Order-Sensitive Sequential Interventions on Ideal Lattices

When a student must learn concepts connected by prerequisite dependencies, when does the order of instruction matter, and what does it cost to find the best one? We study instructional sequencing as a stochastic shortest-path problem in which attempting a concept succeeds with a state-dependent probability and failure leaves the learner state unchanged. We first prove that this stochasticity can be eliminated exactly: the problem collapses to a deterministic shortest-path problem on the lattice of prerequisite order ideals, preserving optimal values and actions. The collapse removes stochastic complexity but not combinatorial complexity: optimal sequencing remains NP-hard -- via reduction from feedback arc set in tournaments -- even with no prerequisite edges, unit costs, uniform binary nonnegative transfer, and success probabilities at least $1/2$. Hardness is not uniform: when realizable transfer preferences remain jointly acyclic with the prerequisites, any topological order of the residual joint graph is optimal, and fixed prerequisite width yields polynomial-time exact dynamic programming. A computable diagnostic, $mΔ$, bounds the value of sequencing before optimization. On 70,893 interactions from an introductory CS course, the diagnostic certifies a doubly easy regime -- little value to optimize and little space to search -- while constructed transfer instances realize the challenging regime, where myopic sequencing suffers large regret yet exact A* with a consistent heuristic expands only linearly many states on that family.

Many ranking and agent trace datasets are recorded as linear orders even though their latent structure is only partially ordered. This is especially common in agent and workflow traces, where observed order may reflect arbitrary linearization rather than true prerequisites. We introduce a differentiable relaxation for latent partial-order inference from such traces. Starting from a hard frontier-constrained model of noisy linear extensions, we replace discontinuous product-order precedence and binary frontier feasibility with smooth surrogates, yielding a continuous posterior that preserves closure-level partial-order semantics and supports gradient-based MCMC and variational inference. We prove soft transitivity, sharp-limit frontier recovery, and convergence to the hard likelihood. Experiments on synthetic data, records of social dominance relations, and cloud-agent traces show close posterior fidelity to hard MCMC on small instances and improved runtime--accuracy trade-offs on larger problems.

We study an optimal intervention problem for linear threshold models. This is a popular class of dynamical network systems whereby a number of agents, identified with the nodes of a graph, strategically change their binary action (0 or 1) according to a threshold rule. Specifically, an agent adopts action 1 if and only if the fraction of its neighbors in the interaction graph that do so is greater than or equal to a prescribed threshold. Assuming that a planner can modify the agents' thresholds at a cost equal to the aggregate threshold increase, we study the minimum intervention cost needed to ensure global convergence to the all-1 configuration. Our main contribution is the introduction of a new graph-theoretic quantity, called oriented path number, that is the minimum number of disjoint paths needed to cover the graph that can be oriented to form a directed acyclic graph. When thresholds are all equal to 1/2, the optimal cost is shown to coincide with the oriented path number, whereas, in the general case, it turns out to be the main ingredient of a bound on the optimal intervention cost.