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Fix #290: [Model] IntegralFlowWithMultipliers (#736)
* Add plan for #290: [Model] IntegralFlowWithMultipliers * Add IntegralFlowWithMultipliers model * Polish IntegralFlowWithMultipliers paper and tests * chore: remove plan file after implementation * Fix formatting after merge conflict resolution Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com> * Fix paper solve command: add --solver brute-force (no ILP path for this model) 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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@@ -131,6 +131,7 @@
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"ConsecutiveBlockMinimization": [Consecutive Block Minimization],
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"ConsecutiveOnesSubmatrix": [Consecutive Ones Submatrix],
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"DirectedTwoCommodityIntegralFlow": [Directed Two-Commodity Integral Flow],
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"IntegralFlowWithMultipliers": [Integral Flow With Multipliers],
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"MinMaxMulticenter": [Min-Max Multicenter],
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"FlowShopScheduling": [Flow Shop Scheduling],
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"MinimumCutIntoBoundedSets": [Minimum Cut Into Bounded Sets],
@@ -5251,6 +5252,71 @@ A classical NP-complete problem from Garey and Johnson @garey1979[Ch.~3, p.~76],
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) <fig:d2cif>
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]
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#{
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let x = load-model-example("IntegralFlowWithMultipliers")
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let config = x.optimal_config
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[
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#problem-def("IntegralFlowWithMultipliers")[
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Given a directed graph $G = (V, A)$, distinguished vertices $s, t in V$, arc capacities $c: A -> ZZ^+$, vertex multipliers $h: V backslash {s, t} -> ZZ^+$, and a requirement $R in ZZ^+$, determine whether there exists an integral flow function $f: A -> ZZ_(>= 0)$ such that (1) $f(a) <= c(a)$ for every $a in A$, (2) for each nonterminal vertex $v in V backslash {s, t}$, the value $h(v)$ times the total inflow into $v$ equals the total outflow from $v$, and (3) the net flow into $t$ is at least $R$.
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][
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Integral Flow With Multipliers is Garey and Johnson's gain/loss network problem ND33 @garey1979. Sahni includes the same integral vertex-multiplier formulation among his computationally related problems, where partition-style reductions show that adding discrete gain factors destroys the ordinary max-flow structure @sahni1974. The key wrinkle is that conservation is no longer symmetric: one unit entering a vertex may force several units to leave, so the feasible integral solutions behave more like multiplicative gadgets than classical flow balances.
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When every multiplier equals $1$, the model collapses to ordinary single-commodity max flow and becomes polynomial-time solvable by the standard network-flow machinery summarized in Garey and Johnson @garey1979. Jewell studies a different continuous flow-with-gains model in which gain factors live on arcs and the flow may be fractional @jewell1962. That continuous relaxation remains polynomially tractable, so it should not be conflated with the NP-complete integral vertex-multiplier decision problem catalogued here. In this implementation the witness stores one bounded integer variable per arc, giving the direct exact-search bound $O((C + 1)^m)$ where $m = |A|$ and $C = max_(a in A) c(a)$.
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*Example.* The canonical fixture encodes the Partition multiset ${2, 3, 4, 5, 6, 4}$ using source $s = v_0$, sink $t = v_7$, six unit-capacity arcs out of $s$, six sink arcs with capacities $(2, 3, 4, 5, 6, 4)$, and multipliers $(2, 3, 4, 5, 6, 4)$ on the intermediate vertices. Setting the source arcs to $v_1$, $v_3$, and $v_5$ to $1$ forces outgoing sink arcs of $2$, $4$, and $6$, respectively. The sink therefore receives net inflow $2 + 4 + 6 = 12$, exactly meeting the requirement $R = 12$.
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#pred-commands(
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"pred create --example IntegralFlowWithMultipliers -o integral-flow-with-multipliers.json",
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"pred solve integral-flow-with-multipliers.json --solver brute-force",
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"pred evaluate integral-flow-with-multipliers.json --config " + config.map(str).join(","),
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)
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#figure(
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canvas(length: 0.9cm, {
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import draw: *
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let blue = graph-colors.at(0)
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let gray = luma(180)
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let source = (0, 0)
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let sink = (6, 0)
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let mids = (
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(2.4, 2.5),
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(2.4, 1.5),
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(2.4, 0.5),
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(2.4, -0.5),
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(2.4, -1.5),
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(2.4, -2.5),
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)
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let labels = (
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[$v_1, h = 2$],
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[$v_2, h = 3$],
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[$v_3, h = 4$],
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[$v_4, h = 5$],
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[$v_5, h = 6$],
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[$v_6, h = 4$],
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)
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let active = (0, 2, 4)
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for (i, pos) in mids.enumerate() {
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let chosen = active.contains(i)
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let color = if chosen { blue } else { gray }
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let thickness = if chosen { 1.3pt } else { 0.6pt }
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line(source, pos, stroke: (paint: color, thickness: thickness), mark: (end: "straight", scale: 0.45))
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line(pos, sink, stroke: (paint: color, thickness: thickness), mark: (end: "straight", scale: 0.45))
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circle(pos, radius: 0.22, fill: if chosen { blue.lighten(75%) } else { white }, stroke: 0.6pt)
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content((pos.at(0) + 0.85, pos.at(1)), text(6.5pt, labels.at(i)))
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}
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circle(source, radius: 0.24, fill: blue.lighten(75%), stroke: 0.6pt)
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circle(sink, radius: 0.24, fill: blue.lighten(75%), stroke: 0.6pt)
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content(source, text(7pt, [$s = v_0$]))
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content(sink, text(7pt, [$t = v_7$]))
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}),
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caption: [Integral Flow With Multipliers: the blue branches send one unit from $s$ into $v_1$, $v_3$, and $v_5$, forcing sink inflow $2 + 4 + 6 = 12$ at $t$.],
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) <fig:ifwm>
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]
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]
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}
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#problem-def("AdditionalKey")[
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Given a set $A$ of attribute names, a collection $F$ of functional dependencies on $A$,
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a subset $R subset.eq A$, and a set $K$ of candidate keys for the relational scheme $chevron.l R, F chevron.r$,

‎docs/paper/references.bib‎

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@@ -255,6 +255,27 @@ @article{evenItaiShamir1976
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doi = {10.1137/0205048}
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}
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@article{sahni1974,
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author = {Sartaj Sahni},
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title = {Computationally Related Problems},
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journal = {SIAM Journal on Computing},
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volume = {3},
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number = {4},
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pages = {262--279},
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year = {1974}
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}
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@article{jewell1962,
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author = {William S. Jewell},
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title = {Optimal Flow Through Networks with Gains},
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journal = {Operations Research},
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volume = {10},
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number = {4},
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pages = {476--499},
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year = {1962},
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doi = {10.1287/opre.10.4.476}
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}
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@article{abdelWahabKameda1978,
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author = {H. M. Abdel-Wahab and T. Kameda},
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title = {Scheduling to Minimize Maximum Cumulative Cost Subject to Series-Parallel Precedence Constraints},

‎problemreductions-cli/src/cli.rs‎

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@@ -229,6 +229,7 @@ Flags by problem type:
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PartitionIntoTriangles --graph
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GraphPartitioning --graph
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GeneralizedHex --graph, --source, --sink
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IntegralFlowWithMultipliers --arcs, --capacities, --source, --sink, --multipliers, --requirement
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MinimumCutIntoBoundedSets --graph, --edge-weights, --source, --sink, --size-bound, --cut-bound
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HamiltonianCircuit, HC --graph
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LongestCircuit --graph, --edge-weights, --bound
@@ -313,6 +314,7 @@ Examples:
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pred create SAT --num-vars 3 --clauses \"1,2;-1,3\"
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pred create QUBO --matrix \"1,0.5;0.5,2\"
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pred create GeneralizedHex --graph 0-1,0-2,0-3,1-4,2-4,3-4,4-5 --source 0 --sink 5
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pred create IntegralFlowWithMultipliers --arcs \"0>1,0>2,1>3,2>3\" --capacities 1,1,2,2 --source 0 --sink 3 --multipliers 1,2,3,1 --requirement 2
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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
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pred create StringToStringCorrection --source-string \"0,1,2,3,1,0\" --target-string \"0,1,3,2,1\" --bound 2 | pred solve - --solver brute-force
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pred create MIS/KingsSubgraph --positions \"0,0;1,0;1,1;0,1\"
@@ -356,12 +358,18 @@ pub struct CreateArgs {
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/// Edge capacities for multicommodity flow problems (e.g., 1,1,2)
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#[arg(long)]
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pub capacities: Option<String>,
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/// Vertex multipliers in vertex order (e.g., 1,2,3,1)
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#[arg(long)]
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pub multipliers: Option<String>,
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/// Source vertex for path-based graph problems and MinimumCutIntoBoundedSets
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#[arg(long)]
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pub source: Option<usize>,
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/// Sink vertex for path-based graph problems and MinimumCutIntoBoundedSets
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#[arg(long)]
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pub sink: Option<usize>,
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/// Required sink inflow for IntegralFlowWithMultipliers
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#[arg(long)]
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pub requirement: Option<u64>,
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/// Required number of paths for LengthBoundedDisjointPaths
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#[arg(long)]
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pub num_paths_required: Option<usize>,

‎problemreductions-cli/src/commands/create.rs‎

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@@ -50,8 +50,10 @@ fn all_data_flags_empty(args: &CreateArgs) -> bool {
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&& args.edge_weights.is_none()
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&& args.edge_lengths.is_none()
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&& args.capacities.is_none()
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&& args.multipliers.is_none()
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&& args.source.is_none()
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&& args.sink.is_none()
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&& args.requirement.is_none()
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&& args.num_paths_required.is_none()
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&& args.couplings.is_none()
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&& args.fields.is_none()
@@ -513,6 +515,9 @@ fn example_for(canonical: &str, graph_type: Option<&str>) -> &'static str {
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"KClique" => "--graph 0-1,0-2,1-3,2-3,2-4,3-4 --k 3",
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"GraphPartitioning" => "--graph 0-1,1-2,2-3,0-2,1-3,0-3",
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"GeneralizedHex" => "--graph 0-1,0-2,0-3,1-4,2-4,3-4,4-5 --source 0 --sink 5",
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"IntegralFlowWithMultipliers" => {
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"--arcs \"0>1,0>2,1>3,2>3\" --capacities 1,1,2,2 --source 0 --sink 3 --multipliers 1,2,3,1 --requirement 2"
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}
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"MinimumCutIntoBoundedSets" => {
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"--graph 0-1,1-2,2-3 --edge-weights 1,1,1 --source 0 --sink 3 --size-bound 3 --cut-bound 1"
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}
@@ -1116,6 +1121,95 @@ pub fn create(args: &CreateArgs, out: &OutputConfig) -> Result<()> {
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)
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}
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// IntegralFlowWithMultipliers (directed arcs + capacities + source/sink + multipliers + requirement)
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"IntegralFlowWithMultipliers" => {
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let usage = "Usage: pred create IntegralFlowWithMultipliers --arcs \"0>1,0>2,1>3,2>3\" --capacities 1,1,2,2 --source 0 --sink 3 --multipliers 1,2,3,1 --requirement 2";
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let arcs_str = args.arcs.as_deref().ok_or_else(|| {
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anyhow::anyhow!("IntegralFlowWithMultipliers requires --arcs\n\n{usage}")
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})?;
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let (graph, num_arcs) = parse_directed_graph(arcs_str, args.num_vertices)
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.map_err(|e| anyhow::anyhow!("{e}\n\n{usage}"))?;
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let capacities_str = args.capacities.as_deref().ok_or_else(|| {
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anyhow::anyhow!("IntegralFlowWithMultipliers requires --capacities\n\n{usage}")
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})?;
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let capacities: Vec<u64> = util::parse_comma_list(capacities_str)
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.map_err(|e| anyhow::anyhow!("{e}\n\n{usage}"))?;
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if capacities.len() != num_arcs {
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bail!(
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"Expected {} capacities but got {}\n\n{}",
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num_arcs,
1141+
capacities.len(),
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usage
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);
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}
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for (arc_index, &capacity) in capacities.iter().enumerate() {
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let fits = usize::try_from(capacity)
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.ok()
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.and_then(|value| value.checked_add(1))
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.is_some();
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if !fits {
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bail!(
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"capacity {} at arc index {} is too large for this platform\n\n{}",
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capacity,
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arc_index,
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usage
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);
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}
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}
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let num_vertices = graph.num_vertices();
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let source = args.source.ok_or_else(|| {
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anyhow::anyhow!("IntegralFlowWithMultipliers requires --source\n\n{usage}")
1163+
})?;
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let sink = args.sink.ok_or_else(|| {
1165+
anyhow::anyhow!("IntegralFlowWithMultipliers requires --sink\n\n{usage}")
1166+
})?;
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validate_vertex_index("source", source, num_vertices, usage)?;
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validate_vertex_index("sink", sink, num_vertices, usage)?;
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if source == sink {
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bail!(
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"IntegralFlowWithMultipliers requires distinct --source and --sink\n\n{}",
1172+
usage
1173+
);
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}
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1176+
let multipliers_str = args.multipliers.as_deref().ok_or_else(|| {
1177+
anyhow::anyhow!("IntegralFlowWithMultipliers requires --multipliers\n\n{usage}")
1178+
})?;
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let multipliers: Vec<u64> = util::parse_comma_list(multipliers_str)
1180+
.map_err(|e| anyhow::anyhow!("{e}\n\n{usage}"))?;
1181+
if multipliers.len() != num_vertices {
1182+
bail!(
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"Expected {} multipliers but got {}\n\n{}",
1184+
num_vertices,
1185+
multipliers.len(),
1186+
usage
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);
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}
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if multipliers
1190+
.iter()
1191+
.enumerate()
1192+
.any(|(vertex, &multiplier)| vertex != source && vertex != sink && multiplier == 0)
1193+
{
1194+
bail!("non-terminal multipliers must be positive\n\n{usage}");
1195+
}
1196+
1197+
let requirement = args.requirement.ok_or_else(|| {
1198+
anyhow::anyhow!("IntegralFlowWithMultipliers requires --requirement\n\n{usage}")
1199+
})?;
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(
1201+
ser(IntegralFlowWithMultipliers::new(
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graph,
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source,
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sink,
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multipliers,
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capacities,
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requirement,
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))?,
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resolved_variant.clone(),
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)
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}
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// Minimum cut into bounded sets (graph + edge weights + s/t/B/K)
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"MinimumCutIntoBoundedSets" => {
11211215
let (graph, _) = parse_graph(args).map_err(|e| {
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edge_weights: None,
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edge_lengths: None,
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capacities: None,
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multipliers: None,
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source: None,
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sink: None,
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requirement: None,
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num_paths_required: None,
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couplings: None,
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fields: None,

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