d4-conf-atal-2014-0033

The paper has a plausible continuous mirror, but it has no named computational-complexity result of the kind your anchor rule requires. Its only numbered result, Theorem 1, proves that the allocation procedure in Algorithms 4 and 5 is monotonic, assuming agents do not misreport by arriving early or departing late; the proof adapts an argument from [12]. That is a mechanism-design property, not a result about tractability or hardness. The closest complexity claim is unnumbered prose in Section 3.1: the offline problem with known jobs and supply is computationally hard because it generalizes NP-hard parallel-machine scheduling, citing [9]. The paper does not prove that claim as a numbered result. Its other algorithms and efficiency findings are pseudocode and empirical results, not named complexity results. So there is no qualifying anchor to restate, and the candidate below cannot count as a per-anchor answer under your rules.

The most convincing regime is a demand-response programme with millions of households enrolled in standard appliance schemes. A household’s job type records its value for running the appliance, power use, duration, and allowed start window. Households in the same programme cohort share those parameters; there might be millions of jobs but only tens or hundreds of distinct types. Let each type’s mass be its fraction of participating households. Supply and conventional backup capacity are expressed per household too. A mass of \(0.1\) of one type started at 6 p.m. means that fraction of households runs its full, non-preemptive job then—not that any individual runs a fraction of a job. Standard contracts and appliance programmes make the shared-value assumption plausible, though the paper’s strategic, privately valued agents make that assumption less comfortable.

One candidate problem is Finite-Scenario Continuous Online Scheduling. An instance gives a finite horizon, a finite set of job types \(\theta=(v_\theta,r_\theta,l_\theta,a_\theta,d_\theta)\), and an explicit scenario tree. Each tree node reveals the per-capita mass \(A_{u\theta}\) of newly arriving jobs of each type and renewable supply \(x_u\); branches have rational probabilities. Conventional supply is unlimited at linear cost \(b>0\) per unit. The question is to find a nonanticipative policy: at each node \(u\), choose how much available mass \(z_{u\theta}\) of each type to start. Each job must start within its arrival and deadline window, may be started at most once, and, once started, consumes \(r_\theta\) for \(l_\theta\) consecutive periods. The policy maximizes expected per-capita welfare: value from jobs started minus conventional-generation cost over all periods. A decision version asks whether a policy with expected welfare at least \(K\) exists.

For an explicitly listed scenario tree, that aggregate problem is an LP: mass availability and deadlines give linear constraints, and auxiliary variables can represent conventional generation above renewable supply. It should therefore be in Class A, with running time polynomial in the tree size, the number of types \(\tau\), the horizon, and the rational input length. This is a natural high-multiplicity version of the paper’s underlying online welfare problem: it retains uncertainty, deadlines, values, fixed job profiles, and non-preemption for every unit of mass. It does change the computational picture by allowing the aggregate to divide across start times. That is the point of high multiplicity, but it is also why the mirror needs careful scrutiny.

The proposed question covers optimal scheduling under an explicit finite scenario representation, not the paper’s particular consensus and expectation heuristics, their experimental \(85\%\)-plus efficiency, or the incentive-compatible mechanism. For a compactly represented stochastic process such as the paper’s hidden Markov model, the explicit scenario tree may be exponentially large; whether its structure permits an efficient algorithm is a further question, not established by this LP observation. Other natural questions are how accurately a discrete high-multiplicity instance can be approximated from its mass solution, and whether any useful analogue of the paper’s incentive guarantees survives in an atomless population.

My weakest point is therefore twofold: the mirror has no qualifying numbered result in this paper to anchor it, and its clean tractability claim depends on explicitly representing uncertainty. The scenario itself is credible, and the offline hardness claim makes the prospect of a continuous tractability gain worth investigating, but that claim is cited from elsewhere and does not meet the requested anchor standard.