Dynamic Interval Scheduling with Random Start and End Times

We study sequential interval scheduling when task start and end times are random. The set of tasks and their weights are known in advance, while each task’s start and end times are drawn from known discrete distributions and revealed only upon commitment; this also eliminates tasks that conflict with the committed task, and remaining tasks are those that do not conflict. The objective is to maximize the expected weight of a conflict-free schedule. We propose two models that differ in how conflicts are enforced, develop LP relaxations and bounds for each, and present a computational study.

💡 Research Summary

The paper introduces a novel stochastic extension of the classic interval scheduling problem, where each task’s start and end times are not fixed but drawn from known discrete probability distributions. The set of tasks and their weights are known a priori, but the actual intervals become revealed only when a task is committed to. Upon committing to a task, its realized interval may conflict with the remaining tasks; the way conflicts are handled gives rise to two distinct models.

The first model, Dynamic Scheduling with Random Start and End times (DSRSE), treats conflicts probabilistically. After a task i is scheduled and its interval (s_i, e_i) is realized, each remaining task j computes a conflict probability p_{ij} based on the distributions of both tasks. A Bernoulli trial B_{ij}∼Bernoulli(p_{ij}) decides whether j is eliminated (B_{ij}=1) or stays in the pool (B_{ij}=0). The process repeats until no tasks remain, and the objective is to maximize the expected total weight of the selected tasks.

The second model, Conservative DSRSE (CDSRSE), adopts a more pessimistic rule: if any slot in the support of a remaining task overlaps with a slot already occupied by a committed task, the remaining task is deleted outright, regardless of the actual realized interval. This eliminates the need to update distributions after each decision, simplifying the state space.

Both models are shown to be NP‑hard by reduction from the dynamic maximum stable set problem on a star graph. The authors then develop linear‑programming (LP) relaxations for each model. For DSRSE they introduce stage‑indexed variables x^{t}{s,ik}, x^{t}{e,ik}, and x^{t}_{o,ik} representing the probability that task i starts, ends, or occupies slot k at stage t. The objective becomes Σ_t Σ_i w_i x^{t}i, where x^{t}i = Σ_k x^{t}{s,ik} = Σ_k x^{t}{e,ik}. The key feasibility constraint is that the total probability of any slot being occupied at any stage does not exceed one. This leads to a fairly intricate set of constraints (equations (1) and (2) in the paper) that capture the relationship between start, end, and occupancy probabilities. The resulting primal LP (DLP‑P) and its dual (DLP‑D) form a primal‑dual pair that generalizes the classic clique‑cover formulation for interval graphs.

For CDSRSE the distribution updates disappear, allowing the occupancy constraint to be expressed simply as Σ_i P^{o}{ik} x_i ≤ 1 for each slot k, where P^{o}{ik} is the marginal probability that task i occupies slot k. The corresponding LP (CDLP‑P) and dual (CDLP‑D) are much more compact. By defining p* = min_{i,r : P^{o}{ir}>0} P^{o}{ir}, the authors obtain the bound α_{pes} ≤ OPT ≤ α_{pes}/p*, where α_{pes} is the optimal value of the minimum clique‑cover LP on the “pessimistic” interval graph G_{pes} (each task occupies its widest possible interval). They acknowledge that this bound can be weak when p* is small.

The paper also provides several structural insights. When every task occupies a single slot (s_i = e_i), a simple greedy algorithm that always picks the highest‑weight remaining task is optimal. For the uniform‑weight case, they show that the optimal schedule can be related to a minimum clique cover J of size α_{pes}, and derive tighter additive bounds involving the probabilities of non‑occupancy (1‑P^{o}_{ik}).

A computational study evaluates the quality of the LP relaxations. Instances are generated for various (n,m) pairs ranging from (8,12) up to (80,120). For each task, start and end supports are sampled uniformly, and the actual start/end times are drawn from uniform distributions over those supports; weights are drawn uniformly from