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Fix #295: Add DirectedTwoCommodityIntegralFlow model (#660)
* Add plan for #295: DirectedTwoCommodityIntegralFlow Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com> * Implement #295: Add DirectedTwoCommodityIntegralFlow model Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com> * chore: remove plan file after implementation * fix: address PR #660 review follow-ups * fix: improve D2CIF CLI guidance * fix: remove duplicate CLI fields from merge (capacities, source/sink, requirements) Both DirectedTwoCommodityIntegralFlow and UndirectedTwoCommodityIntegralFlow share the same CLI fields. The merge with main created duplicates. Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com> --------- Co-authored-by: Claude Opus 4.6 (1M context) <noreply@anthropic.com> Co-authored-by: zazabap <sweynan@icloud.com>
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‎docs/paper/reductions.typ‎

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"PartitionIntoTriangles": [Partition Into Triangles],
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"FlowShopScheduling": [Flow Shop Scheduling],
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"MinimumTardinessSequencing": [Minimum Tardiness Sequencing],
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"DirectedTwoCommodityIntegralFlow": [Directed Two-Commodity Integral Flow],
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)
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// Definition label: "def:<ProblemName>" — each definition block must have a matching label
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]
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}
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#problem-def("DirectedTwoCommodityIntegralFlow")[
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Given a directed graph $G = (V, A)$ with arc capacities $c: A -> ZZ^+$, two source-sink pairs $(s_1, t_1)$ and $(s_2, t_2)$, and requirements $R_1, R_2 in ZZ^+$, determine whether there exist two integral flow functions $f_1, f_2: A -> ZZ_(>= 0)$ such that (1) $f_1(a) + f_2(a) <= c(a)$ for all $a in A$, (2) flow $f_i$ is conserved at every vertex except $s_1, s_2, t_1, t_2$, and (3) the net flow into $t_i$ under $f_i$ is at least $R_i$ for $i in {1, 2}$.
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][
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Directed Two-Commodity Integral Flow is a fundamental NP-complete problem in multicommodity flow theory, catalogued as ND38 in Garey & Johnson @garey1979. While single-commodity max-flow is solvable in polynomial time and fractional multicommodity flow reduces to linear programming, requiring integral flows with just two commodities makes the problem NP-complete.
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NP-completeness was proved by Even, Itai, and Shamir via reduction from 3-SAT @even1976. The problem remains NP-complete even when all arc capacities are 1 and $R_1 = 1$. No sub-exponential exact algorithm is known; brute-force enumeration over $(C + 1)^(2|A|)$ flow assignments dominates, where $C = max_(a in A) c(a)$.#footnote[No algorithm improving on brute-force is known for Directed Two-Commodity Integral Flow.]
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*Example.* Consider a directed graph with 6 vertices and 8 arcs (all with unit capacity), sources $s_1 = 0$, $s_2 = 1$, sinks $t_1 = 4$, $t_2 = 5$, and requirements $R_1 = R_2 = 1$. Commodity 1 routes along the path $0 -> 2 -> 4$ and commodity 2 along $1 -> 3 -> 5$, satisfying all capacity and conservation constraints.
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#figure(
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canvas(length: 1cm, {
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import draw: *
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let positions = (
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(0, 1), // 0 = s1
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(0, -1), // 1 = s2
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(2, 1), // 2
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(2, -1), // 3
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(4, 1), // 4 = t1
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(4, -1), // 5 = t2
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)
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let labels = ($s_1$, $s_2$, $2$, $3$, $t_1$, $t_2$)
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let arcs = ((0, 2), (0, 3), (1, 2), (1, 3), (2, 4), (2, 5), (3, 4), (3, 5))
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// Commodity 1 path: arcs 0 (0->2) and 4 (2->4)
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let c1-arcs = (0, 4)
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// Commodity 2 path: arcs 3 (1->3) and 7 (3->5)
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let c2-arcs = (3, 7)
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// Draw arcs
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for (idx, (u, v)) in arcs.enumerate() {
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let from = positions.at(u)
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let to = positions.at(v)
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let color = if c1-arcs.contains(idx) { blue } else if c2-arcs.contains(idx) { red } else { gray.darken(20%) }
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let thickness = if c1-arcs.contains(idx) or c2-arcs.contains(idx) { 1.2pt } else { 0.6pt }
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line(from, to, stroke: (paint: color, thickness: thickness), mark: (end: "straight", scale: 0.5))
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}
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// Draw vertices
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for (k, pos) in positions.enumerate() {
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let fill = if k == 0 or k == 4 { blue.lighten(70%) } else if k == 1 or k == 5 { red.lighten(70%) } else { white }
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circle(pos, radius: 0.3, fill: fill, stroke: 0.6pt, name: str(k))
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content(pos, text(8pt, labels.at(k)))
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}
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}),
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caption: [Two-commodity flow: commodity 1 (blue, $s_1 -> 2 -> t_1$) and commodity 2 (red, $s_2 -> 3 -> t_2$).],
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) <fig:d2cif>
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]
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// Completeness check: warn about problem types in JSON but missing from paper
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#{
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let json-models = {

‎docs/paper/references.bib‎

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doi = {10.1109/TPAMI.2004.75}
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}
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@article{even1976,
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author = {Shimon Even and Alon Itai and Adi Shamir},
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title = {On the Complexity of Timetable and Multicommodity Flow Problems},
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journal = {SIAM Journal on Computing},
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volume = {5},
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number = {4},
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pages = {691--703},
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year = {1976},
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doi = {10.1137/0205048}
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}
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@article{papadimitriou1982,
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author = {Christos H. Papadimitriou and Mihalis Yannakakis},
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title = {The Complexity of Restricted Spanning Tree Problems},

‎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": "DirectedTwoCommodityIntegralFlow",
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"description": "Two-commodity integral flow feasibility on a directed graph",
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"fields": [
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{
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"name": "graph",
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"type_name": "DirectedGraph",
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"description": "Directed graph G = (V, A)"
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},
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{
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"name": "capacities",
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"type_name": "Vec<u64>",
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"description": "Capacity c(a) for each arc"
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},
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{
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"name": "source_1",
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"type_name": "usize",
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"description": "Source vertex s_1 for commodity 1"
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},
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{
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"name": "sink_1",
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"type_name": "usize",
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"description": "Sink vertex t_1 for commodity 1"
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},
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{
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"name": "source_2",
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"type_name": "usize",
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"description": "Source vertex s_2 for commodity 2"
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},
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{
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"name": "sink_2",
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"type_name": "usize",
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"description": "Sink vertex t_2 for commodity 2"
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},
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{
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"name": "requirement_1",
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"type_name": "u64",
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"description": "Flow requirement R_1 for commodity 1"
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},
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{
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"name": "requirement_2",
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"type_name": "u64",
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"description": "Flow requirement R_2 for commodity 2"
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}
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]
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},
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{
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"name": "ExactCoverBy3Sets",
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"description": "Determine if a collection of 3-element subsets contains an exact cover",

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