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Merged
zazabap merged 7 commits into from
Mar 16, 2026
Merged

Fix #253: [Model] MultipleChoiceBranching #656

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bbe4ec6
Add plan for #253: [Model] MultipleChoiceBranching
GiggleLiu Mar 16, 2026
305f112
Implement #253: [Model] MultipleChoiceBranching
GiggleLiu Mar 16, 2026
ed2bf69
chore: remove plan file after implementation
GiggleLiu Mar 16, 2026
2f24524
Merge remote-tracking branch 'origin/main' into review-pr-656-issue-2…
GiggleLiu Mar 16, 2026
cf5688b
fix: address MultipleChoiceBranching review feedback
GiggleLiu Mar 16, 2026
f6f56b2
fix: harden MultipleChoiceBranching review follow-ups
GiggleLiu Mar 16, 2026
32fd9b5
Merge main and fix coverage for MultipleChoiceBranching
zazabap Mar 16, 2026
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41 changes: 41 additions & 0 deletions docs/paper/reductions.typ
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Expand Up @@ -101,6 +101,7 @@
"SubsetSum": [Subset Sum],
"MinimumFeedbackArcSet": [Minimum Feedback Arc Set],
"MinimumFeedbackVertexSet": [Minimum Feedback Vertex Set],
"MultipleChoiceBranching": [Multiple Choice Branching],
"ShortestCommonSupersequence": [Shortest Common Supersequence],
"MinimumSumMulticenter": [Minimum Sum Multicenter],
"SteinerTree": [Steiner Tree],
Expand Down Expand Up @@ -1945,6 +1946,46 @@ NP-completeness was established by Garey, Johnson, and Stockmeyer @gareyJohnsonS
*Example.* Consider $G$ with $V = {0, 1, 2, 3, 4, 5}$ and arcs $(0 arrow 1), (1 arrow 2), (2 arrow 0), (1 arrow 3), (3 arrow 4), (4 arrow 1), (2 arrow 5), (5 arrow 3), (3 arrow 0)$. This graph contains four directed cycles: $0 arrow 1 arrow 2 arrow 0$, $1 arrow 3 arrow 4 arrow 1$, $0 arrow 1 arrow 3 arrow 0$, and $2 arrow 5 arrow 3 arrow 0 arrow 1 arrow 2$. Removing $A' = {(0 arrow 1), (3 arrow 4)}$ breaks all four cycles (vertex 0 becomes a sink in the residual graph), giving a minimum FAS of size 2.
]

#{
let x = load-model-example("MultipleChoiceBranching")
let nv = graph-num-vertices(x.instance)
let arcs = x.instance.graph.inner.edges.map(e => (e.at(0), e.at(1)))
let sol = x.samples.at(0)
let chosen = sol.config.enumerate().filter(((i, v)) => v == 1).map(((i, _)) => i)
[
#problem-def("MultipleChoiceBranching")[
Given a directed graph $G = (V, A)$, arc weights $w: A -> ZZ^+$, a partition $A_1, A_2, dots, A_m$ of $A$, and a threshold $K in ZZ^+$, determine whether there exists a subset $A' subset.eq A$ with $sum_(a in A') w(a) >= K$ such that every vertex has in-degree at most one in $(V, A')$, the selected subgraph $(V, A')$ is acyclic, and $|A' inter A_i| <= 1$ for every partition group.
][
Multiple Choice Branching is the directed-graph problem ND11 in Garey & Johnson @garey1979. The partition constraint turns the polynomial-time maximum branching setting into an NP-complete decision problem: Garey and Johnson note that the problem remains NP-complete even when the digraph is strongly connected and all weights are equal, while the special case in which every partition group has size 1 reduces to ordinary maximum branching and becomes polynomial-time solvable @garey1979.

A conservative exact algorithm enumerates all $2^{|A|}$ arc subsets and checks the partition, in-degree, acyclicity, and threshold constraints in polynomial time. This is the brute-force search space used by the implementation.#footnote[We use the registry complexity bound $O^*(2^{|A|})$ for the full partitioned problem.]

*Example.* Consider the digraph on $n = #nv$ vertices with arcs $(0 arrow 1), (0 arrow 2), (1 arrow 3), (2 arrow 3), (1 arrow 4), (3 arrow 5), (4 arrow 5), (2 arrow 4)$, partition groups $A_1 = {(0 arrow 1), (0 arrow 2)}$, $A_2 = {(1 arrow 3), (2 arrow 3)}$, $A_3 = {(1 arrow 4), (2 arrow 4)}$, $A_4 = {(3 arrow 5), (4 arrow 5)}$, and threshold $K = 10$. The highlighted selection $A' = {(0 arrow 1), (1 arrow 3), (2 arrow 4), (3 arrow 5)}$ has total weight $3 + 4 + 3 + 3 = 13 >= 10$, uses exactly one arc from each partition group, and gives in-degrees 1 at vertices $1, 3, 4,$ and $5$. Because every selected arc points strictly left-to-right in the drawing, the selected subgraph is acyclic. The canonical fixture contains #x.optimal.len() satisfying selections for this instance; the figure highlights one of them.

#figure({
let verts = ((0, 1.6), (1.3, 2.3), (1.3, 0.9), (3.0, 2.3), (3.0, 0.9), (4.6, 1.6))
canvas(length: 1cm, {
for (idx, arc) in arcs.enumerate() {
let (u, v) = arc
let selected = chosen.contains(idx)
draw.line(
verts.at(u),
verts.at(v),
stroke: if selected { 2pt + graph-colors.at(0) } else { 0.9pt + luma(180) },
mark: (end: "straight", scale: if selected { 0.5 } else { 0.4 }),
)
}
for (k, pos) in verts.enumerate() {
g-node(pos, name: "v" + str(k), label: [$v_#k$])
}
})
},
caption: [Directed graph for Multiple Choice Branching. Blue arcs show the satisfying branching $(0 arrow 1), (1 arrow 3), (2 arrow 4), (3 arrow 5)$ of total weight 13; gray arcs are available but unselected.],
) <fig:mcb-example>
]
]
}

#problem-def("FlowShopScheduling")[
Given $m$ processors and a set $J$ of $n$ jobs, where each job $j in J$ consists of $m$ tasks $t_1 [j], t_2 [j], dots, t_m [j]$ with lengths $ell(t_i [j]) in ZZ^+_0$, and a deadline $D in ZZ^+$, determine whether there exists a permutation schedule $pi$ of the jobs such that all jobs complete by time $D$. Each job must be processed on machines $1, 2, dots, m$ in order, and job $j$ cannot start on machine $i+1$ until its task on machine $i$ is completed.
][
Expand Down
26 changes: 26 additions & 0 deletions docs/src/reductions/problem_schemas.json
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Expand Up @@ -575,6 +575,32 @@
}
]
},
{
"name": "MultipleChoiceBranching",
"description": "Find a branching with partition constraints and weight at least K",
"fields": [
{
"name": "graph",
"type_name": "DirectedGraph",
"description": "The directed graph G=(V,A)"
},
{
"name": "weights",
"type_name": "Vec<W>",
"description": "Arc weights w(a) for each arc a in A"
},
{
"name": "partition",
"type_name": "Vec<Vec<usize>>",
"description": "Partition of arc indices; each arc index must appear in exactly one group"
},
{
"name": "threshold",
"type_name": "W::Sum",
"description": "Weight threshold K"
}
]
},
{
"name": "OptimalLinearArrangement",
"description": "Find a vertex ordering on a line with total edge length at most K",
Expand Down
61 changes: 35 additions & 26 deletions docs/src/reductions/reduction_graph.json
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Expand Up @@ -429,6 +429,15 @@
"doc_path": "models/graph/struct.MinimumVertexCover.html",
"complexity": "1.1996^num_vertices"
},
{
"name": "MultipleChoiceBranching",
"variant": {
"weight": "i32"
},
"category": "graph",
"doc_path": "models/graph/struct.MultipleChoiceBranching.html",
"complexity": "2^num_arcs"
},
{
"name": "OptimalLinearArrangement",
"variant": {
Expand Down Expand Up @@ -599,7 +608,7 @@
},
{
"source": 4,
"target": 57,
"target": 58,
"overhead": [
{
"field": "num_spins",
Expand Down Expand Up @@ -659,7 +668,7 @@
},
{
"source": 13,
"target": 51,
"target": 52,
"overhead": [
{
"field": "num_vars",
Expand Down Expand Up @@ -700,7 +709,7 @@
},
{
"source": 20,
"target": 51,
"target": 52,
"overhead": [
{
"field": "num_vars",
Expand All @@ -726,7 +735,7 @@
},
{
"source": 21,
"target": 51,
"target": 52,
"overhead": [
{
"field": "num_vars",
Expand All @@ -752,7 +761,7 @@
},
{
"source": 22,
"target": 51,
"target": 52,
"overhead": [
{
"field": "num_vars",
Expand All @@ -763,7 +772,7 @@
},
{
"source": 22,
"target": 61,
"target": 62,
"overhead": [
{
"field": "num_elements",
Expand All @@ -774,7 +783,7 @@
},
{
"source": 23,
"target": 53,
"target": 54,
"overhead": [
{
"field": "num_clauses",
Expand All @@ -793,7 +802,7 @@
},
{
"source": 24,
"target": 51,
"target": 52,
"overhead": [
{
"field": "num_vars",
Expand All @@ -819,7 +828,7 @@
},
{
"source": 27,
"target": 57,
"target": 58,
"overhead": [
{
"field": "num_spins",
Expand Down Expand Up @@ -1134,7 +1143,7 @@
},
{
"source": 39,
"target": 51,
"target": 52,
"overhead": [
{
"field": "num_vars",
Expand Down Expand Up @@ -1249,7 +1258,7 @@
"doc_path": "rules/minimumvertexcover_minimumsetcovering/index.html"
},
{
"source": 51,
"source": 52,
"target": 13,
"overhead": [
{
Expand All @@ -1264,8 +1273,8 @@
"doc_path": "rules/qubo_ilp/index.html"
},
{
"source": 51,
"target": 56,
"source": 52,
"target": 57,
"overhead": [
{
"field": "num_spins",
Expand All @@ -1275,7 +1284,7 @@
"doc_path": "rules/spinglass_qubo/index.html"
},
{
"source": 53,
"source": 54,
"target": 4,
"overhead": [
{
Expand All @@ -1290,7 +1299,7 @@
"doc_path": "rules/sat_circuitsat/index.html"
},
{
"source": 53,
"source": 54,
"target": 17,
"overhead": [
{
Expand All @@ -1305,7 +1314,7 @@
"doc_path": "rules/sat_coloring/index.html"
},
{
"source": 53,
"source": 54,
"target": 22,
"overhead": [
{
Expand All @@ -1320,7 +1329,7 @@
"doc_path": "rules/sat_ksat/index.html"
},
{
"source": 53,
"source": 54,
"target": 32,
"overhead": [
{
Expand All @@ -1335,7 +1344,7 @@
"doc_path": "rules/sat_maximumindependentset/index.html"
},
{
"source": 53,
"source": 54,
"target": 41,
"overhead": [
{
Expand All @@ -1350,8 +1359,8 @@
"doc_path": "rules/sat_minimumdominatingset/index.html"
},
{
"source": 56,
"target": 51,
"source": 57,
"target": 52,
"overhead": [
{
"field": "num_vars",
Expand All @@ -1361,7 +1370,7 @@
"doc_path": "rules/spinglass_qubo/index.html"
},
{
"source": 57,
"source": 58,
"target": 27,
"overhead": [
{
Expand All @@ -1376,8 +1385,8 @@
"doc_path": "rules/spinglass_maxcut/index.html"
},
{
"source": 57,
"target": 56,
"source": 58,
"target": 57,
"overhead": [
{
"field": "num_spins",
Expand All @@ -1391,7 +1400,7 @@
"doc_path": "rules/spinglass_casts/index.html"
},
{
"source": 62,
"source": 63,
"target": 13,
"overhead": [
{
Expand All @@ -1406,8 +1415,8 @@
"doc_path": "rules/travelingsalesman_ilp/index.html"
},
{
"source": 62,
"target": 51,
"source": 63,
"target": 52,
"overhead": [
{
"field": "num_vars",
Expand Down
7 changes: 6 additions & 1 deletion problemreductions-cli/src/cli.rs
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Expand Up @@ -240,6 +240,7 @@ Flags by problem type:
CVP --basis, --target-vec [--bounds]
OptimalLinearArrangement --graph, --bound
RuralPostman (RPP) --graph, --edge-weights, --required-edges, --bound
MultipleChoiceBranching --arcs [--weights] --partition --bound [--num-vertices]
SubgraphIsomorphism --graph (host), --pattern (pattern)
LCS --strings
FAS --arcs [--weights] [--num-vertices]
Expand All @@ -264,6 +265,7 @@ Examples:
pred create MIS --graph 0-1,1-2,2-3 --weights 1,1,1
pred create SAT --num-vars 3 --clauses \"1,2;-1,3\"
pred create QUBO --matrix \"1,0.5;0.5,2\"
pred create MultipleChoiceBranching/i32 --arcs \"0>1,0>2,1>3,2>3,1>4,3>5,4>5,2>4\" --weights 3,2,4,1,2,3,1,3 --partition \"0,1;2,3;4,7;5,6\" --bound 10
pred create MIS/KingsSubgraph --positions \"0,0;1,0;1,1;0,1\"
pred create MIS/UnitDiskGraph --positions \"0,0;1,0;0.5,0.8\" --radius 1.5
pred create MIS --random --num-vertices 10 --edge-prob 0.3
Expand Down Expand Up @@ -380,6 +382,9 @@ pub struct CreateArgs {
/// Sets for SetPacking/SetCovering (semicolon-separated, e.g., "0,1;1,2;0,2")
#[arg(long)]
pub sets: Option<String>,
/// Partition groups for arc-index partitions (semicolon-separated, e.g., "0,1;2,3")
#[arg(long)]
pub partition: Option<String>,
/// Universe size for MinimumSetCovering
#[arg(long)]
pub universe: Option<usize>,
Expand Down Expand Up @@ -413,7 +418,7 @@ pub struct CreateArgs {
/// Required edge indices for RuralPostman (comma-separated, e.g., "0,2,4")
#[arg(long)]
pub required_edges: Option<String>,
/// Upper bound or length bound (for LengthBoundedDisjointPaths, OptimalLinearArrangement, RuralPostman, or SCS)
/// Upper bound or length bound (for LengthBoundedDisjointPaths, MultipleChoiceBranching, OptimalLinearArrangement, RuralPostman, or SCS)
#[arg(long, allow_hyphen_values = true)]
pub bound: Option<i64>,
/// Pattern graph edge list for SubgraphIsomorphism (e.g., 0-1,1-2,2-0)
Expand Down
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