diff --git a/docs/paper/reductions.typ b/docs/paper/reductions.typ
index 707822b0..76cddb44 100644
--- a/docs/paper/reductions.typ
+++ b/docs/paper/reductions.typ
@@ -108,6 +108,7 @@
   "PartitionIntoTriangles": [Partition Into Triangles],
   "FlowShopScheduling": [Flow Shop Scheduling],
   "MinimumTardinessSequencing": [Minimum Tardiness Sequencing],
+  "DirectedTwoCommodityIntegralFlow": [Directed Two-Commodity Integral Flow],
 )
 
 // Definition label: "def:<ProblemName>" — each definition block must have a matching label
@@ -2091,6 +2092,53 @@ NP-completeness was established by Garey, Johnson, and Stockmeyer @gareyJohnsonS
   ]
 }
 
+#problem-def("DirectedTwoCommodityIntegralFlow")[
+  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 $i in {1, 2}$.
+][
+  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.
+
+  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.]
+
+  *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.
+
+  #figure(
+    canvas(length: 1cm, {
+      import draw: *
+      let positions = (
+        (0, 1),    // 0 = s1
+        (0, -1),   // 1 = s2
+        (2, 1),    // 2
+        (2, -1),   // 3
+        (4, 1),    // 4 = t1
+        (4, -1),   // 5 = t2
+      )
+      let labels = ($s_1$, $s_2$, $2$, $3$, $t_1$, $t_2$)
+      let arcs = ((0, 2), (0, 3), (1, 2), (1, 3), (2, 4), (2, 5), (3, 4), (3, 5))
+      // Commodity 1 path: arcs 0 (0->2) and 4 (2->4)
+      let c1-arcs = (0, 4)
+      // Commodity 2 path: arcs 3 (1->3) and 7 (3->5)
+      let c2-arcs = (3, 7)
+
+      // Draw arcs
+      for (idx, (u, v)) in arcs.enumerate() {
+        let from = positions.at(u)
+        let to = positions.at(v)
+        let color = if c1-arcs.contains(idx) { blue } else if c2-arcs.contains(idx) { red } else { gray.darken(20%) }
+        let thickness = if c1-arcs.contains(idx) or c2-arcs.contains(idx) { 1.2pt } else { 0.6pt }
+        line(from, to, stroke: (paint: color, thickness: thickness), mark: (end: "straight", scale: 0.5))
+      }
+
+      // Draw vertices
+      for (k, pos) in positions.enumerate() {
+        let fill = if k == 0 or k == 4 { blue.lighten(70%) } else if k == 1 or k == 5 { red.lighten(70%) } else { white }
+        circle(pos, radius: 0.3, fill: fill, stroke: 0.6pt, name: str(k))
+        content(pos, text(8pt, labels.at(k)))
+      }
+    }),
+    caption: [Two-commodity flow: commodity 1 (blue, $s_1 -> 2 -> t_1$) and commodity 2 (red, $s_2 -> 3 -> t_2$).],
+  ) <fig:d2cif>
+]
+
 // Completeness check: warn about problem types in JSON but missing from paper
 #{
   let json-models = {
diff --git a/docs/paper/references.bib b/docs/paper/references.bib
index 05740229..6482b48a 100644
--- a/docs/paper/references.bib
+++ b/docs/paper/references.bib
@@ -682,6 +682,17 @@ @inproceedings{cordella2004
   doi       = {10.1109/TPAMI.2004.75}
 }
 
+@article{even1976,
+  author  = {Shimon Even and Alon Itai and Adi Shamir},
+  title   = {On the Complexity of Timetable and Multicommodity Flow Problems},
+  journal = {SIAM Journal on Computing},
+  volume  = {5},
+  number  = {4},
+  pages   = {691--703},
+  year    = {1976},
+  doi     = {10.1137/0205048}
+}
+
 @article{papadimitriou1982,
   author  = {Christos H. Papadimitriou and Mihalis Yannakakis},
   title   = {The Complexity of Restricted Spanning Tree Problems},
diff --git a/docs/src/reductions/problem_schemas.json b/docs/src/reductions/problem_schemas.json
index 7d60308b..a69a1275 100644
--- a/docs/src/reductions/problem_schemas.json
+++ b/docs/src/reductions/problem_schemas.json
@@ -89,6 +89,52 @@
       }
     ]
   },
+  {
+    "name": "DirectedTwoCommodityIntegralFlow",
+    "description": "Two-commodity integral flow feasibility on a directed graph",
+    "fields": [
+      {
+        "name": "graph",
+        "type_name": "DirectedGraph",
+        "description": "Directed graph G = (V, A)"
+      },
+      {
+        "name": "capacities",
+        "type_name": "Vec<u64>",
+        "description": "Capacity c(a) for each arc"
+      },
+      {
+        "name": "source_1",
+        "type_name": "usize",
+        "description": "Source vertex s_1 for commodity 1"
+      },
+      {
+        "name": "sink_1",
+        "type_name": "usize",
+        "description": "Sink vertex t_1 for commodity 1"
+      },
+      {
+        "name": "source_2",
+        "type_name": "usize",
+        "description": "Source vertex s_2 for commodity 2"
+      },
+      {
+        "name": "sink_2",
+        "type_name": "usize",
+        "description": "Sink vertex t_2 for commodity 2"
+      },
+      {
+        "name": "requirement_1",
+        "type_name": "u64",
+        "description": "Flow requirement R_1 for commodity 1"
+      },
+      {
+        "name": "requirement_2",
+        "type_name": "u64",
+        "description": "Flow requirement R_2 for commodity 2"
+      }
+    ]
+  },
   {
     "name": "ExactCoverBy3Sets",
     "description": "Determine if a collection of 3-element subsets contains an exact cover",
diff --git a/docs/src/reductions/reduction_graph.json b/docs/src/reductions/reduction_graph.json
index eb200845..5429b350 100644
--- a/docs/src/reductions/reduction_graph.json
+++ b/docs/src/reductions/reduction_graph.json
@@ -57,6 +57,13 @@
       "doc_path": "models/algebraic/struct.ClosestVectorProblem.html",
       "complexity": "2^num_basis_vectors"
     },
+    {
+      "name": "DirectedTwoCommodityIntegralFlow",
+      "variant": {},
+      "category": "graph",
+      "doc_path": "models/graph/struct.DirectedTwoCommodityIntegralFlow.html",
+      "complexity": "(max_capacity + 1)^(2 * num_arcs)"
+    },
     {
       "name": "ExactCoverBy3Sets",
       "variant": {},
@@ -562,7 +569,7 @@
   "edges": [
     {
       "source": 3,
-      "target": 12,
+      "target": 13,
       "overhead": [
         {
           "field": "num_vars",
@@ -577,7 +584,7 @@
     },
     {
       "source": 4,
-      "target": 12,
+      "target": 13,
       "overhead": [
         {
           "field": "num_vars",
@@ -592,7 +599,7 @@
     },
     {
       "source": 4,
-      "target": 56,
+      "target": 57,
       "overhead": [
         {
           "field": "num_spins",
@@ -606,7 +613,7 @@
       "doc_path": "rules/circuit_spinglass/index.html"
     },
     {
-      "source": 8,
+      "source": 9,
       "target": 4,
       "overhead": [
         {
@@ -621,8 +628,8 @@
       "doc_path": "rules/factoring_circuit/index.html"
     },
     {
-      "source": 8,
-      "target": 13,
+      "source": 9,
+      "target": 14,
       "overhead": [
         {
           "field": "num_vars",
@@ -636,8 +643,8 @@
       "doc_path": "rules/factoring_ilp/index.html"
     },
     {
-      "source": 12,
-      "target": 13,
+      "source": 13,
+      "target": 14,
       "overhead": [
         {
           "field": "num_vars",
@@ -651,8 +658,8 @@
       "doc_path": "rules/ilp_bool_ilp_i32/index.html"
     },
     {
-      "source": 12,
-      "target": 50,
+      "source": 13,
+      "target": 51,
       "overhead": [
         {
           "field": "num_vars",
@@ -662,8 +669,8 @@
       "doc_path": "rules/ilp_qubo/index.html"
     },
     {
-      "source": 16,
-      "target": 19,
+      "source": 17,
+      "target": 20,
       "overhead": [
         {
           "field": "num_vertices",
@@ -677,8 +684,8 @@
       "doc_path": "rules/kcoloring_casts/index.html"
     },
     {
-      "source": 19,
-      "target": 12,
+      "source": 20,
+      "target": 13,
       "overhead": [
         {
           "field": "num_vars",
@@ -692,8 +699,8 @@
       "doc_path": "rules/coloring_ilp/index.html"
     },
     {
-      "source": 19,
-      "target": 50,
+      "source": 20,
+      "target": 51,
       "overhead": [
         {
           "field": "num_vars",
@@ -703,8 +710,8 @@
       "doc_path": "rules/coloring_qubo/index.html"
     },
     {
-      "source": 20,
-      "target": 22,
+      "source": 21,
+      "target": 23,
       "overhead": [
         {
           "field": "num_vars",
@@ -718,8 +725,8 @@
       "doc_path": "rules/ksatisfiability_casts/index.html"
     },
     {
-      "source": 20,
-      "target": 50,
+      "source": 21,
+      "target": 51,
       "overhead": [
         {
           "field": "num_vars",
@@ -729,8 +736,8 @@
       "doc_path": "rules/ksatisfiability_qubo/index.html"
     },
     {
-      "source": 21,
-      "target": 22,
+      "source": 22,
+      "target": 23,
       "overhead": [
         {
           "field": "num_vars",
@@ -744,8 +751,8 @@
       "doc_path": "rules/ksatisfiability_casts/index.html"
     },
     {
-      "source": 21,
-      "target": 50,
+      "source": 22,
+      "target": 51,
       "overhead": [
         {
           "field": "num_vars",
@@ -755,8 +762,8 @@
       "doc_path": "rules/ksatisfiability_qubo/index.html"
     },
     {
-      "source": 21,
-      "target": 60,
+      "source": 22,
+      "target": 61,
       "overhead": [
         {
           "field": "num_elements",
@@ -766,8 +773,8 @@
       "doc_path": "rules/ksatisfiability_subsetsum/index.html"
     },
     {
-      "source": 22,
-      "target": 52,
+      "source": 23,
+      "target": 53,
       "overhead": [
         {
           "field": "num_clauses",
@@ -785,8 +792,8 @@
       "doc_path": "rules/sat_ksat/index.html"
     },
     {
-      "source": 23,
-      "target": 50,
+      "source": 24,
+      "target": 51,
       "overhead": [
         {
           "field": "num_vars",
@@ -796,8 +803,8 @@
       "doc_path": "rules/knapsack_qubo/index.html"
     },
     {
-      "source": 25,
-      "target": 12,
+      "source": 26,
+      "target": 13,
       "overhead": [
         {
           "field": "num_vars",
@@ -811,8 +818,8 @@
       "doc_path": "rules/longestcommonsubsequence_ilp/index.html"
     },
     {
-      "source": 26,
-      "target": 56,
+      "source": 27,
+      "target": 57,
       "overhead": [
         {
           "field": "num_spins",
@@ -826,8 +833,8 @@
       "doc_path": "rules/spinglass_maxcut/index.html"
     },
     {
-      "source": 28,
-      "target": 12,
+      "source": 29,
+      "target": 13,
       "overhead": [
         {
           "field": "num_vars",
@@ -841,8 +848,8 @@
       "doc_path": "rules/maximumclique_ilp/index.html"
     },
     {
-      "source": 28,
-      "target": 32,
+      "source": 29,
+      "target": 33,
       "overhead": [
         {
           "field": "num_vertices",
@@ -856,8 +863,8 @@
       "doc_path": "rules/maximumclique_maximumindependentset/index.html"
     },
     {
-      "source": 29,
-      "target": 30,
+      "source": 30,
+      "target": 31,
       "overhead": [
         {
           "field": "num_vertices",
@@ -871,8 +878,8 @@
       "doc_path": "rules/maximumindependentset_casts/index.html"
     },
     {
-      "source": 29,
-      "target": 34,
+      "source": 30,
+      "target": 35,
       "overhead": [
         {
           "field": "num_vertices",
@@ -886,8 +893,8 @@
       "doc_path": "rules/maximumindependentset_casts/index.html"
     },
     {
-      "source": 30,
-      "target": 35,
+      "source": 31,
+      "target": 36,
       "overhead": [
         {
           "field": "num_vertices",
@@ -901,8 +908,8 @@
       "doc_path": "rules/maximumindependentset_casts/index.html"
     },
     {
-      "source": 31,
-      "target": 29,
+      "source": 32,
+      "target": 30,
       "overhead": [
         {
           "field": "num_vertices",
@@ -916,8 +923,8 @@
       "doc_path": "rules/maximumindependentset_gridgraph/index.html"
     },
     {
-      "source": 31,
-      "target": 32,
+      "source": 32,
+      "target": 33,
       "overhead": [
         {
           "field": "num_vertices",
@@ -931,8 +938,8 @@
       "doc_path": "rules/maximumindependentset_casts/index.html"
     },
     {
-      "source": 31,
-      "target": 33,
+      "source": 32,
+      "target": 34,
       "overhead": [
         {
           "field": "num_vertices",
@@ -946,8 +953,8 @@
       "doc_path": "rules/maximumindependentset_triangular/index.html"
     },
     {
-      "source": 31,
-      "target": 37,
+      "source": 32,
+      "target": 38,
       "overhead": [
         {
           "field": "num_sets",
@@ -961,8 +968,8 @@
       "doc_path": "rules/maximumindependentset_maximumsetpacking/index.html"
     },
     {
-      "source": 32,
-      "target": 28,
+      "source": 33,
+      "target": 29,
       "overhead": [
         {
           "field": "num_vertices",
@@ -976,8 +983,8 @@
       "doc_path": "rules/maximumindependentset_maximumclique/index.html"
     },
     {
-      "source": 32,
-      "target": 39,
+      "source": 33,
+      "target": 40,
       "overhead": [
         {
           "field": "num_sets",
@@ -991,8 +998,8 @@
       "doc_path": "rules/maximumindependentset_maximumsetpacking/index.html"
     },
     {
-      "source": 32,
-      "target": 46,
+      "source": 33,
+      "target": 47,
       "overhead": [
         {
           "field": "num_vertices",
@@ -1006,8 +1013,8 @@
       "doc_path": "rules/minimumvertexcover_maximumindependentset/index.html"
     },
     {
-      "source": 33,
-      "target": 35,
+      "source": 34,
+      "target": 36,
       "overhead": [
         {
           "field": "num_vertices",
@@ -1021,8 +1028,8 @@
       "doc_path": "rules/maximumindependentset_casts/index.html"
     },
     {
-      "source": 34,
-      "target": 31,
+      "source": 35,
+      "target": 32,
       "overhead": [
         {
           "field": "num_vertices",
@@ -1036,8 +1043,8 @@
       "doc_path": "rules/maximumindependentset_casts/index.html"
     },
     {
-      "source": 34,
-      "target": 35,
+      "source": 35,
+      "target": 36,
       "overhead": [
         {
           "field": "num_vertices",
@@ -1051,8 +1058,8 @@
       "doc_path": "rules/maximumindependentset_casts/index.html"
     },
     {
-      "source": 35,
-      "target": 32,
+      "source": 36,
+      "target": 33,
       "overhead": [
         {
           "field": "num_vertices",
@@ -1066,8 +1073,8 @@
       "doc_path": "rules/maximumindependentset_casts/index.html"
     },
     {
-      "source": 36,
-      "target": 12,
+      "source": 37,
+      "target": 13,
       "overhead": [
         {
           "field": "num_vars",
@@ -1081,8 +1088,8 @@
       "doc_path": "rules/maximummatching_ilp/index.html"
     },
     {
-      "source": 36,
-      "target": 39,
+      "source": 37,
+      "target": 40,
       "overhead": [
         {
           "field": "num_sets",
@@ -1096,8 +1103,8 @@
       "doc_path": "rules/maximummatching_maximumsetpacking/index.html"
     },
     {
-      "source": 37,
-      "target": 31,
+      "source": 38,
+      "target": 32,
       "overhead": [
         {
           "field": "num_vertices",
@@ -1111,8 +1118,8 @@
       "doc_path": "rules/maximumindependentset_maximumsetpacking/index.html"
     },
     {
-      "source": 37,
-      "target": 39,
+      "source": 38,
+      "target": 40,
       "overhead": [
         {
           "field": "num_sets",
@@ -1126,8 +1133,8 @@
       "doc_path": "rules/maximumsetpacking_casts/index.html"
     },
     {
-      "source": 38,
-      "target": 50,
+      "source": 39,
+      "target": 51,
       "overhead": [
         {
           "field": "num_vars",
@@ -1137,8 +1144,8 @@
       "doc_path": "rules/maximumsetpacking_qubo/index.html"
     },
     {
-      "source": 39,
-      "target": 12,
+      "source": 40,
+      "target": 13,
       "overhead": [
         {
           "field": "num_vars",
@@ -1152,8 +1159,8 @@
       "doc_path": "rules/maximumsetpacking_ilp/index.html"
     },
     {
-      "source": 39,
-      "target": 32,
+      "source": 40,
+      "target": 33,
       "overhead": [
         {
           "field": "num_vertices",
@@ -1167,8 +1174,8 @@
       "doc_path": "rules/maximumindependentset_maximumsetpacking/index.html"
     },
     {
-      "source": 39,
-      "target": 38,
+      "source": 40,
+      "target": 39,
       "overhead": [
         {
           "field": "num_sets",
@@ -1182,8 +1189,8 @@
       "doc_path": "rules/maximumsetpacking_casts/index.html"
     },
     {
-      "source": 40,
-      "target": 12,
+      "source": 41,
+      "target": 13,
       "overhead": [
         {
           "field": "num_vars",
@@ -1197,8 +1204,8 @@
       "doc_path": "rules/minimumdominatingset_ilp/index.html"
     },
     {
-      "source": 43,
-      "target": 12,
+      "source": 44,
+      "target": 13,
       "overhead": [
         {
           "field": "num_vars",
@@ -1212,8 +1219,8 @@
       "doc_path": "rules/minimumsetcovering_ilp/index.html"
     },
     {
-      "source": 46,
-      "target": 32,
+      "source": 47,
+      "target": 33,
       "overhead": [
         {
           "field": "num_vertices",
@@ -1227,8 +1234,8 @@
       "doc_path": "rules/minimumvertexcover_maximumindependentset/index.html"
     },
     {
-      "source": 46,
-      "target": 43,
+      "source": 47,
+      "target": 44,
       "overhead": [
         {
           "field": "num_sets",
@@ -1242,8 +1249,8 @@
       "doc_path": "rules/minimumvertexcover_minimumsetcovering/index.html"
     },
     {
-      "source": 50,
-      "target": 12,
+      "source": 51,
+      "target": 13,
       "overhead": [
         {
           "field": "num_vars",
@@ -1257,8 +1264,8 @@
       "doc_path": "rules/qubo_ilp/index.html"
     },
     {
-      "source": 50,
-      "target": 55,
+      "source": 51,
+      "target": 56,
       "overhead": [
         {
           "field": "num_spins",
@@ -1268,7 +1275,7 @@
       "doc_path": "rules/spinglass_qubo/index.html"
     },
     {
-      "source": 52,
+      "source": 53,
       "target": 4,
       "overhead": [
         {
@@ -1283,8 +1290,8 @@
       "doc_path": "rules/sat_circuitsat/index.html"
     },
     {
-      "source": 52,
-      "target": 16,
+      "source": 53,
+      "target": 17,
       "overhead": [
         {
           "field": "num_vertices",
@@ -1298,8 +1305,8 @@
       "doc_path": "rules/sat_coloring/index.html"
     },
     {
-      "source": 52,
-      "target": 21,
+      "source": 53,
+      "target": 22,
       "overhead": [
         {
           "field": "num_clauses",
@@ -1313,8 +1320,8 @@
       "doc_path": "rules/sat_ksat/index.html"
     },
     {
-      "source": 52,
-      "target": 31,
+      "source": 53,
+      "target": 32,
       "overhead": [
         {
           "field": "num_vertices",
@@ -1328,8 +1335,8 @@
       "doc_path": "rules/sat_maximumindependentset/index.html"
     },
     {
-      "source": 52,
-      "target": 40,
+      "source": 53,
+      "target": 41,
       "overhead": [
         {
           "field": "num_vertices",
@@ -1343,8 +1350,8 @@
       "doc_path": "rules/sat_minimumdominatingset/index.html"
     },
     {
-      "source": 55,
-      "target": 50,
+      "source": 56,
+      "target": 51,
       "overhead": [
         {
           "field": "num_vars",
@@ -1354,8 +1361,8 @@
       "doc_path": "rules/spinglass_qubo/index.html"
     },
     {
-      "source": 56,
-      "target": 26,
+      "source": 57,
+      "target": 27,
       "overhead": [
         {
           "field": "num_vertices",
@@ -1369,8 +1376,8 @@
       "doc_path": "rules/spinglass_maxcut/index.html"
     },
     {
-      "source": 56,
-      "target": 55,
+      "source": 57,
+      "target": 56,
       "overhead": [
         {
           "field": "num_spins",
@@ -1384,8 +1391,8 @@
       "doc_path": "rules/spinglass_casts/index.html"
     },
     {
-      "source": 61,
-      "target": 12,
+      "source": 62,
+      "target": 13,
       "overhead": [
         {
           "field": "num_vars",
@@ -1399,8 +1406,8 @@
       "doc_path": "rules/travelingsalesman_ilp/index.html"
     },
     {
-      "source": 61,
-      "target": 50,
+      "source": 62,
+      "target": 51,
       "overhead": [
         {
           "field": "num_vars",
diff --git a/problemreductions-cli/src/cli.rs b/problemreductions-cli/src/cli.rs
index a289e915..32d58428 100644
--- a/problemreductions-cli/src/cli.rs
+++ b/problemreductions-cli/src/cli.rs
@@ -247,6 +247,7 @@ Flags by problem type:
   FlowShopScheduling              --task-lengths, --deadline [--num-processors]
   MinimumTardinessSequencing      --n, --deadlines [--precedence-pairs]
   SCS                             --strings, --bound [--alphabet-size]
+  D2CIF                           --arcs, --capacities, --source-1, --sink-1, --source-2, --sink-2, --requirement-1, --requirement-2
   ILP, CircuitSAT                 (via reduction only)
 
 Geometry graph variants (use slash notation, e.g., MIS/KingsSubgraph):
diff --git a/problemreductions-cli/src/commands/create.rs b/problemreductions-cli/src/commands/create.rs
index a6261ac0..dc1aaca6 100644
--- a/problemreductions-cli/src/commands/create.rs
+++ b/problemreductions-cli/src/commands/create.rs
@@ -76,6 +76,13 @@ fn all_data_flags_empty(args: &CreateArgs) -> bool {
         && args.deadline.is_none()
         && args.num_processors.is_none()
         && args.alphabet_size.is_none()
+        && args.capacities.is_none()
+        && args.source_1.is_none()
+        && args.sink_1.is_none()
+        && args.source_2.is_none()
+        && args.sink_2.is_none()
+        && args.requirement_1.is_none()
+        && args.requirement_2.is_none()
 }
 
 fn emit_problem_output(output: &ProblemJsonOutput, out: &OutputConfig) -> Result<()> {
@@ -277,6 +284,9 @@ fn example_for(canonical: &str, graph_type: Option<&str>) -> &'static str {
         "Factoring" => "--target 15 --m 4 --n 4",
         "SteinerTree" => "--graph 0-1,1-2,1-3,3-4 --edge-weights 2,2,1,1 --terminals 0,2,4",
         "OptimalLinearArrangement" => "--graph 0-1,1-2,2-3 --bound 5",
+        "DirectedTwoCommodityIntegralFlow" => {
+            "--arcs \"0>2,0>3,1>2,1>3,2>4,2>5,3>4,3>5\" --capacities 1,1,1,1,1,1,1,1 --source-1 0 --sink-1 4 --source-2 1 --sink-2 5 --requirement-1 1 --requirement-2 1"
+        }
         "MinimumFeedbackArcSet" => "--arcs \"0>1,1>2,2>0\"",
         "RuralPostman" => {
             "--graph 0-1,1-2,2-3,3-0 --edge-weights 1,1,1,1 --required-edges 0,2 --bound 4"
@@ -1155,6 +1165,67 @@ pub fn create(args: &CreateArgs, out: &OutputConfig) -> Result<()> {
             )
         }
 
+        // DirectedTwoCommodityIntegralFlow
+        "DirectedTwoCommodityIntegralFlow" => {
+            let arcs_str = args.arcs.as_deref().ok_or_else(|| {
+                anyhow::anyhow!(
+                    "DirectedTwoCommodityIntegralFlow requires --arcs\n\n\
+                     Usage: pred create DirectedTwoCommodityIntegralFlow \
+                     --arcs \"0>2,0>3,1>2,1>3,2>4,2>5,3>4,3>5\" \
+                     --capacities 1,1,1,1,1,1,1,1 \
+                     --source-1 0 --sink-1 4 --source-2 1 --sink-2 5 \
+                     --requirement-1 1 --requirement-2 1"
+                )
+            })?;
+            let (graph, num_arcs) = parse_directed_graph(arcs_str, args.num_vertices)?;
+            let capacities: Vec<u64> = if let Some(ref s) = args.capacities {
+                util::parse_comma_list(s)?
+            } else {
+                vec![1; num_arcs]
+            };
+            anyhow::ensure!(
+                capacities.len() == num_arcs,
+                "capacities length ({}) must match number of arcs ({num_arcs})",
+                capacities.len()
+            );
+            let n = graph.num_vertices();
+            let source_1 = args.source_1.ok_or_else(|| {
+                anyhow::anyhow!("DirectedTwoCommodityIntegralFlow requires --source-1")
+            })?;
+            let sink_1 = args.sink_1.ok_or_else(|| {
+                anyhow::anyhow!("DirectedTwoCommodityIntegralFlow requires --sink-1")
+            })?;
+            let source_2 = args.source_2.ok_or_else(|| {
+                anyhow::anyhow!("DirectedTwoCommodityIntegralFlow requires --source-2")
+            })?;
+            let sink_2 = args.sink_2.ok_or_else(|| {
+                anyhow::anyhow!("DirectedTwoCommodityIntegralFlow requires --sink-2")
+            })?;
+            for (name, idx) in [
+                ("source_1", source_1),
+                ("sink_1", sink_1),
+                ("source_2", source_2),
+                ("sink_2", sink_2),
+            ] {
+                anyhow::ensure!(idx < n, "{name} ({idx}) >= num_vertices ({n})");
+            }
+            let requirement_1 = args.requirement_1.unwrap_or(1);
+            let requirement_2 = args.requirement_2.unwrap_or(1);
+            (
+                ser(DirectedTwoCommodityIntegralFlow::new(
+                    graph,
+                    capacities,
+                    source_1,
+                    sink_1,
+                    source_2,
+                    sink_2,
+                    requirement_1,
+                    requirement_2,
+                ))?,
+                resolved_variant.clone(),
+            )
+        }
+
         // MinimumFeedbackArcSet
         "MinimumFeedbackArcSet" => {
             let arcs_str = args.arcs.as_deref().ok_or_else(|| {
diff --git a/problemreductions-cli/src/dispatch.rs b/problemreductions-cli/src/dispatch.rs
index 659bba48..938d8643 100644
--- a/problemreductions-cli/src/dispatch.rs
+++ b/problemreductions-cli/src/dispatch.rs
@@ -79,8 +79,12 @@ impl LoadedProblem {
             }
         }
 
-        let reduction_path =
-            best_path.ok_or_else(|| anyhow::anyhow!("No reduction path from {} to ILP", name))?;
+        let reduction_path = best_path.ok_or_else(|| {
+            anyhow::anyhow!(
+                "No reduction path from {} to ILP. Try `--solver brute-force`, or reduce to a problem that supports ILP.",
+                name
+            )
+        })?;
 
         let chain = graph
             .reduce_along_path(&reduction_path, self.as_any())
diff --git a/problemreductions-cli/tests/cli_tests.rs b/problemreductions-cli/tests/cli_tests.rs
index 419e690a..8d481fcc 100644
--- a/problemreductions-cli/tests/cli_tests.rs
+++ b/problemreductions-cli/tests/cli_tests.rs
@@ -824,6 +824,100 @@ fn test_create_x3c_alias() {
     std::fs::remove_file(&output_file).ok();
 }
 
+#[test]
+fn test_create_d2cif_alias() {
+    let output_file = std::env::temp_dir().join("pred_test_create_d2cif.json");
+    let output = pred()
+        .args([
+            "-o",
+            output_file.to_str().unwrap(),
+            "create",
+            "D2CIF",
+            "--arcs",
+            "0>2,0>3,1>2,1>3,2>4,2>5,3>4,3>5",
+            "--capacities",
+            "1,1,1,1,1,1,1,1",
+            "--source-1",
+            "0",
+            "--sink-1",
+            "4",
+            "--source-2",
+            "1",
+            "--sink-2",
+            "5",
+            "--requirement-1",
+            "1",
+            "--requirement-2",
+            "1",
+        ])
+        .output()
+        .unwrap();
+    assert!(
+        output.status.success(),
+        "stderr: {}",
+        String::from_utf8_lossy(&output.stderr)
+    );
+    assert!(output_file.exists());
+
+    let content = std::fs::read_to_string(&output_file).unwrap();
+    let json: serde_json::Value = serde_json::from_str(&content).unwrap();
+    assert_eq!(json["type"], "DirectedTwoCommodityIntegralFlow");
+
+    std::fs::remove_file(&output_file).ok();
+}
+
+#[test]
+fn test_solve_d2cif_default_solver_suggests_bruteforce() {
+    let output_file = std::env::temp_dir().join("pred_test_solve_d2cif.json");
+    let create_output = pred()
+        .args([
+            "-o",
+            output_file.to_str().unwrap(),
+            "create",
+            "D2CIF",
+            "--arcs",
+            "0>2,0>3,1>2,1>3,2>4,2>5,3>4,3>5",
+            "--capacities",
+            "1,1,1,1,1,1,1,1",
+            "--source-1",
+            "0",
+            "--sink-1",
+            "4",
+            "--source-2",
+            "1",
+            "--sink-2",
+            "5",
+            "--requirement-1",
+            "1",
+            "--requirement-2",
+            "1",
+        ])
+        .output()
+        .unwrap();
+    assert!(
+        create_output.status.success(),
+        "stderr: {}",
+        String::from_utf8_lossy(&create_output.stderr)
+    );
+
+    let solve_output = pred()
+        .args(["solve", output_file.to_str().unwrap()])
+        .output()
+        .unwrap();
+    assert!(
+        !solve_output.status.success(),
+        "stdout: {}",
+        String::from_utf8_lossy(&solve_output.stdout)
+    );
+    let stderr = String::from_utf8_lossy(&solve_output.stderr);
+    assert!(
+        stderr.contains("--solver brute-force"),
+        "expected brute-force hint, got: {stderr}"
+    );
+
+    std::fs::remove_file(&output_file).ok();
+}
+
 #[test]
 fn test_create_x3c_rejects_duplicate_subset_elements() {
     let output = pred()
diff --git a/src/example_db/fixtures/examples.json b/src/example_db/fixtures/examples.json
index 891fe8e7..39501490 100644
--- a/src/example_db/fixtures/examples.json
+++ b/src/example_db/fixtures/examples.json
@@ -4,6 +4,7 @@
     {"problem":"BicliqueCover","variant":{},"instance":{"graph":{"edges":[[0,0],[0,1],[1,1],[1,2]],"left_size":2,"right_size":3},"k":2},"samples":[{"config":[1,0,0,1,1,0,1,1,0,1],"metric":{"Valid":6}}],"optimal":[{"config":[0,1,0,1,0,1,0,1,0,1],"metric":{"Valid":5}},{"config":[1,0,1,0,1,0,1,0,1,0],"metric":{"Valid":5}}]},
     {"problem":"CircuitSAT","variant":{},"instance":{"circuit":{"assignments":[{"expr":{"op":{"And":[{"op":{"Var":"x1"}},{"op":{"Var":"x2"}}]}},"outputs":["a"]},{"expr":{"op":{"Or":[{"op":{"Var":"x1"}},{"op":{"Var":"x2"}}]}},"outputs":["b"]},{"expr":{"op":{"Xor":[{"op":{"Var":"a"}},{"op":{"Var":"b"}}]}},"outputs":["c"]}]},"variables":["a","b","c","x1","x2"]},"samples":[{"config":[0,1,1,0,1],"metric":true},{"config":[0,1,1,1,0],"metric":true}],"optimal":[{"config":[0,0,0,0,0],"metric":true},{"config":[0,1,1,0,1],"metric":true},{"config":[0,1,1,1,0],"metric":true},{"config":[1,1,0,1,1],"metric":true}]},
     {"problem":"ClosestVectorProblem","variant":{"weight":"i32"},"instance":{"basis":[[2,0],[1,2]],"bounds":[{"lower":-2,"upper":4},{"lower":-2,"upper":4}],"target":[2.8,1.5]},"samples":[{"config":[3,3],"metric":{"Valid":0.5385164807134505}}],"optimal":[{"config":[3,3],"metric":{"Valid":0.5385164807134505}}]},
+    {"problem":"DirectedTwoCommodityIntegralFlow","variant":{},"instance":{"capacities":[1,1,1,1,1,1,1,1],"graph":{"inner":{"edge_property":"directed","edges":[[0,2,null],[0,3,null],[1,2,null],[1,3,null],[2,4,null],[2,5,null],[3,4,null],[3,5,null]],"node_holes":[],"nodes":[null,null,null,null,null,null]}},"requirement_1":1,"requirement_2":1,"sink_1":4,"sink_2":5,"source_1":0,"source_2":1},"samples":[{"config":[1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1],"metric":true}],"optimal":[{"config":[0,0,0,1,0,0,1,0,0,0,1,0,0,1,0,0],"metric":true},{"config":[0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1],"metric":true},{"config":[0,0,0,1,0,0,1,0,0,1,1,0,0,1,0,1],"metric":true},{"config":[0,0,0,1,0,0,1,0,0,1,1,0,1,0,0,1],"metric":true},{"config":[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0],"metric":true},{"config":[0,0,0,1,0,0,1,0,1,0,1,0,1,1,0,0],"metric":true},{"config":[0,0,0,1,0,0,1,0,1,1,0,0,0,1,0,1],"metric":true},{"config":[0,0,0,1,0,0,1,0,1,1,0,0,1,0,0,1],"metric":true},{"config":[0,0,0,1,0,0,1,0,1,1,1,0,1,1,0,1],"metric":true},{"config":[0,0,1,0,1,0,0,0,0,0,0,1,0,0,0,1],"metric":true},{"config":[0,0,1,0,1,0,0,0,0,1,0,0,0,0,0,1],"metric":true},{"config":[0,0,1,0,1,0,0,0,0,1,0,1,0,0,1,1],"metric":true},{"config":[0,0,1,0,1,0,0,0,1,0,0,0,0,1,0,0],"metric":true},{"config":[0,0,1,0,1,0,0,0,1,0,0,1,0,1,0,1],"metric":true},{"config":[0,0,1,0,1,0,0,0,1,0,0,1,0,1,1,0],"metric":true},{"config":[0,0,1,0,1,0,0,0,1,1,0,0,0,1,0,1],"metric":true},{"config":[0,0,1,0,1,0,0,0,1,1,0,0,0,1,1,0],"metric":true},{"config":[0,0,1,0,1,0,0,0,1,1,0,1,0,1,1,1],"metric":true},{"config":[0,0,1,1,0,1,1,0,0,1,0,0,0,0,0,1],"metric":true},{"config":[0,0,1,1,0,1,1,0,1,1,0,0,1,0,0,1],"metric":true},{"config":[0,0,1,1,1,0,0,1,1,0,0,0,0,1,0,0],"metric":true},{"config":[0,0,1,1,1,0,0,1,1,1,0,0,0,1,1,0],"metric":true},{"config":[0,0,1,1,1,0,1,0,0,1,0,0,0,0,0,1],"metric":true},{"config":[0,0,1,1,1,0,1,0,1,0,0,0,0,1,0,0],"metric":true},{"config":[0,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1],"metric":true},{"config":[0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,1],"metric":true},{"config":[0,1,0,0,0,0,1,0,0,0,1,0,0,1,0,0],"metric":true},{"config":[0,1,0,0,0,0,1,0,0,0,1,1,0,1,0,1],"metric":true},{"config":[0,1,0,0,0,0,1,0,0,0,1,1,1,0,0,1],"metric":true},{"config":[0,1,0,0,0,0,1,0,1,0,0,0,0,1,0,0],"metric":true},{"config":[0,1,0,0,0,0,1,0,1,0,0,1,0,1,0,1],"metric":true},{"config":[0,1,0,0,0,0,1,0,1,0,0,1,1,0,0,1],"metric":true},{"config":[0,1,0,0,0,0,1,0,1,0,1,0,1,1,0,0],"metric":true},{"config":[0,1,0,0,0,0,1,0,1,0,1,1,1,1,0,1],"metric":true},{"config":[0,1,0,1,0,0,1,1,0,0,1,0,0,1,0,0],"metric":true},{"config":[0,1,0,1,0,0,1,1,1,0,0,0,0,1,0,0],"metric":true},{"config":[0,1,0,1,0,0,1,1,1,0,1,0,1,1,0,0],"metric":true},{"config":[0,1,1,0,0,1,1,0,0,0,0,1,0,0,0,1],"metric":true},{"config":[0,1,1,0,0,1,1,0,1,0,0,1,1,0,0,1],"metric":true},{"config":[0,1,1,0,1,0,0,1,1,0,0,0,0,1,0,0],"metric":true},{"config":[0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0],"metric":true},{"config":[0,1,1,0,1,0,1,0,0,0,0,1,0,0,0,1],"metric":true},{"config":[0,1,1,0,1,0,1,0,1,0,0,0,0,1,0,0],"metric":true},{"config":[0,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1],"metric":true},{"config":[0,1,1,1,1,0,1,1,1,0,0,0,0,1,0,0],"metric":true},{"config":[1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1],"metric":true},{"config":[1,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],"metric":true},{"config":[1,0,0,0,1,0,0,0,0,0,1,1,0,1,0,1],"metric":true},{"config":[1,0,0,0,1,0,0,0,0,0,1,1,0,1,1,0],"metric":true},{"config":[1,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1],"metric":true},{"config":[1,0,0,0,1,0,0,0,0,1,0,1,0,0,1,1],"metric":true},{"config":[1,0,0,0,1,0,0,0,0,1,1,0,0,1,0,1],"metric":true},{"config":[1,0,0,0,1,0,0,0,0,1,1,0,0,1,1,0],"metric":true},{"config":[1,0,0,0,1,0,0,0,0,1,1,1,0,1,1,1],"metric":true},{"config":[1,0,0,1,0,1,1,0,0,1,0,0,0,0,0,1],"metric":true},{"config":[1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1],"metric":true},{"config":[1,0,0,1,1,0,0,1,0,0,1,0,0,1,0,0],"metric":true},{"config":[1,0,0,1,1,0,0,1,0,1,1,0,0,1,1,0],"metric":true},{"config":[1,0,0,1,1,0,1,0,0,0,1,0,0,1,0,0],"metric":true},{"config":[1,0,0,1,1,0,1,0,0,1,0,0,0,0,0,1],"metric":true},{"config":[1,0,0,1,1,0,1,0,0,1,1,0,0,1,0,1],"metric":true},{"config":[1,0,1,0,1,1,0,0,0,0,0,1,0,0,0,1],"metric":true},{"config":[1,0,1,0,1,1,0,0,0,1,0,0,0,0,0,1],"metric":true},{"config":[1,0,1,0,1,1,0,0,0,1,0,1,0,0,1,1],"metric":true},{"config":[1,0,1,1,1,1,1,0,0,1,0,0,0,0,0,1],"metric":true},{"config":[1,1,0,0,0,1,1,0,0,0,0,1,0,0,0,1],"metric":true},{"config":[1,1,0,0,0,1,1,0,0,0,1,1,1,0,0,1],"metric":true},{"config":[1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0],"metric":true},{"config":[1,1,0,0,1,0,0,1,0,0,1,1,0,1,1,0],"metric":true},{"config":[1,1,0,0,1,0,1,0,0,0,0,1,0,0,0,1],"metric":true},{"config":[1,1,0,0,1,0,1,0,0,0,1,0,0,1,0,0],"metric":true},{"config":[1,1,0,0,1,0,1,0,0,0,1,1,0,1,0,1],"metric":true},{"config":[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0],"metric":true},{"config":[1,1,1,0,1,1,1,0,0,0,0,1,0,0,0,1],"metric":true}]},
     {"problem":"ExactCoverBy3Sets","variant":{},"instance":{"subsets":[[0,1,2],[0,2,4],[3,4,5],[3,5,7],[6,7,8],[1,4,6],[2,5,8]],"universe_size":9},"samples":[{"config":[1,0,1,0,1,0,0],"metric":true}],"optimal":[{"config":[1,0,1,0,1,0,0],"metric":true}]},
     {"problem":"Factoring","variant":{},"instance":{"m":2,"n":3,"target":15},"samples":[{"config":[1,1,1,0,1],"metric":{"Valid":0}}],"optimal":[{"config":[1,1,1,0,1],"metric":{"Valid":0}}]},
     {"problem":"HamiltonianPath","variant":{"graph":"SimpleGraph"},"instance":{"graph":{"inner":{"edge_property":"undirected","edges":[[0,1,null],[0,2,null],[1,3,null],[2,3,null],[3,4,null],[3,5,null],[4,2,null],[5,1,null]],"node_holes":[],"nodes":[null,null,null,null,null,null]}}},"samples":[{"config":[0,2,4,3,1,5],"metric":true}],"optimal":[{"config":[0,1,5,3,2,4],"metric":true},{"config":[0,1,5,3,4,2],"metric":true},{"config":[0,2,4,3,1,5],"metric":true},{"config":[0,2,4,3,5,1],"metric":true},{"config":[1,0,2,4,3,5],"metric":true},{"config":[1,5,3,4,2,0],"metric":true},{"config":[2,0,1,5,3,4],"metric":true},{"config":[2,4,3,5,1,0],"metric":true},{"config":[3,4,2,0,1,5],"metric":true},{"config":[3,5,1,0,2,4],"metric":true},{"config":[4,2,0,1,3,5],"metric":true},{"config":[4,2,0,1,5,3],"metric":true},{"config":[4,2,3,5,1,0],"metric":true},{"config":[4,3,2,0,1,5],"metric":true},{"config":[4,3,5,1,0,2],"metric":true},{"config":[5,1,0,2,3,4],"metric":true},{"config":[5,1,0,2,4,3],"metric":true},{"config":[5,1,3,4,2,0],"metric":true},{"config":[5,3,1,0,2,4],"metric":true},{"config":[5,3,4,2,0,1],"metric":true}]},
diff --git a/src/lib.rs b/src/lib.rs
index acbd1d76..80e37f7f 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -45,8 +45,9 @@ pub mod prelude {
     pub use crate::models::algebraic::{BMF, QUBO};
     pub use crate::models::formula::{CNFClause, CircuitSAT, KSatisfiability, Satisfiability};
     pub use crate::models::graph::{
-        BicliqueCover, GraphPartitioning, HamiltonianPath, IsomorphicSpanningTree,
-        LengthBoundedDisjointPaths, SpinGlass, SteinerTree, SubgraphIsomorphism,
+        BicliqueCover, DirectedTwoCommodityIntegralFlow, GraphPartitioning, HamiltonianPath,
+        IsomorphicSpanningTree, LengthBoundedDisjointPaths, SpinGlass, SteinerTree,
+        SubgraphIsomorphism,
     };
     pub use crate::models::graph::{
         KColoring, MaxCut, MaximalIS, MaximumClique, MaximumIndependentSet, MaximumMatching,
diff --git a/src/models/graph/directed_two_commodity_integral_flow.rs b/src/models/graph/directed_two_commodity_integral_flow.rs
new file mode 100644
index 00000000..ea6f8962
--- /dev/null
+++ b/src/models/graph/directed_two_commodity_integral_flow.rs
@@ -0,0 +1,305 @@
+//! Directed Two-Commodity Integral Flow problem implementation.
+//!
+//! Given a directed graph with arc capacities and two source-sink pairs with
+//! flow requirements, determine whether two integral flow functions exist that
+//! jointly satisfy capacity, conservation, and requirement constraints.
+//!
+//! NP-complete even with unit capacities (Even, Itai, and Shamir, 1976).
+
+use crate::registry::{FieldInfo, ProblemSchemaEntry};
+use crate::topology::DirectedGraph;
+use crate::traits::{Problem, SatisfactionProblem};
+use serde::{Deserialize, Serialize};
+
+inventory::submit! {
+    ProblemSchemaEntry {
+        name: "DirectedTwoCommodityIntegralFlow",
+        display_name: "Directed Two-Commodity Integral Flow",
+        aliases: &["D2CIF"],
+        dimensions: &[],
+        module_path: module_path!(),
+        description: "Two-commodity integral flow feasibility on a directed graph",
+        fields: &[
+            FieldInfo { name: "graph", type_name: "DirectedGraph", description: "Directed graph G = (V, A)" },
+            FieldInfo { name: "capacities", type_name: "Vec<u64>", description: "Capacity c(a) for each arc" },
+            FieldInfo { name: "source_1", type_name: "usize", description: "Source vertex s_1 for commodity 1" },
+            FieldInfo { name: "sink_1", type_name: "usize", description: "Sink vertex t_1 for commodity 1" },
+            FieldInfo { name: "source_2", type_name: "usize", description: "Source vertex s_2 for commodity 2" },
+            FieldInfo { name: "sink_2", type_name: "usize", description: "Sink vertex t_2 for commodity 2" },
+            FieldInfo { name: "requirement_1", type_name: "u64", description: "Flow requirement R_1 for commodity 1" },
+            FieldInfo { name: "requirement_2", type_name: "u64", description: "Flow requirement R_2 for commodity 2" },
+        ],
+    }
+}
+
+/// Directed Two-Commodity Integral Flow problem.
+///
+/// Given a directed graph G = (V, A) with arc capacities c(a), two source-sink
+/// pairs (s_1, t_1) and (s_2, t_2), and requirements R_1, R_2, determine
+/// whether two integral flow functions f_1, f_2: A -> Z_0^+ exist such that:
+/// 1. Joint capacity: f_1(a) + f_2(a) <= c(a) for all a in A
+/// 2. Flow conservation: for each commodity i, flow is conserved at every
+///    vertex except the four terminals
+/// 3. Requirements: net flow into t_i under f_i is at least R_i
+///
+/// # Variables
+///
+/// 2|A| variables: first |A| for commodity 1's flow on each arc,
+/// next |A| for commodity 2's flow on each arc. Variable j for arc a
+/// of commodity i ranges over {0, ..., c(a)}.
+///
+/// # Example
+///
+/// ```
+/// use problemreductions::models::graph::DirectedTwoCommodityIntegralFlow;
+/// use problemreductions::topology::DirectedGraph;
+/// use problemreductions::{Problem, Solver, BruteForce};
+///
+/// // 6-vertex network: s1=0, s2=1, t1=4, t2=5
+/// let graph = DirectedGraph::new(6, vec![
+///     (0, 2), (0, 3), (1, 2), (1, 3),
+///     (2, 4), (2, 5), (3, 4), (3, 5),
+/// ]);
+/// let problem = DirectedTwoCommodityIntegralFlow::new(
+///     graph, vec![1; 8], 0, 4, 1, 5, 1, 1,
+/// );
+/// let solver = BruteForce::new();
+/// assert!(solver.find_satisfying(&problem).is_some());
+/// ```
+#[derive(Debug, Clone, Serialize, Deserialize)]
+pub struct DirectedTwoCommodityIntegralFlow {
+    /// The directed graph G = (V, A).
+    graph: DirectedGraph,
+    /// Capacity c(a) for each arc.
+    capacities: Vec<u64>,
+    /// Source vertex s_1 for commodity 1.
+    source_1: usize,
+    /// Sink vertex t_1 for commodity 1.
+    sink_1: usize,
+    /// Source vertex s_2 for commodity 2.
+    source_2: usize,
+    /// Sink vertex t_2 for commodity 2.
+    sink_2: usize,
+    /// Flow requirement R_1 for commodity 1.
+    requirement_1: u64,
+    /// Flow requirement R_2 for commodity 2.
+    requirement_2: u64,
+}
+
+impl DirectedTwoCommodityIntegralFlow {
+    /// Create a new Directed Two-Commodity Integral Flow problem.
+    ///
+    /// # Panics
+    ///
+    /// Panics if:
+    /// - `capacities.len() != graph.num_arcs()`
+    /// - Any terminal vertex index >= `graph.num_vertices()`
+    #[allow(clippy::too_many_arguments)]
+    pub fn new(
+        graph: DirectedGraph,
+        capacities: Vec<u64>,
+        source_1: usize,
+        sink_1: usize,
+        source_2: usize,
+        sink_2: usize,
+        requirement_1: u64,
+        requirement_2: u64,
+    ) -> Self {
+        let n = graph.num_vertices();
+        assert_eq!(
+            capacities.len(),
+            graph.num_arcs(),
+            "capacities length must match graph num_arcs"
+        );
+        assert!(source_1 < n, "source_1 ({source_1}) >= num_vertices ({n})");
+        assert!(sink_1 < n, "sink_1 ({sink_1}) >= num_vertices ({n})");
+        assert!(source_2 < n, "source_2 ({source_2}) >= num_vertices ({n})");
+        assert!(sink_2 < n, "sink_2 ({sink_2}) >= num_vertices ({n})");
+        Self {
+            graph,
+            capacities,
+            source_1,
+            sink_1,
+            source_2,
+            sink_2,
+            requirement_1,
+            requirement_2,
+        }
+    }
+
+    /// Get a reference to the underlying directed graph.
+    pub fn graph(&self) -> &DirectedGraph {
+        &self.graph
+    }
+
+    /// Get a reference to the capacities.
+    pub fn capacities(&self) -> &[u64] {
+        &self.capacities
+    }
+
+    /// Get source vertex for commodity 1.
+    pub fn source_1(&self) -> usize {
+        self.source_1
+    }
+
+    /// Get sink vertex for commodity 1.
+    pub fn sink_1(&self) -> usize {
+        self.sink_1
+    }
+
+    /// Get source vertex for commodity 2.
+    pub fn source_2(&self) -> usize {
+        self.source_2
+    }
+
+    /// Get sink vertex for commodity 2.
+    pub fn sink_2(&self) -> usize {
+        self.sink_2
+    }
+
+    /// Get requirement for commodity 1.
+    pub fn requirement_1(&self) -> u64 {
+        self.requirement_1
+    }
+
+    /// Get requirement for commodity 2.
+    pub fn requirement_2(&self) -> u64 {
+        self.requirement_2
+    }
+
+    /// Get the number of vertices.
+    pub fn num_vertices(&self) -> usize {
+        self.graph.num_vertices()
+    }
+
+    /// Get the number of arcs.
+    pub fn num_arcs(&self) -> usize {
+        self.graph.num_arcs()
+    }
+
+    /// Get the maximum capacity across all arcs.
+    pub fn max_capacity(&self) -> u64 {
+        self.capacities.iter().copied().max().unwrap_or(0)
+    }
+
+    /// Check whether a flow assignment is feasible.
+    ///
+    /// `config` has 2*|A| entries: first |A| for commodity 1, next |A| for commodity 2.
+    pub fn is_feasible(&self, config: &[usize]) -> bool {
+        let m = self.graph.num_arcs();
+        if config.len() != 2 * m {
+            return false;
+        }
+        let arcs = self.graph.arcs();
+        let terminals = [self.source_1, self.sink_1, self.source_2, self.sink_2];
+
+        // (1) Joint capacity constraint
+        for a in 0..m {
+            let f1 = config[a] as u64;
+            let f2 = config[m + a] as u64;
+            if f1 + f2 > self.capacities[a] {
+                return false;
+            }
+        }
+
+        // (2) Flow conservation for each commodity at non-terminal vertices
+        let n = self.graph.num_vertices();
+        let mut balances = [vec![0_i128; n], vec![0_i128; n]];
+        for (a, &(u, w)) in arcs.iter().enumerate() {
+            let flow_1 = config[a] as i128;
+            let flow_2 = config[m + a] as i128;
+
+            balances[0][u] -= flow_1;
+            balances[0][w] += flow_1;
+            balances[1][u] -= flow_2;
+            balances[1][w] += flow_2;
+        }
+
+        for (commodity, commodity_balances) in balances.iter().enumerate() {
+            for (v, &balance) in commodity_balances.iter().enumerate() {
+                if !terminals.contains(&v) && balance != 0 {
+                    return false;
+                }
+            }
+
+            let snk = if commodity == 0 {
+                self.sink_1
+            } else {
+                self.sink_2
+            };
+            let req = if commodity == 0 {
+                self.requirement_1
+            } else {
+                self.requirement_2
+            };
+
+            if commodity_balances[snk] < i128::from(req) {
+                return false;
+            }
+        }
+
+        true
+    }
+}
+
+impl Problem for DirectedTwoCommodityIntegralFlow {
+    const NAME: &'static str = "DirectedTwoCommodityIntegralFlow";
+    type Metric = bool;
+
+    fn dims(&self) -> Vec<usize> {
+        self.capacities
+            .iter()
+            .chain(self.capacities.iter())
+            .map(|&c| (c as usize) + 1)
+            .collect()
+    }
+
+    fn evaluate(&self, config: &[usize]) -> bool {
+        self.is_feasible(config)
+    }
+
+    fn variant() -> Vec<(&'static str, &'static str)> {
+        crate::variant_params![]
+    }
+}
+
+impl SatisfactionProblem for DirectedTwoCommodityIntegralFlow {}
+
+crate::declare_variants! {
+    default sat DirectedTwoCommodityIntegralFlow => "(max_capacity + 1)^(2 * num_arcs)",
+}
+
+#[cfg(feature = "example-db")]
+pub(crate) fn canonical_model_example_specs() -> Vec<crate::example_db::specs::ModelExampleSpec> {
+    vec![crate::example_db::specs::ModelExampleSpec {
+        id: "directed_two_commodity_integral_flow",
+        build: || {
+            let graph = DirectedGraph::new(
+                6,
+                vec![
+                    (0, 2),
+                    (0, 3),
+                    (1, 2),
+                    (1, 3),
+                    (2, 4),
+                    (2, 5),
+                    (3, 4),
+                    (3, 5),
+                ],
+            );
+            let problem =
+                DirectedTwoCommodityIntegralFlow::new(graph, vec![1; 8], 0, 4, 1, 5, 1, 1);
+            // Solution: commodity 1 path 0->2->4 (arcs 0,4), commodity 2 path 1->3->5 (arcs 3,7)
+            // config = [f1(a0)..f1(a7), f2(a0)..f2(a7)]
+            //        = [1,0,0,0,1,0,0,0, 0,0,0,1,0,0,0,1]
+            crate::example_db::specs::satisfaction_example(
+                problem,
+                vec![vec![1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1]],
+            )
+        },
+    }]
+}
+
+#[cfg(test)]
+#[path = "../../unit_tests/models/graph/directed_two_commodity_integral_flow.rs"]
+mod tests;
diff --git a/src/models/graph/mod.rs b/src/models/graph/mod.rs
index bde7b663..ef940c03 100644
--- a/src/models/graph/mod.rs
+++ b/src/models/graph/mod.rs
@@ -24,9 +24,11 @@
 //! - [`RuralPostman`]: Rural Postman (circuit covering required edges)
 //! - [`SteinerTree`]: Minimum-weight tree spanning all required terminals
 //! - [`SubgraphIsomorphism`]: Subgraph isomorphism (decision problem)
+//! - [`DirectedTwoCommodityIntegralFlow`]: Directed two-commodity integral flow (satisfaction)
 //! - [`UndirectedTwoCommodityIntegralFlow`]: Two-commodity integral flow on undirected graphs
 
 pub(crate) mod biclique_cover;
+pub(crate) mod directed_two_commodity_integral_flow;
 pub(crate) mod graph_partitioning;
 pub(crate) mod hamiltonian_path;
 pub(crate) mod isomorphic_spanning_tree;
@@ -52,6 +54,7 @@ pub(crate) mod traveling_salesman;
 pub(crate) mod undirected_two_commodity_integral_flow;
 
 pub use biclique_cover::BicliqueCover;
+pub use directed_two_commodity_integral_flow::DirectedTwoCommodityIntegralFlow;
 pub use graph_partitioning::GraphPartitioning;
 pub use hamiltonian_path::HamiltonianPath;
 pub use isomorphic_spanning_tree::IsomorphicSpanningTree;
@@ -97,6 +100,7 @@ pub(crate) fn canonical_model_example_specs() -> Vec<crate::example_db::specs::M
     specs.extend(biclique_cover::canonical_model_example_specs());
     specs.extend(partition_into_triangles::canonical_model_example_specs());
     specs.extend(steiner_tree::canonical_model_example_specs());
+    specs.extend(directed_two_commodity_integral_flow::canonical_model_example_specs());
     specs.extend(undirected_two_commodity_integral_flow::canonical_model_example_specs());
     specs
 }
diff --git a/src/models/mod.rs b/src/models/mod.rs
index 3031a2d9..94614519 100644
--- a/src/models/mod.rs
+++ b/src/models/mod.rs
@@ -12,12 +12,12 @@ pub mod set;
 pub use algebraic::{ClosestVectorProblem, BMF, ILP, QUBO};
 pub use formula::{CNFClause, CircuitSAT, KSatisfiability, Satisfiability};
 pub use graph::{
-    BicliqueCover, GraphPartitioning, HamiltonianPath, IsomorphicSpanningTree, KColoring,
-    LengthBoundedDisjointPaths, MaxCut, MaximalIS, MaximumClique, MaximumIndependentSet,
-    MaximumMatching, MinimumDominatingSet, MinimumFeedbackArcSet, MinimumFeedbackVertexSet,
-    MinimumSumMulticenter, MinimumVertexCover, OptimalLinearArrangement, PartitionIntoTriangles,
-    RuralPostman, SpinGlass, SteinerTree, SubgraphIsomorphism, TravelingSalesman,
-    UndirectedTwoCommodityIntegralFlow,
+    BicliqueCover, DirectedTwoCommodityIntegralFlow, GraphPartitioning, HamiltonianPath,
+    IsomorphicSpanningTree, KColoring, LengthBoundedDisjointPaths, MaxCut, MaximalIS,
+    MaximumClique, MaximumIndependentSet, MaximumMatching, MinimumDominatingSet,
+    MinimumFeedbackArcSet, MinimumFeedbackVertexSet, MinimumSumMulticenter, MinimumVertexCover,
+    OptimalLinearArrangement, PartitionIntoTriangles, RuralPostman, SpinGlass, SteinerTree,
+    SubgraphIsomorphism, TravelingSalesman, UndirectedTwoCommodityIntegralFlow,
 };
 pub use misc::{
     BinPacking, Factoring, FlowShopScheduling, Knapsack, LongestCommonSubsequence,
diff --git a/src/unit_tests/models/graph/directed_two_commodity_integral_flow.rs b/src/unit_tests/models/graph/directed_two_commodity_integral_flow.rs
new file mode 100644
index 00000000..57ea7952
--- /dev/null
+++ b/src/unit_tests/models/graph/directed_two_commodity_integral_flow.rs
@@ -0,0 +1,197 @@
+use super::*;
+use crate::solvers::{BruteForce, Solver};
+use crate::topology::DirectedGraph;
+use crate::traits::Problem;
+
+/// YES instance: 6 vertices, 8 arcs (all capacity 1).
+/// s1=0, t1=4, s2=1, t2=5, R1=1, R2=1.
+fn yes_instance() -> DirectedTwoCommodityIntegralFlow {
+    let graph = DirectedGraph::new(
+        6,
+        vec![
+            (0, 2),
+            (0, 3),
+            (1, 2),
+            (1, 3),
+            (2, 4),
+            (2, 5),
+            (3, 4),
+            (3, 5),
+        ],
+    );
+    DirectedTwoCommodityIntegralFlow::new(graph, vec![1; 8], 0, 4, 1, 5, 1, 1)
+}
+
+/// NO instance: 4 vertices, 3 arcs (all capacity 1).
+/// s1=0, t1=3, s2=1, t2=3, R1=1, R2=1.
+/// Bottleneck at arc (2,3) with capacity 1.
+fn no_instance() -> DirectedTwoCommodityIntegralFlow {
+    let graph = DirectedGraph::new(4, vec![(0, 2), (1, 2), (2, 3)]);
+    DirectedTwoCommodityIntegralFlow::new(graph, vec![1; 3], 0, 3, 1, 3, 1, 1)
+}
+
+#[test]
+fn test_directed_two_commodity_integral_flow_creation() {
+    let problem = yes_instance();
+    assert_eq!(problem.num_vertices(), 6);
+    assert_eq!(problem.num_arcs(), 8);
+    assert_eq!(problem.dims().len(), 16); // 2 * 8
+    assert!(problem.dims().iter().all(|&d| d == 2)); // capacity 1 -> domain {0,1}
+    assert_eq!(problem.source_1(), 0);
+    assert_eq!(problem.sink_1(), 4);
+    assert_eq!(problem.source_2(), 1);
+    assert_eq!(problem.sink_2(), 5);
+    assert_eq!(problem.requirement_1(), 1);
+    assert_eq!(problem.requirement_2(), 1);
+    assert_eq!(problem.max_capacity(), 1);
+}
+
+#[test]
+fn test_directed_two_commodity_integral_flow_evaluation_satisfying() {
+    let problem = yes_instance();
+    // Commodity 1: path 0->2->4 (arcs 0,4)
+    // Commodity 2: path 1->3->5 (arcs 3,7)
+    // config = [f1(a0..a7), f2(a0..a7)]
+    let config = vec![1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1];
+    assert!(problem.evaluate(&config));
+}
+
+#[test]
+fn test_directed_two_commodity_integral_flow_evaluation_unsatisfying() {
+    let problem = no_instance();
+    // All zeros: no flow at all
+    let config = vec![0, 0, 0, 0, 0, 0];
+    assert!(!problem.evaluate(&config));
+}
+
+#[test]
+fn test_directed_two_commodity_integral_flow_capacity_violation() {
+    let problem = yes_instance();
+    // Try sending both commodities through the same arc (arc 0: 0->2)
+    // f1(a0)=1, f2(a0)=1 -> violates capacity 1
+    let mut config = vec![0; 16];
+    config[0] = 1; // f1 on arc 0
+    config[8] = 1; // f2 on arc 0
+    assert!(!problem.evaluate(&config));
+}
+
+#[test]
+fn test_directed_two_commodity_integral_flow_conservation_violation() {
+    let problem = yes_instance();
+    // f1 sends flow into vertex 2 but not out
+    let mut config = vec![0; 16];
+    config[0] = 1; // f1 on arc 0 (0->2): flow into vertex 2
+                   // No outgoing flow from vertex 2 for commodity 1
+    assert!(!problem.evaluate(&config));
+}
+
+#[test]
+fn test_directed_two_commodity_integral_flow_negative_net_flow_at_sink_is_infeasible() {
+    let graph = DirectedGraph::new(3, vec![(1, 2)]);
+    let problem = DirectedTwoCommodityIntegralFlow::new(graph, vec![1], 0, 1, 2, 2, 1, 0);
+
+    // Commodity 1 sends flow out of its sink with no incoming flow.
+    let config = vec![1, 0];
+    assert!(!problem.evaluate(&config));
+}
+
+#[test]
+fn test_directed_two_commodity_integral_flow_solver_yes() {
+    let problem = yes_instance();
+    let solver = BruteForce::new();
+    let solution = solver.find_satisfying(&problem);
+    assert!(solution.is_some());
+    let sol = solution.unwrap();
+    assert!(problem.evaluate(&sol));
+}
+
+#[test]
+fn test_directed_two_commodity_integral_flow_solver_no() {
+    let problem = no_instance();
+    let solver = BruteForce::new();
+    let solution = solver.find_satisfying(&problem);
+    assert!(solution.is_none());
+}
+
+#[test]
+fn test_directed_two_commodity_integral_flow_serialization() {
+    let problem = yes_instance();
+    let json = serde_json::to_string(&problem).unwrap();
+    let deserialized: DirectedTwoCommodityIntegralFlow = serde_json::from_str(&json).unwrap();
+    assert_eq!(deserialized.num_vertices(), 6);
+    assert_eq!(deserialized.num_arcs(), 8);
+    assert_eq!(deserialized.source_1(), 0);
+    assert_eq!(deserialized.sink_1(), 4);
+    assert_eq!(deserialized.source_2(), 1);
+    assert_eq!(deserialized.sink_2(), 5);
+    assert_eq!(deserialized.requirement_1(), 1);
+    assert_eq!(deserialized.requirement_2(), 1);
+}
+
+#[test]
+fn test_directed_two_commodity_integral_flow_problem_name() {
+    assert_eq!(
+        <DirectedTwoCommodityIntegralFlow as Problem>::NAME,
+        "DirectedTwoCommodityIntegralFlow"
+    );
+}
+
+#[test]
+fn test_directed_two_commodity_integral_flow_accessors() {
+    let problem = yes_instance();
+    assert_eq!(problem.graph().num_vertices(), 6);
+    assert_eq!(problem.graph().num_arcs(), 8);
+    assert_eq!(problem.capacities(), &[1, 1, 1, 1, 1, 1, 1, 1]);
+}
+
+#[test]
+fn test_directed_two_commodity_integral_flow_paper_example() {
+    let problem = yes_instance();
+    let solver = BruteForce::new();
+
+    // Verify the known solution evaluates to true
+    let config = vec![1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1];
+    assert!(problem.evaluate(&config));
+
+    // Find all satisfying solutions and verify count
+    let all_solutions = solver.find_all_satisfying(&problem);
+    assert!(!all_solutions.is_empty());
+
+    // Each solution must evaluate to true
+    for sol in &all_solutions {
+        assert!(problem.evaluate(sol));
+    }
+}
+
+#[test]
+fn test_directed_two_commodity_integral_flow_wrong_config_length() {
+    let problem = yes_instance();
+    // Config with wrong length should return false (infeasible)
+    assert!(!problem.evaluate(&[0; 15])); // too short
+    assert!(!problem.evaluate(&[0; 17])); // too long
+    assert!(!problem.evaluate(&[])); // empty
+}
+
+#[test]
+fn test_directed_two_commodity_integral_flow_higher_capacity() {
+    // Test with capacity 2: two paths can share an arc
+    let graph = DirectedGraph::new(3, vec![(0, 1), (1, 2)]);
+    let problem = DirectedTwoCommodityIntegralFlow::new(
+        graph,
+        vec![2, 2], // capacity 2 on both arcs
+        0,
+        2,
+        0,
+        2,
+        1,
+        1,
+    );
+    assert_eq!(problem.dims(), vec![3, 3, 3, 3]); // each variable in {0,1,2}
+
+    // Both commodities can share: f1=1, f2=1 on both arcs
+    let config = vec![1, 1, 1, 1];
+    assert!(problem.evaluate(&config));
+
+    let solver = BruteForce::new();
+    assert!(solver.find_satisfying(&problem).is_some());
+}
diff --git a/src/unit_tests/trait_consistency.rs b/src/unit_tests/trait_consistency.rs
index e5117e24..ef0f09f6 100644
--- a/src/unit_tests/trait_consistency.rs
+++ b/src/unit_tests/trait_consistency.rs
@@ -106,6 +106,19 @@ fn test_all_problems_implement_trait_correctly() {
         ),
         "MinimumFeedbackArcSet",
     );
+    check_problem_trait(
+        &DirectedTwoCommodityIntegralFlow::new(
+            DirectedGraph::new(4, vec![(0, 1), (1, 2), (2, 3)]),
+            vec![1; 3],
+            0,
+            3,
+            0,
+            3,
+            1,
+            1,
+        ),
+        "DirectedTwoCommodityIntegralFlow",
+    );
     check_problem_trait(
         &MinimumSumMulticenter::new(
             SimpleGraph::new(3, vec![(0, 1), (1, 2)]),