Interval-Based, Learning-Augmented Scheduling

• The paper introduces a learning-augmented framework that integrates predictions into online interval scheduling, achieving a balance between optimality and robustness.
• The methodology rigorously quantifies prediction errors using normalized metrics and competitive ratios to navigate the consistency–robustness trade-off.
• Empirical analyses on HPC workloads demonstrate that schemes like Trust-and-Greedy sustain near-optimal performance even with moderate prediction noise.

Interval-based, learning-augmented scheduling combines classical online interval scheduling with predictions, typically supplied by a learning algorithm or external oracle, to improve performance in settings where future requests are uncertain. The framework is motivated by scenarios where anticipatory information, possibly error-prone, can be incorporated while retaining robustness guarantees. Recent advances rigorously analyze the impact of prediction errors and design algorithms that interpolate between optimality under perfect prediction and worst-case guarantees against adversarial inputs (Boyar et al., 2023).

1. Formal Problem Definition

The online interval scheduling problem on a single machine, or equivalently a path graph of length $m$, receives as input an online sequence %%%%1%%%% where each $i = (r_i, d_i)$ is an interval with integer release time $r_i$ and deadline $d_i > r_i$. Upon presentation, each interval must be irrevocably accepted or rejected, subject to the constraint that accepted intervals are pairwise non-overlapping (touching at endpoints is allowed). The offline optimum is

$\mathrm{OPT}(I) = \max\{\,|S|:\ S\subseteq I,\ \text{$S$ is pairwise non-overlapping}\,\}.$

In the learning-augmented variant, a prediction $P \subseteq U$ (with $U$ the set of all possible intervals) is provided before input begins. Prediction errors take two forms:

• False positives: $P \setminus I$ (predicted intervals never arriving);
• False negatives: $I \setminus P$ (unpredicted intervals that do arrive).

The size of the prediction error is

$$\eta(P, I) = \mathrm{OPT}\big( (P \setminus I) \cup (I \setminus P) \big),$$

measuring the largest feasible set from incorrectly predicted intervals. The normalized error is $\gamma(P,I) = \eta(P,I) / \mathrm{OPT}(I)$, ranging in $[0,1]$.

2. Performance Metrics and Consistency–Robustness Trade-off

Algorithmic performance is quantified by the competitive ratio as a function of the prediction error. For an algorithm $A$ and prediction error $\varepsilon$,

$$\mathrm{CR}_A(\varepsilon) = \inf_{(P,I) : \eta(P,I) = \varepsilon} \frac{A(P, I)}{\mathrm{OPT}(I)},$$

where $A(P, I)$ is the set size accepted by $A$ on input $I$ given prediction $P$. Two principal benchmarks arise:

• Consistency: $\mathrm{CR}_A(0)$, i.e., competitive ratio with perfect predictions.
• Robustness: $$\liminf_{\varepsilon \to \infty} \mathrm{CR}_A(\varepsilon)$$, i.e., performance when predictions are essentially adversarial.

A central objective is to design algorithms parametrized to navigate the achievable trade-off between $(\alpha, \beta)$: consistency $\alpha$ and robustness $\beta$.

3. Algorithmic Strategies and Theoretical Guarantees

Several algorithms exemplify the spectrum of approaches:

Summary of Algorithms

Algorithm	Competitive Ratio Bound	Key Property
Trust	$\geq 1 - 2\gamma$	Simple; follows prediction
Trust-and-Greedy (TG)	$\geq 1 - \gamma$	Matches best-possible
Level-based	$O(1/\log m)$ competitive w/o predictions	Classical robust baseline
RobustTrust$_\alpha$	Consistency $\geq\alpha$, Robustness $\geq(1-\alpha)/\lceil \log m \rceil$	Mixture of TG and level-based

Trust Algorithm:

Computes $\mathrm{OPT}(P)$ and accepts future arrivals $i \in P \cap I$ that fit into this offline plan; rejects everything else. This yields $$A_\text{Trust}(P, I) \geq \mathrm{OPT}(I) - 2\eta(P, I)$$, so $\mathrm{CR}_\text{Trust}(\gamma) \geq 1 - 2\gamma$ (Theorem 5). Instances exist matching this bound.

Trust-and-Greedy (TG) Algorithm:

Initializes an evolving plan $A = \mathrm{OPT}(P)$. Upon arrival of interval $i$:

• If $i \notin P$, immediately reject.
• Else, if $i$ does not overlap already accepted intervals and can replace at most one interval $j \in A$ (not yet accepted, overlapping $i$, $j$ ends no earlier than $i$), accept $i$ and, if needed, replace $j$ in $A$ with $i$; otherwise reject.

TG achieves $A_\text{TG}(P,I) \geq \mathrm{OPT}(I) - \eta(P,I)$, thus $\mathrm{CR}_\text{TG}(\gamma) \geq 1-\gamma$, which is optimal for deterministic algorithms (Theorem 14).

Lower Bounds:

Any deterministic algorithm $A$ satisfies $$\mathrm{CR}_A(\epsilon) \leq 1 - \epsilon/\mathrm{OPT}(I)$$ (Theorem 11); TG achieves this bound.

Randomized Consistency–Robustness Pareto Frontier:

Writing $r = \lfloor \log m\rfloor - 1$, any (randomized) algorithm with consistency $\alpha$ and robustness $\beta$ must satisfy $\alpha + \frac{1}{2} r \beta \leq 1$ (Theorem 17). A mixture, dubbed RobustTrust$_\alpha$, runs TG with probability $\alpha$ and the level-based algorithm otherwise, achieving consistency $\geq \alpha$ and robustness $\geq (1-\alpha)/\lceil\log m \rceil$.

4. Empirical Analysis on Real-World Data

Extensive validation employs four HPC traces: LLNL-uBGL-2006, NASA-iPSC-1993, CTC-SP2-1996, and SDSC-DS-2004, filtered to create interval-scheduling instances. For each workload, a random half-sample of $N$ intervals forms the online sequence $I$, and predictions $P$ are formed by adding/removing $d$ intervals, varying $d$ from $0$ to $n = \lfloor N/2 \rfloor$. Normalized error $\gamma$ and payoff ratio $\mathrm{payoff}(A) / \mathrm{OPT}(I)$ are measured as a function of $\gamma$.

Findings:

• TG sustains near-optimal performance for $\gamma \lesssim 1.5$–$2.0$.
• Trust's ratio degrades linearly and falls rapidly below TG as $\gamma$ increases.
• TG outperforms Trust for all $\gamma>0$ even in heavy-overlap scenarios (e.g., SDSC).
• TG also dominates Trust and naïve greedy whenever either false positives or false negatives are absent.

5. Properties of the Error Measures

The error metric $\eta(P, I)$ exhibits desirable algebraic properties:

• Lipschitz property: Small changes in prediction do not cause disproportionately large increases in error.
• Monotonicity: Adding redundant ("dummy") intervals to the prediction $P$ does not artificially decrease the measured error.

These ensure that a moderately noisy prediction will not catastrophically degrade algorithmic decisions and that attempts to manipulate error metrics via spurious intervals are ineffective.

6. Practical Guidelines and Domain Implications

Application guidance depends on estimated domain prediction quality. For reliability $\hat\gamma$, the recommended mixture sets $\alpha \approx 1 - \hat\gamma$, yielding consistency near $1$ and robustness $\approx \hat\gamma / \lfloor \log m \rfloor$. In typical practice, TG alone suffices for $\hat\gamma$ up to approximately 0.4. For noisier predictions ($\hat\gamma > 0.5$), a gradual shift to the classical $O(\log m)$-competitive approach is warranted.

A plausible implication is that in practical deployments, as long as prediction quality is moderate or better, learning-augmented strategies such as TG robustly outperform both "trust-only" and non-predictive algorithms, gracefully interpolating between the empirical benefits of predictions and worst-case guarantees as prediction quality varies (Boyar et al., 2023).