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feat: add IntegerExpressionMembership model (#552) (#814)
* feat: add IntegerExpressionMembership model (#552) Add the Integer Expression Membership problem: given a recursive integer expression tree over union and Minkowski sum operations on singleton positive integers, and a target K, decide whether K belongs to the set represented by the expression. Includes model, tests, CLI support, canonical example, and paper entry. Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com> * Fix agentic review issues: overflow safety, positive-integer validation - Use checked_add for u64 addition to prevent overflow panic (evaluates to Or(false) on overflow instead of crashing) - Validate all Atom values > 0 and target > 0 in constructor - Add CLI-side validation with clean error messages - Add boundary tests: overflow safety, zero atom/target rejection Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com> --------- Co-authored-by: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
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‎docs/paper/reductions.typ‎

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"FlowShopScheduling": [Flow Shop Scheduling],
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"JobShopScheduling": [Job-Shop Scheduling],
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"GroupingBySwapping": [Grouping by Swapping],
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"IntegerExpressionMembership": [Integer Expression Membership],
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"MinimumCutIntoBoundedSets": [Minimum Cut Into Bounded Sets],
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"MinimumDummyActivitiesPert": [Minimum Dummy Activities in PERT Networks],
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"MinimumSumMulticenter": [Minimum Sum Multicenter],
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]
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}
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#{
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let x = load-model-example("IntegerExpressionMembership")
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[
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#problem-def("IntegerExpressionMembership")[
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Given a recursive integer expression $e$ over union ($union$) and Minkowski sum ($+$) operations on singleton positive integers, and a target $K in NN^+$, determine whether $K in "eval"(e)$, where $"eval"("Atom"(n)) = \{n\}$, $"eval"(F union G) = "eval"(F) union "eval"(G)$, and $"eval"(F + G) = \{m + n : m in "eval"(F), n in "eval"(G)\}$.
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][
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Integer Expression Membership asks whether a specific integer is reachable by selecting one branch at each union node in a recursive expression tree. Each configuration assigns a binary choice (left or right) at every union node in depth-first order; with all unions resolved, the expression reduces to a chain of sums and atoms that evaluates to a single integer.#footnote[No algorithm improving on brute-force enumeration of all $2^u$ union-branch combinations (where $u$ is the number of union nodes) is known for the general case.]
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*Example.* Consider $e = (1 union 4) + (3 union 6) + (2 union 5)$ with $K = 12$. There are $u = 3$ union nodes, giving $2^3 = 8$ configurations. The reachable set is $\{6, 9, 12, 15\}$. Since $12 in \{6, 9, 12, 15\}$, the answer is YES. One witness: choose right ($4$), right ($6$), left ($2$) at the three union nodes, giving $4 + 6 + 2 = 12 = K$.
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#pred-commands(
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"pred create --example " + problem-spec(x) + " -o iem.json",
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"pred solve iem.json --solver brute-force",
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"pred evaluate iem.json --config " + x.optimal_config.map(str).join(","),
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)
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#figure(
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canvas(length: 0.7cm, {
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import draw: *
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// Draw expression tree: Sum( Sum( Union(1,4), Union(3,6) ), Union(2,5) )
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let node-r = 0.45
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let level-gap = 1.5
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let spread = 2.0
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// Root: Sum (level 0)
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circle((0, 0), radius: node-r, name: "root", fill: rgb("#e8f0fe"))
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content("root", text(8pt, $+$))
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// Left child: Sum (level 1)
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circle((-spread, -level-gap), radius: node-r, name: "lsum", fill: rgb("#e8f0fe"))
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content("lsum", text(8pt, $+$))
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line("root.south-west", "lsum.north", mark: (end: "straight"))
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// Right child: Union (level 1)
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circle((spread, -level-gap), radius: node-r, name: "ru", fill: rgb("#fce8e6"))
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content("ru", text(8pt, $union$))
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line("root.south-east", "ru.north", mark: (end: "straight"))
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// Left-Left: Union (level 2)
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circle((-spread - 1.2, -2 * level-gap), radius: node-r, name: "llu", fill: rgb("#fce8e6"))
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content("llu", text(8pt, $union$))
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line("lsum.south-west", "llu.north", mark: (end: "straight"))
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// Left-Right: Union (level 2)
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circle((-spread + 1.2, -2 * level-gap), radius: node-r, name: "lru", fill: rgb("#fce8e6"))
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content("lru", text(8pt, $union$))
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line("lsum.south-east", "lru.north", mark: (end: "straight"))
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// Leaves (level 3)
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let leaf-spread = 0.7
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for (parent, vals, xoff) in (
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("llu", (1, 4), -spread - 1.2),
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("lru", (3, 6), -spread + 1.2),
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("ru", (2, 5), spread),
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) {
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circle((xoff - leaf-spread, -3 * level-gap), radius: node-r, name: parent + "l", fill: rgb("#e6f4ea"))
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content(parent + "l", text(8pt, str(vals.at(0))))
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line(parent + ".south-west", (parent + "l") + ".north", mark: (end: "straight"))
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circle((xoff + leaf-spread, -3 * level-gap), radius: node-r, name: parent + "r", fill: rgb("#e6f4ea"))
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content(parent + "r", text(8pt, str(vals.at(1))))
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line(parent + ".south-east", (parent + "r") + ".north", mark: (end: "straight"))
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}
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}),
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caption: [Expression tree $e = (1 union 4) + (3 union 6) + (2 union 5)$ with target $K = 12$. Blue nodes are sums ($+$), red nodes are unions ($union$), green leaves are atoms. Choosing right at the first two unions and left at the third yields $4 + 6 + 2 = 12$.],
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) <fig:integer-expression-membership>
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]
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]
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}
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#{
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let x = load-model-example("FeasibleBasisExtension")
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let A = x.instance.matrix

‎problemreductions-cli/src/cli.rs‎

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@@ -319,6 +319,7 @@ Flags by problem type:
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D2CIF --arcs, --capacities, --source-1, --sink-1, --source-2, --sink-2, --requirement-1, --requirement-2
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MinimumDummyActivitiesPert --arcs [--num-vertices]
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CBQ --domain-size, --relations, --conjuncts-spec
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IntegerExpressionMembership --expression (JSON), --target
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ILP, CircuitSAT (via reduction only)
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Geometry graph variants (use slash notation, e.g., MIS/KingsSubgraph):
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/// Target string for StringToStringCorrection (comma-separated symbol indices, e.g., "0,1,3,2")
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#[arg(long)]
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pub target_string: Option<String>,
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/// Expression tree for IntegerExpressionMembership (JSON, e.g., '{"Sum":[{"Atom":1},{"Atom":2}]}')
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#[arg(long)]
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pub expression: Option<String>,
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/// Coefficient a for QuadraticDiophantineEquations (coefficient of x²)
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#[arg(long)]
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pub coeff_a: Option<u64>,

‎problemreductions-cli/src/commands/create.rs‎

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@@ -23,15 +23,15 @@ use problemreductions::models::graph::{
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use problemreductions::models::misc::{
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AdditionalKey, BinPacking, BoyceCoddNormalFormViolation, CapacityAssignment, CbqRelation,
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ConjunctiveBooleanQuery, ConsistencyOfDatabaseFrequencyTables, EnsembleComputation,
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ExpectedRetrievalCost, FlowShopScheduling, FrequencyTable, GroupingBySwapping,
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JobShopScheduling, KnownValue, KthLargestMTuple, LongestCommonSubsequence,
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MinimumTardinessSequencing, MultiprocessorScheduling, PaintShop, PartiallyOrderedKnapsack,
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ProductionPlanning, QueryArg, RectilinearPictureCompression, ResourceConstrainedScheduling,
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SchedulingToMinimizeWeightedCompletionTime, SchedulingWithIndividualDeadlines,
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SequencingToMinimizeMaximumCumulativeCost, SequencingToMinimizeWeightedCompletionTime,
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SequencingToMinimizeWeightedTardiness, SequencingWithReleaseTimesAndDeadlines,
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SequencingWithinIntervals, ShortestCommonSupersequence, StringToStringCorrection, SubsetSum,
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SumOfSquaresPartition, ThreePartition, TimetableDesign,
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ExpectedRetrievalCost, FlowShopScheduling, FrequencyTable, GroupingBySwapping, IntExpr,
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IntegerExpressionMembership, JobShopScheduling, KnownValue, KthLargestMTuple,
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LongestCommonSubsequence, MinimumTardinessSequencing, MultiprocessorScheduling, PaintShop,
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PartiallyOrderedKnapsack, ProductionPlanning, QueryArg, RectilinearPictureCompression,
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ResourceConstrainedScheduling, SchedulingToMinimizeWeightedCompletionTime,
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SchedulingWithIndividualDeadlines, SequencingToMinimizeMaximumCumulativeCost,
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SequencingToMinimizeWeightedCompletionTime, SequencingToMinimizeWeightedTardiness,
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SequencingWithReleaseTimesAndDeadlines, SequencingWithinIntervals, ShortestCommonSupersequence,
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StringToStringCorrection, SubsetSum, SumOfSquaresPartition, ThreePartition, TimetableDesign,
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};
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use problemreductions::models::BiconnectivityAugmentation;
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use problemreductions::prelude::*;
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&& args.domain_size.is_none()
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&& args.relations.is_none()
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&& args.conjuncts_spec.is_none()
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&& args.expression.is_none()
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&& args.deps.is_none()
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&& args.query.is_none()
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&& args.coeff_a.is_none()
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}
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"IntegerKnapsack" => "--sizes 3,4,5,2,7 --values 4,5,7,3,9 --capacity 15",
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"SubsetSum" => "--sizes 3,7,1,8,2,4 --target 11",
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"IntegerExpressionMembership" => {
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"--expression '{\"Sum\":[{\"Sum\":[{\"Union\":[{\"Atom\":1},{\"Atom\":4}]},{\"Union\":[{\"Atom\":3},{\"Atom\":6}]}]},{\"Union\":[{\"Atom\":2},{\"Atom\":5}]}]}' --target 12"
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}
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"ThreePartition" => "--sizes 4,5,6,4,6,5 --bound 15",
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"KthLargestMTuple" => "--sets \"2,5,8;3,6;1,4,7\" --k 14 --bound 12",
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"QuadraticDiophantineEquations" => "--coeff-a 3 --coeff-b 5 --coeff-c 53",
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)
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}
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// IntegerExpressionMembership
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"IntegerExpressionMembership" => {
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let usage = "Usage: pred create IntegerExpressionMembership --expression '{\"Sum\":[{\"Atom\":1},{\"Atom\":2}]}' --target 3";
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let expr_str = args.expression.as_deref().ok_or_else(|| {
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anyhow::anyhow!(
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"IntegerExpressionMembership requires --expression and --target\n\n{usage}"
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)
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})?;
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let target = args.target.as_deref().ok_or_else(|| {
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anyhow::anyhow!("IntegerExpressionMembership requires --target\n\n{usage}")
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})?;
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let target: u64 = target
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.parse()
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.context("IntegerExpressionMembership --target must be a positive integer")?;
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if target == 0 {
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anyhow::bail!("IntegerExpressionMembership --target must be > 0");
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}
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let expr: IntExpr = serde_json::from_str(expr_str)
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.context("IntegerExpressionMembership --expression must be valid JSON representing an IntExpr tree")?;
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if !expr.all_atoms_positive() {
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anyhow::bail!("IntegerExpressionMembership --expression must contain only positive integers (all Atom values > 0)");
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}
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(
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ser(IntegerExpressionMembership::new(expr, target))?,
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resolved_variant.clone(),
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)
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}
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// ThreePartition
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"ThreePartition" => {
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let sizes_str = args.sizes.as_deref().ok_or_else(|| {
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storage: None,
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quantifiers: None,
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homologous_pairs: None,
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expression: None,
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coeff_a: None,
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coeff_b: None,
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rhs: None,

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