jul 2025). Equivalently, the underlying chain oracle outputs a random spanning chain whose distribution is $M=(N,\mathcal I)$5-balanced for any $M=(N,\mathcal I)$6 with $M=(N,\mathcal I)$7, and combining this with the chain-to-OCRS theorem yields a $M=(N,\mathcal I)$8-selectable OCRS. Taking $M=(N,\mathcal I)$9 and adjusting constants gives the $\rho := \rank(M)=\max\{|I|: I\in\mathcal I\}.$0 guarantee (Feldman et al., 13 Jul 2025).

A concise comparison of the current bounds is as follows.

Result	Samples per variable	Guarantee
Fu et al. (2024) sample-based matroid OCRS	$\rho := \rank(M)=\max\{|I|: I\in\mathcal I\}.$1	$\rho := \rank(M)=\max\{|I|: I\in\mathcal I\}.$2-selectable
Nearly tight matroid sample-based OCRS	$\rho := \rank(M)=\max\{|I|: I\in\mathcal I\}.$3	$\rho := \rank(M)=\max\{|I|: I\in\mathcal I\}.$4-selectable
Lower bound for constant selectability	$\rho := \rank(M)=\max\{|I|: I\in\mathcal I\}.$5	necessary even offline

The improvement is twofold. First, the dependence shifts from $\rho := \rank(M)=\max\{|I|: I\in\mathcal I\}.$6, the number of random variables, to $\rho := \rank(M)=\max\{|I|: I\in\mathcal I\}.$7, the matroid rank, which may be much smaller. Second, the upper bound nearly matches the known lower bound up to a $\rho := \rank(M)=\max\{|I|: I\in\mathcal I\}.$8 factor. The competitiveness is unchanged: $\rho := \rank(M)=\max\{|I|: I\in\mathcal I\}.$9 remains the best known constant in the considered almighty-adversary setting even when the distributions are fully known (Feldman et al., 13 Jul 2025).

The balanced-chain statement used to obtain this result is explicit. If

$x\in[0,1]^N$0

with $x\in[0,1]^N$1, $x\in[0,1]^N$2, and $x\in[0,1]^N$3, then the distribution of $x\in[0,1]^N$4 is a $x\in[0,1]^N$5-balanced spanning-chain distribution for $x\in[0,1]^N$6. The final OCRS therefore inherits constant selectability from balanced freeness rather than from direct estimation of per-element activation probabilities (Feldman et al., 13 Jul 2025).

4. Sample-based chain construction

The technical obstacle in adapting the Feldman–Svensson–Zenklusen chain construction to the sample-based setting is that the full-information algorithm repeatedly tests probabilities of spanning events of the form

$x\in[0,1]^N$7

A naive empirical implementation would estimate too many probabilities: a single link may require up to $x\in[0,1]^N$8 iterations before stabilization, and a long chain compounds both sample complexity and estimation error. The recent nearly tight construction addresses both issues simultaneously by bounding the chain length, truncating link construction at a random time, and controlling estimation error with Chernoff bounds (Feldman et al., 13 Jul 2025).

The single-link subroutine $x\in[0,1]^N$9 proceeds by maintaining sets $P_M$0. It chooses an iteration cap $P_M$1 from a carefully designed distribution on $P_M$2, where

$P_M$3

and in each iteration draws

$P_M$4

samples from $P_M$5. Using empirical frequencies, it adds to $P_M$6 those elements whose estimated spanning probability exceeds the threshold. The crucial asymmetry in the stopping distribution is that for $P_M$7,

$P_M$8

while $P_M$9 is tiny. This ensures that the probability of stopping exactly at the unique iteration where an element is still “bad” is smaller by an $\mathcal D(x)$0 factor than the probability of stopping earlier, when the same element is already “good” (Feldman et al., 13 Jul 2025).

The analysis formalizes this via the notions of $\mathcal D(x)$1-bad and $\mathcal D(x)$2-good. An element $\mathcal D(x)$3 is $\mathcal D(x)$4-bad if $\mathcal D(x)$5, and $\mathcal D(x)$6-good otherwise. If $\mathcal D(x)$7 is the output of

$\mathcal D(x)$8

then for every $\mathcal D(x)$9,

$R(x)$00

This per-link good-versus-bad dominance is the probabilistic core of the balanced-chain argument (Feldman et al., 13 Jul 2025).

Progress of the chain is controlled through rank shrinkage. If $R(x)$01, $R(x)$02, and $R(x)$03 is the output of $R(x)$04, then

$R(x)$05

With $R(x)$06, this becomes a strict multiplicative contraction by a factor below $R(x)$07. Iterating the link procedure on restricted matroids for

$R(x)$08

links yields

$R(x)$09

and the balancedness analysis then upgrades this empty-last-link property to a $R(x)$10-balanced chain distribution (Feldman et al., 13 Jul 2025).

The sample complexity follows by multiplication of the structural parameters: $R(x)$11 links, each with $R(x)$12 iterations and $R(x)$13 samples per iteration, giving

$R(x)$14

samples overall. This is the source of the nearly tight bound (Feldman et al., 13 Jul 2025).

5. Prophet inequalities, lower bounds, and information limits

Sample-based OCRSs are tightly coupled to sample-based prophet inequalities. The standard reduction chooses thresholds $R(x)$15, declares an element active when $R(x)$16, arranges the induced activity probabilities as a point in the matroid polytope, and then runs an OCRS on the active set. Fu et al. extended this reduction to the sample-based regime: a $R(x)$17-selectable sample-based matroid OCRS using $R(x)$18 samples per element implies a $R(x)$19-competitive sample-based matroid prophet inequality using

$R(x)$20

samples per element. Applying this reduction to the nearly tight matroid OCRS yields a sample-based matroid prophet inequality with sample complexity

$R(x)$21

and competitive ratio $R(x)$22 against an almighty adversary (Feldman et al., 13 Jul 2025).

The lower-bound landscape is equally important. Fu et al. proved that even for offline contention resolution under matroid constraints, any algorithm guaranteeing constant selectability must use at least $R(x)$23 samples per variable on some instances, and in their construction $R(x)$24 is essentially $R(x)$25. Consequently, $R(x)$26 samples are necessary for constant-factor contention resolution in the sample-based matroid model. The upper bound $R(x)$27 is therefore nearly tight (Feldman et al., 13 Jul 2025).

More severe impossibility results appear in broader information models. For graphic and transversal matroids, no CRS or OCRS with $R(x)$28 samples can be $R(x)$29-balanced or selectable; in particular, constant-sample information is insufficient to preserve every active candidate with constant probability in those classes (Fu et al., 2021). In a different direction, for arbitrary correlated distributions over subsets, no finite-sample prior-independent universal CRS exists even in the rank-1 setting; the barrier already appears offline and is stronger than the product-distribution matroid model typically used in sample-based prophet inequalities (Dughmi, 2019).

These results delimit what “sample-based OCRS” can mean. For product distributions over matroid constraints, polylogarithmic sample complexity is achievable and nearly optimal. For universal schemes over broader distribution families, or for richer classes under constant-sample information, constant-factor guarantees may be impossible (Feldman et al., 13 Jul 2025).

6. Variants, adversaries, and broader OCRS context

The behavior of OCRSs is highly sensitive to the adversary model. In $R(x)$30-uniform matroids, a simple full-information OCRS achieves $R(x)$31-selectability against a fixed-order adversary, whereas against an almighty adversary no OCRS can be $R(x)$32-selectable; the known greedy algorithm matches this upper bound asymptotically (Dinev et al., 2023). This separation is directly relevant to sample-based OCRS results for general matroids: the current $R(x)$33 competitiveness is notable precisely because it holds against the strongest adversary considered in the literature (Feldman et al., 13 Jul 2025).

A second axis of comparison is the amount of structural commitment made before arrivals. Greedy OCRSs, which pre-sample a down-closed feasible family and then accept every active element that preserves membership in that family, achieve $R(x)$34-selectability for the single-item setting, partition matroids, and transversal matroids, and this is optimal within the greedy class even in the rank-1 case (Livanos, 2021). Oblivious schemes in the single-item setting also achieve the same $R(x)$35 constant, and that constant is optimal for oblivious OCRSs (Fu et al., 2021). These results do not yield sample-complexity bounds for general matroids, but they clarify the interaction between information, precommitment, and adversarial robustness.

Beyond product distributions, contention resolution has a strong connection to arbitrary correlated priors and to secretary-type problems. Offline contention resolution for matroids under correlated input distributions is characterized by weighted-rank inequalities, and universal random-order contention resolution is intimately tied to the matroid secretary problem (Dughmi, 2019). In fact, the matroid secretary conjecture is equivalent to the existence of universal random-order contention resolution schemes that match the offline balance ratio up to a constant factor for every uncontentious correlated distribution (Dughmi, 2021). This does not directly concern the product-distribution sample-based model of recent matroid OCRS work, but it places sample-based OCRS within a larger program: understanding exactly which kinds of distributional information are necessary for online contention resolution, and which can be learned or inferred from limited access.

For product distributions under matroid constraints, the current picture is unusually sharp. Sample-based chain construction reduces the required information from $R(x)$36 samples per variable to $R(x)$37, almost matching the $R(x)$38 lower bound, while preserving the best known $R(x)$39 guarantee against an almighty adversary (Feldman et al., 13 Jul 2025). The main open questions are equally explicit: close the remaining $R(x)$40 gap to a tight $R(x)$41, improve the constants in both selectability and sample complexity, and determine whether matroid prophet inequalities can achieve comparable sample complexity without routing the construction through OCRSs (Feldman et al., 13 Jul 2025).