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Fix #400: [Model] SetBasis (#661)
* Add plan for #400: [Model] SetBasis * Implement #400: [Model] SetBasis * Add CLI coverage for #400 SetBasis * chore: remove plan file after implementation * fix: address SetBasis review feedback - harden SetBasis evaluation against malformed serialized input - validate SetBasis CLI subsets before construction - align problem-specific create help with actual CLI flags * fix: add coverage tests and constructor edge cases for SetBasis Add targeted tests for uncovered code paths: - is_subset returning false (basis set not subset of target) - is_valid_solution wrapper method - k=0 with empty/non-empty collection (edge cases) - Empty collection with k > 0 (trivially satisfiable) Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com> --------- Co-authored-by: zazabap <sweynan@icloud.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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@@ -75,6 +75,7 @@
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"MaximumClique": [Maximum Clique],
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"MaximumSetPacking": [Maximum Set Packing],
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"MinimumSetCovering": [Minimum Set Covering],
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"SetBasis": [Set Basis],
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"SpinGlass": [Spin Glass],
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"QUBO": [QUBO],
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"ILP": [Integer Linear Programming],
@@ -1107,6 +1108,43 @@ NP-completeness was established by Garey, Johnson, and Stockmeyer @gareyJohnsonS
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]
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}
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#{
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let x = load-model-example("SetBasis")
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let coll = x.instance.collection
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let m = coll.len()
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let U-size = x.instance.universe_size
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let k = x.instance.k
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let sample = x.samples.at(0)
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let sat-count = x.optimal.len()
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let basis = range(k).map(i =>
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range(U-size).filter(j => sample.config.at(i * U-size + j) == 1)
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)
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let fmt-set(s) = "${" + s.map(e => str(e + 1)).join(", ") + "}$"
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[
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#problem-def("SetBasis")[
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Given finite set $S$, collection $cal(C)$ of subsets of $S$, and integer $k$, does there exist a family $cal(B) = {B_1, ..., B_k}$ with each $B_i subset.eq S$ such that for every $C in cal(C)$ there exists $cal(B)_C subset.eq cal(B)$ with $union.big_(B in cal(B)_C) B = C$?
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][
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The Set Basis problem was shown NP-complete by Stockmeyer @stockmeyer1975setbasis and appears as SP7 in Garey & Johnson @garey1979. It asks for an exact union-based description of a family of sets, unlike Set Cover which only requires covering the underlying universe. Applications include data compression, database schema design, and Boolean function minimization. The library's decision encoding uses $k |S|$ membership bits, so brute-force over those bits gives an $O^*(2^(k |S|))$ exact algorithm#footnote[This is the direct search bound induced by the encoding implemented here; we are not aware of a faster general exact worst-case algorithm for this representation.].
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*Example.* Let $S = {1, 2, 3, 4}$, $k = #k$, and $cal(C) = {#range(m).map(i => $C_#(i + 1)$).join(", ")}$ with #coll.enumerate().map(((i, s)) => $C_#(i + 1) = #fmt-set(s)$).join(", "). The sample basis from the issue is $cal(B) = {#range(k).map(i => $B_#(i + 1)$).join(", ")}$ with #basis.enumerate().map(((i, s)) => $B_#(i + 1) = #fmt-set(s)$).join(", "). Then $C_1 = B_1 union B_2$, $C_2 = B_2 union B_3$, $C_3 = B_1 union B_3$, and $C_4 = B_1 union B_2 union B_3$. There are #sat-count satisfying encodings in total: the singleton basis can be permuted in $3! = 6$ ways, and the three pair sets $C_1, C_2, C_3$ also form a basis with another six row permutations.
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#figure(
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canvas(length: 1cm, {
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let elems = ((-0.9, 0.2), (0.0, -0.5), (0.9, 0.2), (1.8, -0.5))
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for i in range(k) {
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let positions = basis.at(i).map(e => elems.at(e))
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sregion(positions, pad: 0.28, label: [$B_#(i + 1)$], ..sregion-selected)
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}
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for (idx, pos) in elems.enumerate() {
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selem(pos, label: [#(idx + 1)], fill: if idx < 3 { black } else { luma(160) })
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}
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}),
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caption: [Set Basis example: the singleton basis $cal(B) = {#range(k).map(i => $B_#(i + 1)$).join(", ")}$ reconstructs every target set in $cal(C)$; element $4$ is unused by the target family.],
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) <fig:set-basis>
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]
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]
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}
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== Optimization Problems
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#{

‎docs/paper/references.bib‎

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doi = {10.1145/1502793.1502794}
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}
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@techreport{stockmeyer1975setbasis,
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author = {Larry J. Stockmeyer},
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title = {The Set Basis Problem Is NP-Complete},
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institution = {IBM Thomas J. Watson Research Center},
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number = {RC 5431},
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address = {Yorktown Heights, New York},
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year = {1975}
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}
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@article{cygan2014,
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author = {Marek Cygan and Daniel Lokshtanov and Marcin Pilipczuk and Micha{\l} Pilipczuk and Saket Saurabh},
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title = {Minimum Bisection Is Fixed Parameter Tractable},

‎docs/src/reductions/problem_schemas.json‎

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}
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]
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},
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{
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"name": "SetBasis",
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"description": "Determine whether a collection of sets admits a basis of size k under union",
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"fields": [
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{
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"name": "universe_size",
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"type_name": "usize",
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"description": "Size of the ground set S"
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},
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{
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"name": "collection",
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"type_name": "Vec<Vec<usize>>",
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"description": "Collection C of target subsets of S"
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},
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{
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"name": "k",
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"type_name": "usize",
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"description": "Required number of basis sets"
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}
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]
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},
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{
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"name": "ShortestCommonSupersequence",
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"description": "Find a common supersequence of bounded length for a set of strings",

‎docs/src/reductions/reduction_graph.json‎

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"doc_path": "models/formula/struct.Satisfiability.html",
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"complexity": "2^num_variables"
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},
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{
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"name": "SetBasis",
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"variant": {},
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"category": "set",
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"doc_path": "models/set/struct.SetBasis.html",
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"complexity": "2^(basis_size * universe_size)"
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},
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{
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"name": "ShortestCommonSupersequence",
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"variant": {},
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},
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{
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"source": 4,
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"target": 54,
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"target": 55,
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"overhead": [
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{
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"field": "num_spins",
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},
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{
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"source": 21,
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"target": 58,
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"target": 59,
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"overhead": [
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{
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"field": "num_elements",
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},
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{
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"source": 25,
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"target": 55,
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"overhead": [
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{
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"field": "num_spins",
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},
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{
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"source": 49,
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"target": 53,
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"target": 54,
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"overhead": [
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{
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"field": "num_spins",
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"doc_path": "rules/sat_minimumdominatingset/index.html"
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},
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{
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"source": 53,
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"source": 54,
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"target": 49,
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"overhead": [
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{
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"doc_path": "rules/spinglass_qubo/index.html"
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},
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{
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"source": 54,
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"source": 55,
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"target": 25,
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"overhead": [
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{
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"doc_path": "rules/spinglass_maxcut/index.html"
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},
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{
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"source": 54,
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"source": 55,
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"target": 54,
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"overhead": [
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"field": "num_spins",
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"doc_path": "rules/spinglass_casts/index.html"
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},
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{
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"source": 60,
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"target": 12,
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"overhead": [
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{
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"doc_path": "rules/travelingsalesman_ilp/index.html"
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},
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{
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"source": 60,
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"target": 49,
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"overhead": [
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{

‎problemreductions-cli/src/cli.rs‎

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MaximumSetPacking --sets [--weights]
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MinimumSetCovering --universe, --sets [--weights]
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X3C (ExactCoverBy3Sets) --universe, --sets (3 elements each)
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SetBasis --universe, --sets, --k
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BicliqueCover --left, --right, --biedges, --k
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BMF --matrix (0/1), --rank
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SteinerTree --graph, --edge-weights, --terminals
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pred create MIS/UnitDiskGraph --positions \"0,0;1,0;0.5,0.8\" --radius 1.5
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pred create MIS --random --num-vertices 10 --edge-prob 0.3
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pred create FVS --arcs \"0>1,1>2,2>0\" --weights 1,1,1
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pred create X3C --universe 9 --sets \"0,1,2;0,2,4;3,4,5;3,5,7;6,7,8;1,4,6;2,5,8\"")]
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pred create X3C --universe 9 --sets \"0,1,2;0,2,4;3,4,5;3,5,7;6,7,8;1,4,6;2,5,8\"
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pred create SetBasis --universe 4 --sets \"0,1;1,2;0,2;0,1,2\" --k 3")]
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pub struct CreateArgs {
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/// Problem type (e.g., MIS, QUBO, SAT). Omit when using --example.
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#[arg(value_parser = crate::problem_name::ProblemNameParser)]

‎problemreductions-cli/src/commands/create.rs‎

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}
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}
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fn cli_flag_name(field_name: &str) -> String {
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match field_name {
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"universe_size" => "universe".to_string(),
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"collection" | "subsets" => "sets".to_string(),
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"left_size" => "left".to_string(),
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"right_size" => "right".to_string(),
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"edges" => "biedges".to_string(),
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"vertex_weights" => "weights".to_string(),
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"edge_lengths" => "edge-weights".to_string(),
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"num_tasks" => "n".to_string(),
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"precedences" => "precedence-pairs".to_string(),
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_ => field_name.replace('_', "-"),
228+
}
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}
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fn example_for(canonical: &str, graph_type: Option<&str>) -> &'static str {
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match canonical {
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"MaximumIndependentSet"
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}
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"SubgraphIsomorphism" => "--graph 0-1,1-2,2-0 --pattern 0-1",
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"SubsetSum" => "--sizes 3,7,1,8,2,4 --target 11",
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"SetBasis" => "--universe 4 --sets \"0,1;1,2;0,2;0,1,2\" --k 3",
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"ShortestCommonSupersequence" => "--strings \"0,1,2;1,2,0\" --bound 4",
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_ => "",
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}
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let hint = type_format_hint(&field.type_name, graph_type);
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eprintln!(
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" --{:<16} {} ({})",
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field.name.replace('_', "-"),
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cli_flag_name(&field.name),
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field.description,
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hint
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);
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)
719735
}
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// SetBasis
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"SetBasis" => {
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let universe = args.universe.ok_or_else(|| {
740+
anyhow::anyhow!(
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"SetBasis requires --universe, --sets, and --k\n\n\
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Usage: pred create SetBasis --universe 4 --sets \"0,1;1,2;0,2;0,1,2\" --k 3"
743+
)
744+
})?;
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let k = args.k.ok_or_else(|| {
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anyhow::anyhow!(
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"SetBasis requires --k\n\n\
748+
Usage: pred create SetBasis --universe 4 --sets \"0,1;1,2;0,2;0,1,2\" --k 3"
749+
)
750+
})?;
751+
let sets = parse_sets(args)?;
752+
for (i, set) in sets.iter().enumerate() {
753+
for &element in set {
754+
if element >= universe {
755+
bail!(
756+
"Set {} contains element {} which is outside universe of size {}",
757+
i,
758+
element,
759+
universe
760+
);
761+
}
762+
}
763+
}
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(
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ser(problemreductions::models::set::SetBasis::new(
766+
universe, sets, k,
767+
))?,
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resolved_variant.clone(),
769+
)
770+
}
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// BicliqueCover
722773
"BicliqueCover" => {
723774
let left = args.left.ok_or_else(|| {

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