Online Interval Scheduling with Predictions

TL;DR

This work analyzes online interval scheduling and disjoint-path allocation with predictions, introducing a rigorous prediction-error framework via $\eta$ and $\gamma$ that governs algorithmic performance. It develops the Trust family of algorithms, including the optimal $1-\gamma$-competitive TrustGreedy for interval scheduling and the strictly $1-2\gamma$-competitive Trust for general disjoint-path problems, along with tight lower bounds. The paper also explores consistency-robustness trade-offs, presenting general upper bounds and a RobustTrust randomized scheme that achieves near Pareto-optimal behavior, and validates the theory with experiments on real scheduling workloads. Finally, it shows that the results extend to related problems like online matching and independent set on line graphs, via reductions from star-like constructions.

Abstract

In online interval scheduling, the input is an online sequence of intervals, and the goal is to accept a maximum number of non-overlapping intervals. In the more general disjoint path allocation problem, the input is a sequence of requests, each consisting of pairs of vertices of a known graph, and the goal is to accept a maximum number of requests forming edge-disjoint paths between accepted pairs. We study a setting with a potentially erroneous prediction specifying the set of requests forming the input sequence and provide tight upper and lower bounds on the competitive ratios of online algorithms as a function of the prediction error. We also present asymptotically tight trade-offs between consistency (competitive ratio with error-free predictions) and robustness (competitive ratio with adversarial predictions) of interval scheduling algorithms. Finally, we provide experimental results on real-world scheduling workloads that confirm our theoretical analysis.

Figures (5)

• Figure 1: An illustration of Example \ref{['ex:completeness']}. All requests appear in both $I$ and $\operatorname{\mathit{\hat{I}}}\xspace$, except $\{A_1,A_2\}$, which are false negatives.
• Figure 3: An illustration of $\operatorname{\textsc{TrustGreedy}}$: Suppose $I = \langle S_1, S_2, S_3, S_4, S_5, S_7 \rangle$ and $\operatorname{\mathit{\hat{I}}}\xspace = (I \setminus {S_1}) \cup {S_6}$, so that $S_1$ is a false negative and $S_6$ is a false positive. The optimal solution for $\operatorname{\mathit{\hat{I}}}\xspace$ constructed by $\operatorname{\textsc{TrustGreedy}}$ is $I^* = \{S_2, S_5, S_6\}$. Initially, the set $A$ is initialized as $I^*$. Upon the arrival of $S_1$, it replaces $S_2$, and $A$ becomes $\{S_1, S_5, S_6\}$. Eventually, the algorithm accepts $\{S_1, S_5\}$.
• Figure 4: Profit as a function of normalized error value
• Figure 5: Profit as a function of normalized error value in the absence of false positives (a), (c) and false negatives (b), (d).
• Figure 6: Graphs for matching and independent set

Theorems & Definitions (16)

• Example 1
• Proposition 1
• Theorem 1
• Theorem 2
• Corollary 1
• Theorem 3
• Theorem 4
• Lemma 1
• Lemma 2
• Lemma 3