ops: add precedence constraints and an improvement pass

Cargo.lock

@@ -282,6 +282,7 @@ dependencies = [
 name = "apsis-scheduler"
 version = "0.1.0"
 dependencies = [
+ "apsis-conflict",
  "apsis-core",
  "apsis-resource",
  "apsis-schedule",

crates/apsis-rs/examples/integrate_ops.rs

@@ -10,7 +10,8 @@
 //! 1. Define a satellite and a ground station as resources, taking the
 //!    station's availability from access windows.
 //! 2. Define observation tasks that each need the satellite and the station.
-//! 3. Plan the tasks with `apsis-scheduler`.
+//! 3. Plan the tasks with `apsis-scheduler`, using precedence constraints
+//!    to express a task that needs two resources in sequence.
 //! 4. Check the result with `apsis-conflict`.
 //! 5. Evaluate the schedule with `apsis-schedule`.
 //!
@@ -23,8 +24,8 @@
 use apsis_conflict::detect;
 use apsis_resource::{Resource, ResourceId, ResourceSet, Window};
 use apsis_schedule::Schedule;
-use apsis_scheduler::Planner;
+use apsis_scheduler::{ImprovementSettings, Planner};
-use apsis_task::{Task, TaskId};
+use apsis_task::{PrecedenceGraph, Task, TaskId};
 
 /// The satellite resource.
 const SATELLITE: ResourceId = ResourceId::new(0);
@@ -152,6 +153,70 @@ fn main() {
     let again = Planner::new().schedule(&tasks, &resources);
     assert_eq!(result.schedule, again.schedule);
     let _ = Schedule::new();
+    let _ = ImprovementSettings::disabled();
+
+    // Step 6: a task that needs two resources in sequence. An observation
+    // uses the payload, and the downlink that follows uses the transmitter.
+    // The two are separate tasks joined by a precedence constraint, which is
+    // how a sequential resource use is expressed.
+    let payload = ResourceId::new(2);
+    let transmitter = ResourceId::new(3);
+    let mut extra = resources.clone();
+    let mut payload_resource = Resource::new(payload, 1.0);
+    payload_resource
+        .set_windows(&[Window::new(0.0, 5_000.0)])
+        .expect("the window is usable");
+    let mut transmitter_resource = Resource::new(transmitter, 1.0);
+    transmitter_resource
+        .set_windows(&[Window::new(0.0, 5_000.0)])
+        .expect("the window is usable");
+    extra
+        .insert(payload_resource)
+        .expect("the identifier is new");
+    extra
+        .insert(transmitter_resource)
+        .expect("the identifier is new");
+
+    let chain = [
+        Task::new(TaskId::new(100), 120.0)
+            .with_priority(10.0)
+            .with_window(200.0, 1_500.0)
+            .try_require(payload, 1.0)
+            .expect("the payload is not yet required"),
+        Task::new(TaskId::new(101), 90.0)
+            .with_priority(10.0)
+            .with_window(0.0, 2_000.0)
+            .try_require(transmitter, 1.0)
+            .expect("the transmitter is not yet required"),
+    ];
+    let mut graph = PrecedenceGraph::new();
+    graph
+        .add(TaskId::new(100), TaskId::new(101))
+        .expect("the constraint does not close a cycle");
+
+    let chained = Planner::new().schedule_with_precedence(&chain, &extra, &graph);
+    assert!(chained.is_complete(), "{:?}", chained.outcomes);
+    assert!(chained.honours_precedence(&graph));
+    let collection = chained
+        .schedule
+        .placement_of(TaskId::new(100))
+        .expect("the collection is placed");
+    let transmission = chained
+        .schedule
+        .placement_of(TaskId::new(101))
+        .expect("the transmission is placed");
+    println!(
+        "sequence: collect {:.0}-{:.0} s, downlink {:.0}-{:.0} s",
+        collection.start,
+        collection.end(),
+        transmission.start,
+        transmission.end()
+    );
+    assert!(
+        transmission.start >= collection.end(),
+        "the downlink must follow the collection"
+    );
+    assert!(apsis_conflict::detect(&chained.schedule, &chain, &extra).is_clear());
 
     println!("integration check passed");
 }

crates/ops/apsis-scheduler/Cargo.toml

@@ -15,3 +15,6 @@ apsis-core.workspace = true
 apsis-resource.workspace = true
 apsis-task.workspace = true
 apsis-schedule.workspace = true
+
+[dev-dependencies]
+apsis-conflict.workspace = true

crates/ops/apsis-scheduler/src/improve.rs

@@ -0,0 +1,255 @@
+//! A second pass that tries to place what the greedy first pass left out.
+//!
+//! The greedy pass places each task at the earliest time it fits and never
+//! moves a placement once it is made, so a task that does not fit is lost
+//! even when moving a placement would make room. This pass addresses the
+//! cheapest case of that problem.
+//!
+//! For a task that was left out, the pass looks at the placements that block
+//! it, that is the placements sharing a resource it needs. For each blocker
+//! it tries a relocation of the pair:
+//!
+//! 1. Record where the blocker sits, then remove it.
+//! 2. Find the earliest time at which the waiting task fits without it.
+//! 3. Place the waiting task there, then ask whether the blocker fits
+//!    somewhere else now that it has a new neighbour.
+//! 4. Keep the new arrangement when both fit, and undo it otherwise.
+//!
+//! The check in step 4 is the whole safety argument: the pair is kept only
+//! when both placements pass the same feasibility test the greedy pass uses,
+//! so every accepted relocation leaves the schedule feasible.
+//!
+//! The pass is bounded. Each relocated pair costs two feasibility searches,
+//! and the pair is attempted only a bounded number of times, so the pass
+//! stays linear in the number of tasks. It is still a heuristic: scheduling
+//! with resource constraints is intractable, so the pass reduces the loss
+//! rather than removing it, and the caller continues to see what was left
+//! out.
+
+use crate::greedy::earliest_feasible_start;
+use apsis_resource::ResourceSet;
+use apsis_schedule::Schedule;
+use apsis_task::{Task, TaskId, TaskState};
+
+/// The settings of the improvement pass.
+#[derive(Debug, Clone, Copy, PartialEq)]
+pub struct ImprovementSettings {
+    /// Whether the pass runs.
+    pub enabled: bool,
+    /// The greatest number of blocking placements to try per task.
+    ///
+    /// Each attempt costs two feasibility searches. A small bound keeps the
+    /// pass predictable; a larger one finds more placements on a crowded
+    /// schedule at proportional cost.
+    pub max_attempts_per_task: usize,
+}
+
+impl Default for ImprovementSettings {
+    /// Returns settings that run the pass with four attempts per task.
+    fn default() -> Self {
+        Self {
+            enabled: true,
+            max_attempts_per_task: 4,
+        }
+    }
+}
+
+impl ImprovementSettings {
+    /// Creates settings with the given bound.
+    pub const fn new(max_attempts_per_task: usize) -> Self {
+        Self {
+            enabled: true,
+            max_attempts_per_task,
+        }
+    }
+
+    /// Returns settings that do not run the pass.
+    pub const fn disabled() -> Self {
+        Self {
+            enabled: false,
+            max_attempts_per_task: 0,
+        }
+    }
+}
+
+/// What the improvement pass changed.
+#[derive(Debug, Clone, PartialEq, Default)]
+pub struct ImprovementOutcome {
+    /// The identifiers of the tasks the pass placed.
+    pub placed: Vec<TaskId>,
+    /// The identifiers of the tasks the pass moved to make room.
+    pub moved: Vec<TaskId>,
+}
+
+impl ImprovementOutcome {
+    /// Returns the number of tasks the pass placed.
+    pub fn placed_count(&self) -> usize {
+        self.placed.len()
+    }
+
+    /// Returns `true` when the pass changed nothing.
+    pub fn is_empty(&self) -> bool {
+        self.placed.is_empty() && self.moved.is_empty()
+    }
+}
+
+/// Tries to place deferred tasks by relocating the placements that block them.
+///
+/// `deferred` are the tasks the greedy pass could not place, in the order to
+/// consider them. `states` holds the state of every task indexed by
+/// identifier and is updated in place, so the caller sees the final
+/// assignment. The schedule is modified in place.
+///
+/// Returns a record of what the pass placed and moved.
+pub fn improve(
+    tasks: &[Task],
+    resources: &ResourceSet,
+    deferred: &[TaskId],
+    schedule: &mut Schedule,
+    states: &mut [TaskState],
+    settings: ImprovementSettings,
+) -> ImprovementOutcome {
+    let mut outcome = ImprovementOutcome::default();
+    if !settings.enabled || settings.max_attempts_per_task == 0 {
+        return outcome;
+    }
+
+    for id in deferred {
+        let Some(waiting) = tasks.iter().find(|task| task.id == *id) else {
+            continue;
+        };
+        // A task that is not deferred has been placed by an earlier step.
+        if schedule.is_placed(*id) {
+            continue;
+        }
+
+        for blocker_id in blocking_tasks(waiting, tasks, schedule)
+            .into_iter()
+            .take(settings.max_attempts_per_task)
+        {
+            let Some(blocker) = tasks.iter().find(|task| task.id == blocker_id) else {
+                continue;
+            };
+            if relocate_pair(waiting, blocker, tasks, schedule, resources) {
+                let start = schedule
+                    .placement_of(*id)
+                    .map_or(0.0, |placement| placement.start);
+                set_state(states, *id, TaskState::Scheduled { start });
+                let blocker_start = schedule
+                    .placement_of(blocker_id)
+                    .map_or(0.0, |placement| placement.start);
+                set_state(
+                    states,
+                    blocker_id,
+                    TaskState::Scheduled {
+                        start: blocker_start,
+                    },
+                );
+                outcome.placed.push(*id);
+                outcome.moved.push(blocker_id);
+                break;
+            }
+        }
+    }
+
+    outcome.placed.sort();
+    outcome.placed.dedup();
+    outcome.moved.sort();
+    outcome.moved.dedup();
+    outcome
+}
+
+/// Returns the placed tasks that share a resource with a waiting task, in
+/// time order.
+///
+/// Only these can block it: a placement on a resource the task does not need
+/// consumes capacity that the task was never going to use.
+fn blocking_tasks(waiting: &Task, tasks: &[Task], schedule: &Schedule) -> Vec<TaskId> {
+    let mut blockers: Vec<(f64, TaskId)> = schedule
+        .placements()
+        .iter()
+        .filter(|placement| {
+            tasks
+                .iter()
+                .find(|other| other.id == placement.task)
+                .is_some_and(|other| {
+                    waiting
+                        .requirements
+                        .iter()
+                        .any(|requirement| other.demand_for(requirement.resource).is_some())
+                })
+        })
+        .map(|placement| (placement.start, placement.task))
+        .collect();
+    blockers.sort_by(|first, second| {
+        first
+            .0
+            .partial_cmp(&second.0)
+            .unwrap_or(core::cmp::Ordering::Equal)
+            .then_with(|| first.1.cmp(&second.1))
+    });
+    blockers.into_iter().map(|(_, task)| task).collect()
+}
+
+/// Tries to place a waiting task by moving a blocker, keeping both only when
+/// both fit.
+///
+/// Returns `true` when the pair was rearranged and both tasks are placed. On
+/// a `false` return the schedule is exactly as it was.
+fn relocate_pair(
+    waiting: &Task,
+    blocker: &Task,
+    tasks: &[Task],
+    schedule: &mut Schedule,
+    resources: &ResourceSet,
+) -> bool {
+    let Some(blocker_start) = schedule.placement_of(blocker.id).map(|p| p.start) else {
+        return false;
+    };
+    // Remove the blocker and look for a home for the waiting task.
+    if schedule.unplace(blocker.id).is_err() {
+        return false;
+    }
+    let waiting_start = earliest_feasible_start(waiting, tasks, schedule, resources);
+    if waiting_start.is_none() {
+        // Nothing changed, so restore the blocker.
+        let _ = schedule.place(blocker, blocker_start);
+        return false;
+    }
+    let waiting_start = waiting_start.expect("checked above");
+    if schedule.place(waiting, waiting_start).is_err() {
+        let _ = schedule.place(blocker, blocker_start);
+        return false;
+    }
+
+    // With the waiting task placed, look for a new home for the blocker.
+    match earliest_feasible_start(blocker, tasks, schedule, resources) {
+        Some(blocker_new_start) => {
+            if schedule.place(blocker, blocker_new_start).is_ok() {
+                return true;
+            }
+            // The search said the placement was feasible, so a failure here
+            // can only come from the blocker being placed already, which it
+            // is not. Restore the original arrangement and report failure
+            // rather than leaving a half applied move.
+            let _ = schedule.unplace(waiting.id);
+            let _ = schedule.place(blocker, blocker_start);
+            false
+        }
+        None => {
+            // The blocker has nowhere to go, so the move is abandoned and
+            // the original arrangement restored.
+            let _ = schedule.unplace(waiting.id);
+            let _ = schedule.place(blocker, blocker_start);
+            false
+        }
+    }
+}
+
+/// Updates the recorded state of a task, ignoring an identifier outside the
+/// state list.
+fn set_state(states: &mut [TaskState], id: TaskId, state: TaskState) {
+    if let Some(slot) = states.get_mut(id.index()) {
+        *slot = state;
+    }
+}

crates/ops/apsis-scheduler/src/lib.rs

@@ -44,14 +44,16 @@
 #![deny(missing_docs)]
 
 mod greedy;
+mod improve;
 mod planner;
 
 pub use greedy::{
     demand_at, earliest_feasible_start, feasible_at, infeasible_tasks, resources_free_at,
     resources_have_capacity,
 };
+pub use improve::{ImprovementOutcome, ImprovementSettings, improve};
 pub use planner::{PlanOutcome, Planner, SchedulingResult};
 
 pub use apsis_resource::{ResourceId, ResourceSet};
 pub use apsis_schedule::Schedule;
-pub use apsis_task::{Task, TaskId};
+pub use apsis_task::{PrecedenceGraph, Task, TaskId};

crates/ops/apsis-scheduler/src/planner.rs

@@ -1,9 +1,10 @@
 //! The greedy planner.
 
 use crate::greedy::{earliest_feasible_start, infeasible_tasks};
+use crate::improve::{ImprovementSettings, improve};
 use apsis_resource::ResourceSet;
 use apsis_schedule::Schedule;
-use apsis_task::{Task, TaskId, TaskState, task_order};
+use apsis_task::{PrecedenceGraph, Task, TaskId, TaskState, task_order};
 
 /// What happened to one task during planning.
 ///
@@ -38,7 +39,17 @@ pub struct SchedulingResult {
     /// The schedule that was produced.
     pub schedule: Schedule,
     /// The outcome of every task, in the order the planner considered them.
+    ///
+    /// The order is the order of the task list the planner was given, not the
+    /// order in which the planner placed them, so that a caller can index the
+    /// outcomes by the position of the task in its own list.
     pub outcomes: Vec<PlanOutcome>,
+    /// The identifiers of the tasks that the improvement pass placed.
+    ///
+    /// These tasks were left out by the greedy pass and placed afterwards by
+    /// moving other placements. They are also recorded as scheduled in the
+    /// outcomes.
+    pub improved: Vec<TaskId>,
 }
 
 impl SchedulingResult {
@@ -84,12 +95,31 @@ impl SchedulingResult {
     /// A plan that leaves a mandatory task unplaced is not usable, however
     /// much optional work it found room for.
     pub fn honours_mandatory(&self, tasks: &[Task]) -> bool {
-        let mandatory: Vec<TaskId> = tasks
+        tasks
             .iter()
             .filter(|task| task.mandatory)
-            .map(|task| task.id)
+            .all(|task| self.schedule.is_placed(task.id))
-            .collect();
+    }
-        mandatory.iter().all(|id| self.schedule.is_placed(*id))
+
+    /// Returns `true` when every precedence constraint is satisfied.
+    ///
+    /// A constraint is satisfied when the second task starts at or after the
+    /// end of the first. A constraint whose tasks are not both placed is
+    /// trivially satisfied, because nothing has been promised about them.
+    pub fn honours_precedence(&self, graph: &PrecedenceGraph) -> bool {
+        self.outcomes
+            .iter()
+            .filter(|outcome| outcome.is_scheduled())
+            .all(|outcome| {
+                let Some(after) = self.schedule.placement_of(outcome.task) else {
+                    return true;
+                };
+                graph.predecessors_of(outcome.task).iter().all(|before| {
+                    self.schedule
+                        .placement_of(*before)
+                        .is_none_or(|placement| placement.end() <= after.start)
+                })
+            })
     }
 }
 
@@ -108,22 +138,30 @@ pub struct Planner {
     /// left, while the reverse order could consume the only window the
     /// mandatory task had.
     pub mandatory_first: bool,
+    /// The settings of the improvement pass.
+    pub improvement: ImprovementSettings,
 }
 
 impl Default for Planner {
-    /// Returns a planner that places mandatory tasks first.
+    /// Returns a planner that places mandatory tasks first and runs the
+    /// improvement pass.
     fn default() -> Self {
         Self {
             mandatory_first: true,
+            improvement: ImprovementSettings::default(),
         }
     }
 }
 
 impl Planner {
     /// Creates a planner with the default settings.
+    ///
+    /// The defaults are the ones [`Planner::default`] uses: mandatory tasks
+    /// first and the improvement pass enabled with four attempts per task.
     pub const fn new() -> Self {
         Self {
             mandatory_first: true,
+            improvement: ImprovementSettings::new(4),
         }
     }
 
@@ -133,26 +171,61 @@ impl Planner {
         self
     }
 
+    /// Returns a copy of this planner with the improvement pass configured.
+    pub const fn with_improvement(mut self, improvement: ImprovementSettings) -> Self {
+        self.improvement = improvement;
+        self
+    }
+
     /// Returns the order in which the planner considers the tasks.
     ///
-    /// Mandatory tasks come first when the planner is configured that way,
+    /// Without a precedence graph the order is mandatory tasks first, when
-    /// each group ordered by [`task_order`], which is decreasing weighted
+    /// the planner is configured that way, and then decreasing weighted
     /// value with ties broken by deadline and identifier.
+    ///
+    /// With a precedence graph the order is a topological order of the
+    /// graph, using the value order to break ties. That places a task as
+    /// early as its constraints allow and the most valuable one first among
+    /// the tasks that are free.
     pub fn order(&self, tasks: &[Task]) -> Vec<TaskId> {
-        let ordered = task_order(tasks);
+        self.order_with_precedence(tasks, None)
-        if !self.mandatory_first {
-            return ordered;
     }
+
+    /// Returns the order in which the planner considers the tasks, honouring
+    /// a precedence graph.
+    ///
+    /// A graph whose constraints contain a cycle produces the value order
+    /// unchanged, because the constraints then have no satisfiable
+    /// interpretation. The caller should validate the graph before
+    /// planning; the order is a fallback so that planning still produces
+    /// something rather than failing outright.
+    pub fn order_with_precedence(
+        &self,
+        tasks: &[Task],
+        precedence: Option<&PrecedenceGraph>,
+    ) -> Vec<TaskId> {
+        let ordered = task_order(tasks);
+        let valued = if !self.mandatory_first {
+            ordered
+        } else {
             let mandatory: Vec<TaskId> = tasks
                 .iter()
                 .filter(|task| task.mandatory)
                 .map(|task| task.id)
                 .collect();
-        // The stable partition keeps the relative order within each group.
             let mut result = Vec::with_capacity(ordered.len());
             result.extend(ordered.iter().filter(|id| mandatory.contains(id)).copied());
             result.extend(ordered.iter().filter(|id| !mandatory.contains(id)).copied());
             result
+        };
+
+        let Some(graph) = precedence else {
+            return valued;
+        };
+        if graph.is_empty() {
+            return valued;
+        }
+        graph.topological_order(&valued).unwrap_or(valued)
     }
 
     /// Plans a set of tasks on a set of resources.
@@ -166,13 +239,50 @@ impl Planner {
     /// each placement is checked against the capacity before it is made. The
     /// caller may confirm that with `apsis-conflict`.
     pub fn schedule(&self, tasks: &[Task], resources: &ResourceSet) -> SchedulingResult {
-        let mut schedule = Schedule::new();
+        self.plan(tasks, resources, None)
-        let mut outcomes = Vec::with_capacity(tasks.len());
+    }
 
-        // Tasks that cannot be placed at all are reported without a search.
+    /// Plans a set of tasks, honouring a precedence graph.
+    ///
+    /// A task is placed only after every task that must precede it has been
+    /// placed, so the search for its start begins no earlier than the latest
+    /// end among its predecessors. A predecessor that could not be placed
+    /// makes the successor deferred as well, because the constraint cannot be
+    /// met.
+    ///
+    /// The improvement pass is not applied to a plan with precedence
+    /// constraints, because relocating a placement could break a constraint
+    /// that the pass does not consider. A caller that needs both should
+    /// improve the plan and then verify it with
+    /// [`SchedulingResult::honours_precedence`].
+    pub fn schedule_with_precedence(
+        &self,
+        tasks: &[Task],
+        resources: &ResourceSet,
+        precedence: &PrecedenceGraph,
+    ) -> SchedulingResult {
+        self.plan(tasks, resources, Some(precedence))
+    }
+
+    /// The shared planning routine.
+    fn plan(
+        &self,
+        tasks: &[Task],
+        resources: &ResourceSet,
+        precedence: Option<&PrecedenceGraph>,
+    ) -> SchedulingResult {
+        let mut schedule = Schedule::new();
+        // The state of every task, indexed by identifier, so that a
+        // predecessor's placement can be looked up by index.
+        let mut states = vec![TaskState::Pending; tasks.len()];
+        let mut outcomes = Vec::with_capacity(tasks.len());
+        let mut order: Vec<TaskId> = self.order_with_precedence(tasks, precedence);
+        // A deferred task is one whose search failed, as opposed to one whose
+        // structural preconditions failed.
+        let mut deferred: Vec<TaskId> = Vec::new();
         let infeasible = infeasible_tasks(tasks, resources);
 
-        for id in self.order(tasks) {
+        for id in order.drain(..) {
             let task = tasks
                 .iter()
                 .find(|task| task.id == id)
@@ -184,6 +294,13 @@ impl Planner {
                 } else {
                     format!("the resources cannot hold the demand of {id}")
                 };
+                set_state(
+                    &mut states,
+                    id,
+                    TaskState::Infeasible {
+                        reason: reason.clone(),
+                    },
+                );
                 outcomes.push(PlanOutcome {
                     task: id,
                     state: TaskState::Infeasible { reason },
@@ -191,32 +308,144 @@ impl Planner {
                 continue;
             }
 
-            match earliest_feasible_start(task, tasks, &schedule, resources) {
+            // A task may not start before the latest end among the tasks that
-                Some(start) => {
+            // must precede it.
-                    // The placement was checked to be feasible, so the call
+            let earliest = match self.precedence_floor(id, tasks, precedence, &schedule, &states) {
-                    // succeeds. The schedule may still refuse it if the task
+                Ok(floor) => floor,
-                    // is already placed, which it is not, because each task
+                Err(reason) => {
-                    // is considered once.
+                    set_state(&mut states, id, TaskState::Deferred);
-                    match schedule.place(task, start) {
+                    outcomes.push(PlanOutcome {
-                        Ok(_) => outcomes.push(PlanOutcome {
-                            task: id,
-                            state: TaskState::Scheduled { start },
-                        }),
-                        Err(error) => outcomes.push(PlanOutcome {
-                            task: id,
-                            state: TaskState::Infeasible {
-                                reason: error.to_string(),
-                            },
-                        }),
-                    }
-                }
-                None => outcomes.push(PlanOutcome {
                         task: id,
                         state: TaskState::Deferred,
-                }),
+                    });
+                    let _ = reason;
+                    deferred.push(id);
+                    continue;
+                }
+            };
+
+            // The task with the floor applied, so that the search respects
+            // the constraints.
+            let constrained = match earliest {
+                Some(floor) if floor > task.earliest_start => {
+                    let mut copy = task.clone();
+                    copy.earliest_start = floor;
+                    copy
+                }
+                _ => task.clone(),
+            };
+
+            match earliest_feasible_start(&constrained, tasks, &schedule, resources) {
+                Some(start) => match schedule.place(task, start) {
+                    Ok(_) => {
+                        set_state(&mut states, id, TaskState::Scheduled { start });
+                        outcomes.push(PlanOutcome {
+                            task: id,
+                            state: TaskState::Scheduled { start },
+                        });
+                    }
+                    Err(error) => {
+                        let reason = error.to_string();
+                        set_state(
+                            &mut states,
+                            id,
+                            TaskState::Infeasible {
+                                reason: reason.clone(),
+                            },
+                        );
+                        outcomes.push(PlanOutcome {
+                            task: id,
+                            state: TaskState::Infeasible { reason },
+                        });
+                    }
+                },
+                None => {
+                    set_state(&mut states, id, TaskState::Deferred);
+                    outcomes.push(PlanOutcome {
+                        task: id,
+                        state: TaskState::Deferred,
+                    });
+                    deferred.push(id);
+                }
             }
         }
 
-        SchedulingResult { schedule, outcomes }
+        // The improvement pass runs only for a plan without precedence
+        // constraints, for the reason given on `schedule_with_precedence`.
+        let improved = if precedence.is_some() {
+            Vec::new()
+        } else {
+            let outcome = improve(
+                tasks,
+                resources,
+                &deferred,
+                &mut schedule,
+                &mut states,
+                self.improvement,
+            );
+            // The outcomes that the pass changed must be updated, and the
+            // order of the outcomes follows the task list so that a caller
+            // can index them by position.
+            for id in &outcome.placed {
+                if let Some(slot) = outcomes.iter_mut().find(|entry| entry.task == *id) {
+                    let start = schedule
+                        .placement_of(*id)
+                        .map_or(0.0, |placement| placement.start);
+                    slot.state = TaskState::Scheduled { start };
+                }
+            }
+            outcome.placed
+        };
+
+        // Reorder the outcomes to follow the task list, which is the order a
+        // caller expects when it indexes by its own position.
+        outcomes.sort_by_key(|outcome| outcome.task.index());
+
+        SchedulingResult {
+            schedule,
+            outcomes,
+            improved,
+        }
+    }
+
+    /// Returns the earliest start a task may have, given its predecessors.
+    ///
+    /// Returns `Ok(None)` when there is no constraint, `Ok(Some(t))` when the
+    /// task may start no earlier than `t`, and `Err` when a predecessor was
+    /// not placed, which makes the constraint unsatisfiable.
+    fn precedence_floor(
+        &self,
+        id: TaskId,
+        _tasks: &[Task],
+        precedence: Option<&PrecedenceGraph>,
+        schedule: &Schedule,
+        states: &[TaskState],
+    ) -> Result<Option<f64>, String> {
+        let Some(graph) = precedence else {
+            return Ok(None);
+        };
+        let mut floor: Option<f64> = None;
+        for predecessor in graph.predecessors_of(id) {
+            // A predecessor that was deferred or is infeasible cannot be
+            // waited for.
+            let placed = schedule.placement_of(predecessor);
+            let Some(placement) = placed else {
+                let reason = match states.get(predecessor.index()) {
+                    Some(TaskState::Infeasible { reason }) => reason.clone(),
+                    _ => format!("{predecessor} was not placed"),
+                };
+                return Err(reason);
+            };
+            floor =
+                Some(floor.map_or(placement.end(), |current: f64| current.max(placement.end())));
+        }
+        Ok(floor)
+    }
+}
+
+/// Records the state of a task, ignoring an identifier outside the list.
+fn set_state(states: &mut [TaskState], id: TaskId, state: TaskState) {
+    if let Some(slot) = states.get_mut(id.index()) {
+        *slot = state;
     }
 }

crates/ops/apsis-scheduler/tests/planner.rs

@@ -148,10 +148,12 @@ fn a_mandatory_task_is_placed_before_a_more_valuable_optional_one() {
         .expect("the mandatory task is placed");
     assert_eq!(mandatory_placement.start, 0.0);
 
-    // With the rule turned off, the optional task goes first and the
+    // With the rule turned off, and the improvement pass disabled so that
+    // only the ordering is under test, the optional task goes first and the
     // mandatory one is placed after it, using the rest of its window.
     let without = Planner::new()
         .with_mandatory_first(false)
+        .with_improvement(apsis_scheduler::ImprovementSettings::disabled())
         .schedule(&tasks, &resources);
     let optional_placement = without
         .schedule
@@ -160,6 +162,40 @@ fn a_mandatory_task_is_placed_before_a_more_valuable_optional_one() {
     assert_eq!(optional_placement.start, 0.0);
 }
 
+#[test]
+fn the_improvement_pass_recovers_a_plan_that_the_greedy_order_missed() {
+    // The optional task is worth more, so it is placed first and occupies
+    // the resource from zero to sixty. The mandatory task has a window from
+    // zero to a hundred, so it cannot fit after the optional one and would
+    // be lost. The improvement pass moves the optional task aside and places
+    // both.
+    let resources = ResourceSet::from_resources(vec![resource(0, 1.0)]).expect("one");
+    let optional = task(0, 60.0, &[0])
+        .with_priority(100.0)
+        .with_window(0.0, 1_000.0);
+    let mandatory = task(1, 60.0, &[0])
+        .with_priority(1.0)
+        .with_window(0.0, 100.0)
+        .with_mandatory(true);
+    let tasks = [optional, mandatory];
+
+    let greedy = Planner::new()
+        .with_mandatory_first(false)
+        .with_improvement(apsis_scheduler::ImprovementSettings::disabled())
+        .schedule(&tasks, &resources);
+    // Without the pass the mandatory task is lost.
+    assert!(!greedy.honours_mandatory(&tasks));
+    assert_eq!(greedy.schedule.len(), 1);
+
+    let improved = Planner::new()
+        .with_mandatory_first(false)
+        .schedule(&tasks, &resources);
+    // With the pass both are placed.
+    assert!(improved.is_complete(), "{:?}", improved.outcomes);
+    assert!(improved.honours_mandatory(&tasks));
+    assert!(!improved.improved.is_empty());
+}
+
 #[test]
 fn the_value_ordering_places_the_most_valuable_task_first() {
     let resources = ResourceSet::from_resources(vec![resource(0, 1.0)]).expect("one");

crates/ops/apsis-scheduler/tests/precedence.rs

@@ -0,0 +1,295 @@
+//! Tests for precedence honoured by the planner, and for the improvement
+//! pass.
+
+use apsis_resource::{Resource, ResourceId, ResourceSet, Window};
+use apsis_scheduler::{ImprovementSettings, Planner};
+use apsis_task::{PrecedenceGraph, Task, TaskId};
+
+/// A resource available from 0 to 1 000 seconds.
+fn resource(id: usize, capacity: f64) -> Resource {
+    let mut resource = Resource::new(ResourceId::new(id), capacity);
+    resource
+        .set_windows(&[Window::new(0.0, 1_000.0)])
+        .expect("the window is usable");
+    resource
+}
+
+/// A task requiring one unit of each named resource.
+fn task(id: usize, duration: f64, resources: &[usize]) -> Task {
+    let mut task = Task::new(TaskId::new(id), duration);
+    for resource in resources {
+        task = task
+            .try_require(ResourceId::new(*resource), 1.0)
+            .expect("the resource is new");
+    }
+    task
+}
+
+#[test]
+fn a_chain_of_tasks_is_placed_in_order() {
+    // Three tasks that must run one after another on the same resource.
+    let resources = ResourceSet::from_resources(vec![resource(0, 1.0)]).expect("one");
+    let tasks = [
+        task(0, 60.0, &[0]),
+        task(1, 30.0, &[0]),
+        task(2, 90.0, &[0]),
+    ];
+    let mut graph = PrecedenceGraph::new();
+    graph.add(TaskId::new(0), TaskId::new(1)).expect("no cycle");
+    graph.add(TaskId::new(1), TaskId::new(2)).expect("no cycle");
+
+    let result = Planner::new().schedule_with_precedence(&tasks, &resources, &graph);
+    assert!(result.is_complete(), "{:?}", result.outcomes);
+    assert!(result.honours_precedence(&graph));
+
+    let first = result
+        .schedule
+        .placement_of(TaskId::new(0))
+        .expect("placed");
+    let second = result
+        .schedule
+        .placement_of(TaskId::new(1))
+        .expect("placed");
+    let third = result
+        .schedule
+        .placement_of(TaskId::new(2))
+        .expect("placed");
+    assert_eq!(first.start, 0.0);
+    assert_eq!(second.start, 60.0);
+    assert_eq!(third.start, 90.0);
+}
+
+#[test]
+fn a_precedence_floor_delays_a_task_that_could_start_earlier() {
+    // The second task could start at zero on its own, but must wait for the
+    // first, which runs on a different resource.
+    let resources =
+        ResourceSet::from_resources(vec![resource(0, 1.0), resource(1, 1.0)]).expect("distinct");
+    let tasks = [task(0, 100.0, &[0]), task(1, 60.0, &[1])];
+    let mut graph = PrecedenceGraph::new();
+    graph.add(TaskId::new(0), TaskId::new(1)).expect("no cycle");
+
+    let result = Planner::new().schedule_with_precedence(&tasks, &resources, &graph);
+    assert!(result.is_complete());
+    let second = result
+        .schedule
+        .placement_of(TaskId::new(1))
+        .expect("placed");
+    // The second resource was free from the start, but the constraint delays
+    // the task to the end of the first.
+    assert_eq!(second.start, 100.0);
+    assert!(result.honours_precedence(&graph));
+}
+
+#[test]
+fn a_diamond_precedence_is_honoured() {
+    let resources =
+        ResourceSet::from_resources(vec![resource(0, 1.0), resource(1, 1.0)]).expect("distinct");
+    // The first task feeds two parallel tasks, which both feed a last one.
+    let tasks = [
+        task(0, 100.0, &[0]),
+        task(1, 60.0, &[1]),
+        task(2, 60.0, &[1]),
+        task(3, 30.0, &[0]),
+    ];
+    let mut graph = PrecedenceGraph::new();
+    graph.add(TaskId::new(0), TaskId::new(1)).expect("no cycle");
+    graph.add(TaskId::new(0), TaskId::new(2)).expect("no cycle");
+    graph.add(TaskId::new(1), TaskId::new(3)).expect("no cycle");
+    graph.add(TaskId::new(2), TaskId::new(3)).expect("no cycle");
+
+    let result = Planner::new().schedule_with_precedence(&tasks, &resources, &graph);
+    assert!(result.is_complete(), "{:?}", result.outcomes);
+    assert!(result.honours_precedence(&graph));
+
+    // The last task must follow both of the parallel ones.
+    let last = result
+        .schedule
+        .placement_of(TaskId::new(3))
+        .expect("placed");
+    for parallel in [1_usize, 2] {
+        let placement = result
+            .schedule
+            .placement_of(TaskId::new(parallel))
+            .expect("placed");
+        assert!(
+            placement.end() <= last.start,
+            "{placement:?} against {last:?}"
+        );
+    }
+}
+
+#[test]
+fn a_sequential_observation_and_downlink_is_expressed_as_two_tasks() {
+    // An observation uses the payload and a downlink uses the transmitter,
+    // and the downlink must follow the observation. This is the pattern for
+    // a task that needs two resources in sequence.
+    let resources =
+        ResourceSet::from_resources(vec![resource(0, 1.0), resource(1, 1.0)]).expect("distinct");
+    let observation = task(0, 120.0, &[0]).with_window(50.0, 500.0);
+    let downlink = task(1, 90.0, &[1]).with_window(0.0, 1_000.0);
+    let tasks = [observation, downlink];
+
+    let mut graph = PrecedenceGraph::new();
+    graph.add(TaskId::new(0), TaskId::new(1)).expect("no cycle");
+
+    let result = Planner::new().schedule_with_precedence(&tasks, &resources, &graph);
+    assert!(result.is_complete());
+    assert!(result.honours_precedence(&graph));
+
+    let collection = result
+        .schedule
+        .placement_of(TaskId::new(0))
+        .expect("placed");
+    let transmission = result
+        .schedule
+        .placement_of(TaskId::new(1))
+        .expect("placed");
+    assert_eq!(collection.start, 50.0);
+    // The downlink starts when the collection ends.
+    assert_eq!(transmission.start, 170.0);
+    // The two use different resources, so they may be placed back to back
+    // without a conflict.
+    assert_eq!(collection.end(), transmission.start);
+}
+
+#[test]
+fn a_successor_of_an_unplaced_task_is_deferred() {
+    // The first task cannot be placed, so the constraint cannot be met and
+    // the second is deferred rather than placed early.
+    let resources = ResourceSet::from_resources(vec![resource(0, 1.0)]).expect("one");
+    let impossible = task(0, 90.0, &[0]).with_window(0.0, 60.0);
+    let successor = task(1, 30.0, &[0]);
+    let tasks = [impossible, successor];
+
+    let mut graph = PrecedenceGraph::new();
+    graph.add(TaskId::new(0), TaskId::new(1)).expect("no cycle");
+
+    let result = Planner::new().schedule_with_precedence(&tasks, &resources, &graph);
+    assert!(!result.schedule.is_placed(TaskId::new(1)));
+    assert!(result.deferred().contains(&TaskId::new(1)));
+    // The constraints are trivially satisfied, because nothing was promised.
+    assert!(result.honours_precedence(&graph));
+}
+
+#[test]
+fn the_precedence_aware_order_is_topological() {
+    let tasks = [
+        Task::new(TaskId::new(0), 60.0),
+        Task::new(TaskId::new(1), 60.0),
+        Task::new(TaskId::new(2), 60.0),
+    ];
+    let mut graph = PrecedenceGraph::new();
+    graph.add(TaskId::new(2), TaskId::new(0)).expect("no cycle");
+
+    let order = Planner::new().order_with_precedence(&tasks, Some(&graph));
+    let position = |id: TaskId| {
+        order
+            .iter()
+            .position(|candidate| *candidate == id)
+            .expect("present")
+    };
+    assert!(position(TaskId::new(2)) < position(TaskId::new(0)));
+}
+
+#[test]
+fn a_cycle_in_the_graph_falls_back_to_the_value_order() {
+    // The graph refuses a cycle, so a cycle can only exist if the caller
+    // builds the graph from one task set and orders another. The planner
+    // then falls back to the value order rather than failing.
+    let tasks = [
+        Task::new(TaskId::new(0), 60.0),
+        Task::new(TaskId::new(1), 60.0),
+    ];
+    let graph = PrecedenceGraph::new();
+    let order = Planner::new().order_with_precedence(&tasks, Some(&graph));
+    // An empty graph gives the value order, which for equal values is by
+    // identifier.
+    assert_eq!(order, vec![TaskId::new(0), TaskId::new(1)]);
+}
+
+#[test]
+fn the_improvement_pass_is_skipped_when_precedence_is_used() {
+    let resources = ResourceSet::from_resources(vec![resource(0, 1.0)]).expect("one");
+    let tasks = [task(0, 60.0, &[0]), task(1, 60.0, &[0])];
+    let graph = PrecedenceGraph::new();
+    let result = Planner::new().schedule_with_precedence(&tasks, &resources, &graph);
+    // The pass does not run, so nothing is reported as improved.
+    assert!(result.improved.is_empty());
+}
+
+#[test]
+fn the_improvement_pass_can_be_disabled() {
+    let resources = ResourceSet::from_resources(vec![resource(0, 1.0)]).expect("one");
+    let optional = task(0, 60.0, &[0])
+        .with_priority(100.0)
+        .with_window(0.0, 1_000.0);
+    let mandatory = task(1, 60.0, &[0])
+        .with_priority(1.0)
+        .with_window(0.0, 100.0)
+        .with_mandatory(true);
+    let tasks = [optional, mandatory];
+
+    let planner = Planner::new()
+        .with_mandatory_first(false)
+        .with_improvement(ImprovementSettings::disabled());
+    let result = planner.schedule(&tasks, &resources);
+    assert!(result.improved.is_empty());
+    assert_eq!(result.schedule.len(), 1);
+}
+
+#[test]
+fn the_improvement_pass_preserves_feasibility() {
+    // Whatever the pass decides, the result must remain conflict free.
+    let resources =
+        ResourceSet::from_resources(vec![resource(0, 1.0), resource(1, 2.0)]).expect("distinct");
+    let tasks: Vec<Task> = (0..10)
+        .map(|index| {
+            task(index, 40.0 + f64::from(index as u32) * 5.0, &[index % 2])
+                .with_priority(1.0 + f64::from(index as u32))
+        })
+        .collect();
+
+    let result = Planner::new().schedule(&tasks, &resources);
+    // The schedule must not place two tasks beyond the capacity of a
+    // resource at the same time.
+    let report = apsis_conflict::detect(&result.schedule, &tasks, &resources);
+    assert!(
+        report.is_clear(),
+        "the improvement pass broke feasibility: {:?}",
+        report.conflicts()
+    );
+}
+
+#[test]
+fn the_improvement_pass_never_loses_a_placement() {
+    // Every task that the greedy pass placed must still be placed
+    // afterwards, because the pass only adds placements or moves them.
+    let resources = ResourceSet::from_resources(vec![resource(0, 1.0)]).expect("one");
+    let tasks: Vec<Task> = (0..6)
+        .map(|index| task(index, 60.0, &[0]).with_priority(1.0 + f64::from(index as u32)))
+        .collect();
+
+    let with_pass = Planner::new().schedule(&tasks, &resources);
+    let without_pass = Planner::new()
+        .with_improvement(ImprovementSettings::disabled())
+        .schedule(&tasks, &resources);
+    assert!(
+        with_pass.schedule.len() >= without_pass.schedule.len(),
+        "{} against {}",
+        with_pass.schedule.len(),
+        without_pass.schedule.len()
+    );
+}
+
+#[test]
+fn the_improvement_settings_are_validated_by_use() {
+    let resources = ResourceSet::from_resources(vec![resource(0, 1.0)]).expect("one");
+    let tasks = [task(0, 60.0, &[0])];
+    // A bound of zero disables the move search but leaves the pass enabled,
+    // which behaves as if the pass were off.
+    let planner = Planner::new().with_improvement(ImprovementSettings::new(0));
+    let result = planner.schedule(&tasks, &resources);
+    assert!(result.is_complete());
+    assert!(result.improved.is_empty());
+}

crates/ops/apsis-task/src/error.rs

@@ -22,6 +22,8 @@ pub enum TaskErrorKind {
     NoSuchTask,
     /// A task with the identifier already exists.
     DuplicateTask,
+    /// A precedence constraint would close a cycle.
+    PrecedenceCycle,
 }
 
 impl fmt::Display for TaskErrorKind {
@@ -35,6 +37,7 @@ impl fmt::Display for TaskErrorKind {
             Self::InvalidWeight => "a priority or value is not finite",
             Self::NoSuchTask => "no such task",
             Self::DuplicateTask => "a task with that identifier exists",
+            Self::PrecedenceCycle => "the precedence constraints contain a cycle",
         };
         f.write_str(text)
     }

crates/ops/apsis-task/src/lib.rs

@@ -8,8 +8,9 @@
 //! A task requires all of its resources at the same time, for its whole
 //! duration. That is the model an observation has: the satellite and the
 //! ground station must both be available while the data is taken. A task that
-//! needs resources in sequence is expressed as two tasks with a
+//! needs resources in sequence is expressed as two tasks joined by a
-//! precedence constraint, which the scheduling layer provides.
+//! precedence constraint in a [`PrecedenceGraph`], which says that one task
+//! must finish before the other starts.
 //!
 //! Tasks are immutable once built. The state a scheduler assigns to one is
 //! kept separately, in the schedule, so that the same task can be considered
@@ -37,9 +38,11 @@
 #![deny(missing_docs)]
 
 mod error;
+mod precedence;
 mod task;
 
 pub use error::{TaskError, TaskErrorKind};
+pub use precedence::PrecedenceGraph;
 pub use task::{Requirement, Task, TaskId, TaskSet, TaskState, task_order};
 
 pub use apsis_resource::ResourceId;

crates/ops/apsis-task/src/precedence.rs

@@ -0,0 +1,245 @@
+//! Precedence constraints between tasks.
+//!
+//! A precedence constraint says that one task must finish before another
+//! starts. A set of such constraints forms a directed acyclic graph, and a
+//! scheduler must respect it: a task cannot be placed until every task that
+//! must precede it has been placed and has finished.
+//!
+//! The graph is stored as adjacency lists. It rejects a cycle when an edge is
+//! added, because a cycle is not a set of constraints that can be satisfied
+//! and would otherwise produce an empty topological order that a caller
+//! might mistake for a scheduling failure rather than a modelling error.
+//!
+//! The precedence graph is a separate type from [`Task`](crate::Task)
+//! because the constraints are a property of the set of tasks rather than of
+//! any one task, and because a caller may want to plan the same tasks under
+//! different constraint graphs.
+
+use crate::error::{TaskError, TaskErrorKind};
+use crate::task::TaskId;
+use std::collections::BTreeMap;
+
+/// A directed acyclic graph of precedence constraints.
+///
+/// An edge from one task to another means the first must finish before the
+/// second starts. The order of the tasks themselves is the caller's: the
+/// graph only records which pairs are related.
+#[derive(Debug, Clone, PartialEq, Default)]
+pub struct PrecedenceGraph {
+    /// The successors of each task, in the order the edges were added.
+    successors: BTreeMap<usize, Vec<usize>>,
+    /// The predecessors of each task, in the order the edges were added.
+    predecessors: BTreeMap<usize, Vec<usize>>,
+}
+
+impl PrecedenceGraph {
+    /// Creates an empty graph.
+    pub fn new() -> Self {
+        Self {
+            successors: BTreeMap::new(),
+            predecessors: BTreeMap::new(),
+        }
+    }
+
+    /// Returns the number of edges.
+    pub fn edge_count(&self) -> usize {
+        self.successors.values().map(Vec::len).sum()
+    }
+
+    /// Returns `true` when there are no edges.
+    pub fn is_empty(&self) -> bool {
+        self.successors.values().all(Vec::is_empty)
+    }
+
+    /// Returns `true` when the constraint from `before` to `after` exists.
+    pub fn has_constraint(&self, before: TaskId, after: TaskId) -> bool {
+        self.successors
+            .get(&before.index())
+            .is_some_and(|successors| successors.contains(&after.index()))
+    }
+
+    /// Adds a constraint that `before` must finish before `after` starts.
+    ///
+    /// Returns an error when the two identifiers are the same, because a task
+    /// cannot precede itself, or when the constraint would close a cycle.
+    ///
+    /// Adding a constraint that already exists is accepted and has no effect,
+    /// so a caller may assemble a graph from overlapping sources without
+    /// having to deduplicate.
+    pub fn add(&mut self, before: TaskId, after: TaskId) -> Result<(), TaskError> {
+        if before == after {
+            return Err(TaskError::new(TaskErrorKind::PrecedenceCycle)
+                .with_detail(format!("{before} cannot precede itself")));
+        }
+        if self.has_constraint(before, after) {
+            return Ok(());
+        }
+        // A new edge from `before` to `after` closes a cycle exactly when
+        // `before` is already reachable from `after`. That is checked before
+        // the edge is inserted, so a rejected edge leaves the graph
+        // unchanged.
+        if self.is_reachable(after, before) {
+            return Err(
+                TaskError::new(TaskErrorKind::PrecedenceCycle).with_detail(format!(
+                    "{before} must precede {after}, but {after} already precedes {before}"
+                )),
+            );
+        }
+        self.successors
+            .entry(before.index())
+            .or_default()
+            .push(after.index());
+        self.predecessors
+            .entry(after.index())
+            .or_default()
+            .push(before.index());
+        Ok(())
+    }
+
+    /// Returns `true` when `to` is reachable from `from` following the
+    /// constraints.
+    pub fn is_reachable(&self, from: TaskId, to: TaskId) -> bool {
+        if from == to {
+            return true;
+        }
+        let mut visited: Vec<usize> = Vec::new();
+        let mut stack = vec![from.index()];
+        while let Some(node) = stack.pop() {
+            if node == to.index() {
+                return true;
+            }
+            if visited.contains(&node) {
+                continue;
+            }
+            visited.push(node);
+            if let Some(successors) = self.successors.get(&node) {
+                stack.extend(successors.iter().copied());
+            }
+        }
+        false
+    }
+
+    /// Returns the tasks that must finish before a task starts.
+    ///
+    /// The result is in the order the constraints were added. It is the
+    /// direct predecessors only, not the transitive closure.
+    pub fn predecessors_of(&self, task: TaskId) -> Vec<TaskId> {
+        self.predecessors
+            .get(&task.index())
+            .map(|list| list.iter().map(|index| TaskId::new(*index)).collect())
+            .unwrap_or_default()
+    }
+
+    /// Returns the tasks that must wait for a task to finish.
+    ///
+    /// The result is in the order the constraints were added. It is the
+    /// direct successors only, not the transitive closure.
+    pub fn successors_of(&self, task: TaskId) -> Vec<TaskId> {
+        self.successors
+            .get(&task.index())
+            .map(|list| list.iter().map(|index| TaskId::new(*index)).collect())
+            .unwrap_or_default()
+    }
+
+    /// Returns the tasks with no predecessor.
+    pub fn roots(&self) -> Vec<TaskId> {
+        // Every task that appears as a successor has a predecessor, so the
+        // roots are the tasks that appear only as a predecessor.
+        let mut roots: Vec<usize> = Vec::new();
+        for task in self.successors.keys() {
+            if !self.predecessors.contains_key(task) && !roots.contains(task) {
+                roots.push(*task);
+            }
+        }
+        roots.sort_unstable();
+        roots.into_iter().map(TaskId::new).collect()
+    }
+
+    /// Returns the tasks with no successor.
+    pub fn leaves(&self) -> Vec<TaskId> {
+        let mut leaves: Vec<usize> = Vec::new();
+        for task in self.predecessors.keys() {
+            if !self.successors.contains_key(task) && !leaves.contains(task) {
+                leaves.push(*task);
+            }
+        }
+        leaves.sort_unstable();
+        leaves.into_iter().map(TaskId::new).collect()
+    }
+
+    /// Returns a topological order of the given tasks.
+    ///
+    /// The order respects every constraint: a task appears after all of its
+    /// predecessors. Ties are broken by the position of the task in the
+    /// supplied list, so that a caller can impose its own preference among
+    /// tasks that are otherwise unordered. That is what lets a planner place
+    /// the more valuable tasks first within the freedom the constraints
+    /// allow.
+    ///
+    /// Tasks in the list that have no constraints still appear.
+    ///
+    /// Returns an error when the graph contains a cycle among the given
+    /// tasks. The graph rejects cycles as they are added, so this can only
+    /// happen when a caller builds the graph from one task set and orders
+    /// another, or when the tasks are not all in the graph.
+    pub fn topological_order(&self, tasks: &[TaskId]) -> Result<Vec<TaskId>, TaskError> {
+        // The position of each task in the supplied list, which breaks ties.
+        let mut preference: BTreeMap<usize, usize> = BTreeMap::new();
+        for (position, task) in tasks.iter().enumerate() {
+            preference.insert(task.index(), position);
+        }
+
+        // The number of predecessors of each task that are themselves in the
+        // list. A predecessor outside the list is ignored, because it cannot
+        // be scheduled and so cannot be waited for.
+        let mut pending: BTreeMap<usize, usize> = BTreeMap::new();
+        for task in tasks {
+            let count = self
+                .predecessors_of(*task)
+                .iter()
+                .filter(|predecessor| preference.contains_key(&predecessor.index()))
+                .count();
+            pending.insert(task.index(), count);
+        }
+
+        let mut order: Vec<usize> = Vec::with_capacity(tasks.len());
+        let mut placed: Vec<usize> = Vec::with_capacity(tasks.len());
+
+        while order.len() < tasks.len() {
+            // Among the tasks whose predecessors are all placed, take the
+            // one the caller prefers.
+            let next = tasks
+                .iter()
+                .map(|task| task.index())
+                .filter(|index| !placed.contains(index))
+                .filter(|index| pending.get(index).copied().unwrap_or(0) == 0)
+                .min_by_key(|index| preference.get(index).copied().unwrap_or(usize::MAX));
+
+            let Some(next) = next else {
+                return Err(TaskError::new(TaskErrorKind::PrecedenceCycle)
+                    .with_detail("no task is free of unplaced predecessors"));
+            };
+
+            placed.push(next);
+            order.push(next);
+            for successor in self.successors_of(TaskId::new(next)) {
+                if let Some(count) = pending.get_mut(&successor.index()) {
+                    *count = count.saturating_sub(1);
+                }
+            }
+        }
+
+        Ok(order.into_iter().map(TaskId::new).collect())
+    }
+
+    /// Returns the number of tasks in the graph.
+    pub fn task_count(&self) -> usize {
+        let mut all: Vec<usize> = self.successors.keys().copied().collect();
+        for task in self.predecessors.keys() {
+            if !all.contains(task) {
+                all.push(*task);
+            }
+        }
+        all.len()
+    }
+}

crates/ops/apsis-task/tests/precedence.rs

@@ -0,0 +1,233 @@
+//! Tests for the precedence graph.
+
+use apsis_task::{PrecedenceGraph, TaskErrorKind, TaskId};
+
+/// The identifiers used by the tests.
+fn id(index: usize) -> TaskId {
+    TaskId::new(index)
+}
+
+#[test]
+fn an_empty_graph_has_no_edges() {
+    let graph = PrecedenceGraph::new();
+    assert!(graph.is_empty());
+    assert_eq!(graph.edge_count(), 0);
+    assert_eq!(graph.task_count(), 0);
+    assert!(!graph.has_constraint(id(0), id(1)));
+    assert!(graph.predecessors_of(id(0)).is_empty());
+    assert!(graph.successors_of(id(0)).is_empty());
+}
+
+#[test]
+fn a_constraint_is_recorded_both_ways() {
+    let mut graph = PrecedenceGraph::new();
+    graph.add(id(0), id(1)).expect("no cycle");
+
+    assert!(graph.has_constraint(id(0), id(1)));
+    // The constraint is directed, so the reverse does not exist.
+    assert!(!graph.has_constraint(id(1), id(0)));
+    assert_eq!(graph.successors_of(id(0)), vec![id(1)]);
+    assert_eq!(graph.predecessors_of(id(1)), vec![id(0)]);
+    assert_eq!(graph.edge_count(), 1);
+    assert_eq!(graph.task_count(), 2);
+}
+
+#[test]
+fn a_self_constraint_is_rejected() {
+    let mut graph = PrecedenceGraph::new();
+    let error = graph
+        .add(id(0), id(0))
+        .expect_err("a task cannot precede itself");
+    assert_eq!(error.kind(), TaskErrorKind::PrecedenceCycle);
+    assert!(graph.is_empty());
+}
+
+#[test]
+fn a_duplicate_constraint_is_accepted_and_ignored() {
+    let mut graph = PrecedenceGraph::new();
+    graph.add(id(0), id(1)).expect("no cycle");
+    graph.add(id(0), id(1)).expect("already present");
+    // The edge is recorded once.
+    assert_eq!(graph.edge_count(), 1);
+    assert_eq!(graph.successors_of(id(0)), vec![id(1)]);
+}
+
+#[test]
+fn a_cycle_is_rejected() {
+    let mut graph = PrecedenceGraph::new();
+    graph.add(id(0), id(1)).expect("no cycle");
+    graph.add(id(1), id(2)).expect("no cycle");
+    // Closing the cycle is refused.
+    let error = graph.add(id(2), id(0)).expect_err("this closes a cycle");
+    assert_eq!(error.kind(), TaskErrorKind::PrecedenceCycle);
+    // The graph is unchanged.
+    assert!(!graph.has_constraint(id(2), id(0)));
+    assert_eq!(graph.edge_count(), 2);
+}
+
+#[test]
+fn a_longer_cycle_is_rejected() {
+    let mut graph = PrecedenceGraph::new();
+    for edge in [(0, 1), (1, 2), (2, 3), (3, 4)] {
+        graph.add(id(edge.0), id(edge.1)).expect("no cycle");
+    }
+    assert_eq!(
+        graph
+            .add(id(4), id(1))
+            .expect_err("this closes a cycle")
+            .kind(),
+        TaskErrorKind::PrecedenceCycle
+    );
+}
+
+#[test]
+fn reachability_follows_the_direction() {
+    let mut graph = PrecedenceGraph::new();
+    graph.add(id(0), id(1)).expect("no cycle");
+    graph.add(id(1), id(2)).expect("no cycle");
+
+    // The transitive relation holds forward but not backward.
+    assert!(graph.is_reachable(id(0), id(1)));
+    assert!(graph.is_reachable(id(0), id(2)));
+    assert!(graph.is_reachable(id(1), id(2)));
+    assert!(!graph.is_reachable(id(2), id(0)));
+    // A task reaches itself by a path of no edges.
+    assert!(graph.is_reachable(id(0), id(0)));
+}
+
+#[test]
+fn roots_and_leaves_are_reported() {
+    let mut graph = PrecedenceGraph::new();
+    // A diamond: 0 before 1 and 2, and both before 3.
+    graph.add(id(0), id(1)).expect("no cycle");
+    graph.add(id(0), id(2)).expect("no cycle");
+    graph.add(id(1), id(3)).expect("no cycle");
+    graph.add(id(2), id(3)).expect("no cycle");
+
+    assert_eq!(graph.roots(), vec![id(0)]);
+    assert_eq!(graph.leaves(), vec![id(3)]);
+}
+
+#[test]
+fn a_chain_has_one_root_and_one_leaf() {
+    let mut graph = PrecedenceGraph::new();
+    graph.add(id(0), id(1)).expect("no cycle");
+    graph.add(id(1), id(2)).expect("no cycle");
+    assert_eq!(graph.roots(), vec![id(0)]);
+    assert_eq!(graph.leaves(), vec![id(2)]);
+}
+
+#[test]
+fn the_topological_order_respects_the_constraints() {
+    let mut graph = PrecedenceGraph::new();
+    graph.add(id(0), id(1)).expect("no cycle");
+    graph.add(id(1), id(2)).expect("no cycle");
+
+    let tasks = [id(2), id(1), id(0)];
+    let order = graph
+        .topological_order(&tasks)
+        .expect("the graph is acyclic");
+    // The constraints force 0 before 1 before 2, whatever the input order.
+    assert_eq!(order, vec![id(0), id(1), id(2)]);
+}
+
+#[test]
+fn the_topological_order_breaks_ties_by_position() {
+    let mut graph = PrecedenceGraph::new();
+    // Only 0 is constrained, before 1. Tasks 2 and 3 are free.
+    graph.add(id(0), id(1)).expect("no cycle");
+
+    // The preference order is 3, 2, 0, 1, so among the free tasks 3 comes
+    // first.
+    let tasks = [id(3), id(2), id(0), id(1)];
+    let order = graph
+        .topological_order(&tasks)
+        .expect("the graph is acyclic");
+    assert_eq!(order, vec![id(3), id(2), id(0), id(1)]);
+}
+
+#[test]
+fn the_topological_order_honours_the_diamond() {
+    let mut graph = PrecedenceGraph::new();
+    graph.add(id(0), id(1)).expect("no cycle");
+    graph.add(id(0), id(2)).expect("no cycle");
+    graph.add(id(1), id(3)).expect("no cycle");
+    graph.add(id(2), id(3)).expect("no cycle");
+
+    let tasks = [id(0), id(1), id(2), id(3)];
+    let order = graph
+        .topological_order(&tasks)
+        .expect("the graph is acyclic");
+    assert_eq!(order, vec![id(0), id(1), id(2), id(3)]);
+
+    // The position of each task in the result relative to its constraints.
+    let position_of = |task: TaskId| {
+        order
+            .iter()
+            .position(|candidate| *candidate == task)
+            .expect("present")
+    };
+    assert!(position_of(id(0)) < position_of(id(1)));
+    assert!(position_of(id(0)) < position_of(id(2)));
+    assert!(position_of(id(1)) < position_of(id(3)));
+    assert!(position_of(id(2)) < position_of(id(3)));
+}
+
+#[test]
+fn the_topological_order_includes_unconstrained_tasks() {
+    let mut graph = PrecedenceGraph::new();
+    graph.add(id(0), id(1)).expect("no cycle");
+    // Task 9 has no constraints at all.
+    let tasks = [id(0), id(1), id(9)];
+    let order = graph
+        .topological_order(&tasks)
+        .expect("the graph is acyclic");
+    assert_eq!(order.len(), 3);
+    assert!(order.contains(&id(9)));
+}
+
+#[test]
+fn the_topological_order_ignores_predecessors_outside_the_list() {
+    // A constraint whose predecessor is not in the list cannot be waited
+    // for, so it is ignored rather than blocking the successor.
+    let mut graph = PrecedenceGraph::new();
+    graph.add(id(5), id(1)).expect("no cycle");
+    let tasks = [id(1)];
+    assert_eq!(
+        graph
+            .topological_order(&tasks)
+            .expect("the graph is acyclic"),
+        vec![id(1)]
+    );
+}
+
+#[test]
+fn the_topological_order_of_an_empty_list_is_empty() {
+    let graph = PrecedenceGraph::new();
+    assert!(
+        graph
+            .topological_order(&[])
+            .expect("the graph is acyclic")
+            .is_empty()
+    );
+}
+
+#[test]
+fn a_graph_with_two_independent_chains_orders_both() {
+    let mut graph = PrecedenceGraph::new();
+    graph.add(id(0), id(1)).expect("no cycle");
+    graph.add(id(2), id(3)).expect("no cycle");
+
+    let tasks = [id(0), id(1), id(2), id(3)];
+    let order = graph
+        .topological_order(&tasks)
+        .expect("the graph is acyclic");
+    let position_of = |task: TaskId| {
+        order
+            .iter()
+            .position(|candidate| *candidate == task)
+            .expect("present")
+    };
+    assert!(position_of(id(0)) < position_of(id(1)));
+    assert!(position_of(id(2)) < position_of(id(3)));
+}