Fix #411: [Model] CapacityAssignment

docs/paper/reductions.typ

@@ -117,6 +117,7 @@
   "BoundedComponentSpanningForest": [Bounded Component Spanning Forest],
   "BinPacking": [Bin Packing],
   "BoyceCoddNormalFormViolation": [Boyce-Codd Normal Form Violation],
+  "CapacityAssignment": [Capacity Assignment],
   "ConsistencyOfDatabaseFrequencyTables": [Consistency of Database Frequency Tables],
   "ClosestVectorProblem": [Closest Vector Problem],
   "ConsecutiveSets": [Consecutive Sets],
@@ -5052,6 +5053,39 @@ A classical NP-complete problem from Garey and Johnson @garey1979[Ch.~3, p.~76],
   ]
 }
 
+#{
+  let x = load-model-example("CapacityAssignment")
+  [
+    #problem-def("CapacityAssignment")[
+      Given a finite set $C$ of communication links, an ordered set $M subset ZZ_(> 0)$ of capacities, cost and delay functions $g: C times M -> ZZ_(>= 0)$ and $d: C times M -> ZZ_(>= 0)$ such that for every $c in C$ and $i < j$ in the order of $M$ we have $g(c, i) <= g(c, j)$ and $d(c, i) >= d(c, j)$, and budgets $K, J in ZZ_(>= 0)$, determine whether there exists an assignment $sigma: C -> M$ such that $sum_(c in C) g(c, sigma(c)) <= K$ and $sum_(c in C) d(c, sigma(c)) <= J$.
+    ][
+      Capacity Assignment is the bicriteria communication-network design problem SR7 in Garey & Johnson @garey1979. The original NP-completeness proof, via reduction from Subset Sum, is due to Van Sickle and Chandy @vansicklechandy1977. The model captures discrete provisioning of communication links, where upgrading a link increases installation cost but decreases delay. The direct witness encoding implemented in this repository yields an $O^*(|M|^(|C|))$ exact algorithm by brute-force enumeration#footnote[No algorithm improving on brute-force enumeration is known for the exact witness encoding used in this repository.]. Garey and Johnson also note a pseudo-polynomial dynamic-programming formulation when the budgets are small @garey1979.
+
+      *Example.* Let $C = {c_1, c_2, c_3}$, $M = {1, 2, 3}$, $K = 10$, and $J = 12$. With cost rows $(1, 3, 6)$, $(2, 4, 7)$, $(1, 2, 5)$ and delay rows $(8, 4, 1)$, $(7, 3, 1)$, $(6, 3, 1)$, the assignment $sigma = (2, 2, 2)$ has total cost $3 + 4 + 2 = 9 <= 10$ and total delay $4 + 3 + 3 = 10 <= 12$, so the instance is satisfiable. Brute-force enumeration finds exactly 5 satisfying assignments; for contrast, $sigma = (1, 1, 1)$ violates the delay budget and $sigma = (3, 3, 3)$ violates the cost budget.
+
+      #pred-commands(
+        "pred create --example " + problem-spec(x) + " -o capacity-assignment.json",
+        "pred solve capacity-assignment.json --solver brute-force",
+        "pred evaluate capacity-assignment.json --config " + x.optimal_config.map(str).join(","),
+      )
+
+      #figure({
+        table(
+          columns: (auto, auto, auto),
+          inset: 4pt,
+          align: left,
+          table.header([*Link*], [*Cost row*], [*Delay row*]),
+          [$c_1$], [$(1, 3, 6)$], [$(8, 4, 1)$],
+          [$c_2$], [$(2, 4, 7)$], [$(7, 3, 1)$],
+          [$c_3$], [$(1, 2, 5)$], [$(6, 3, 1)$],
+        )
+      },
+      caption: [Canonical Capacity Assignment instance with budgets $K = 10$ and $J = 12$. Each row lists the cost-delay trade-off for one communication link.],
+      ) <fig:capacity-assignment>
+    ]
+  ]
+}
+
 #{
   let x = load-model-example("PrecedenceConstrainedScheduling")
   let n = x.instance.num_tasks

docs/paper/references.bib

@@ -213,6 +213,14 @@ @article{robertsonSeymour1995
   doi     = {10.1006/jctb.1995.1006}
 }
 
+@inproceedings{vansicklechandy1977,
+  author    = {Lawrence Van Sickle and K. Mani Chandy},
+  title     = {Computational Complexity of Network Design Algorithms},
+  booktitle = {IFIP Congress 77},
+  pages     = {235--239},
+  year      = {1977}
+}
+
 @article{bruckerGareyJohnson1977,
   author  = {Peter Brucker and Michael R. Garey and David S. Johnson},
   title   = {Scheduling equal-length tasks under tree-like precedence constraints to minimize maximum lateness},

problemreductions-cli/src/cli.rs

@@ -246,6 +246,7 @@ Flags by problem type:
   PathConstrainedNetworkFlow      --arcs, --capacities, --source, --sink, --paths, --requirement
   Factoring                       --target, --m, --n
   BinPacking                      --sizes, --capacity
+  CapacityAssignment              --capacities, --cost-matrix, --delay-matrix, --cost-budget, --delay-budget
   SubsetSum                       --sizes, --target
   SumOfSquaresPartition           --sizes, --num-groups, --bound
   PaintShop                       --sequence
@@ -321,6 +322,7 @@ Examples:
   pred create MIS --graph 0-1,1-2,2-3 --weights 1,1,1
   pred create SAT --num-vars 3 --clauses \"1,2;-1,3\"
   pred create QUBO --matrix \"1,0.5;0.5,2\"
+  pred create CapacityAssignment --capacities 1,2,3 --cost-matrix \"1,3,6;2,4,7;1,2,5\" --delay-matrix \"8,4,1;7,3,1;6,3,1\" --cost-budget 10 --delay-budget 12
   pred create GeneralizedHex --graph 0-1,0-2,0-3,1-4,2-4,3-4,4-5 --source 0 --sink 5
   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
   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
@@ -366,15 +368,21 @@ pub struct CreateArgs {
     /// Edge lengths (e.g., 2,3,1) [default: all 1s]
     #[arg(long)]
     pub edge_lengths: Option<String>,
-    /// Edge capacities for multicommodity flow problems (e.g., 1,1,2)
+    /// Capacities (edge capacities for flow problems, capacity levels for CapacityAssignment)
     #[arg(long)]
     pub capacities: Option<String>,
-    /// Edge lower bounds for lower-bounded flow problems (e.g., 1,1,0,0,1,0,1)
-    #[arg(long)]
-    pub lower_bounds: Option<String>,
     /// Bundle capacities for IntegralFlowBundles (e.g., 1,1,1)
     #[arg(long)]
     pub bundle_capacities: Option<String>,
+    /// Cost matrix for CapacityAssignment (semicolon-separated rows, e.g., "1,3,6;2,4,7")
+    #[arg(long)]
+    pub cost_matrix: Option<String>,
+    /// Delay matrix for CapacityAssignment (semicolon-separated rows, e.g., "8,4,1;7,3,1")
+    #[arg(long)]
+    pub delay_matrix: Option<String>,
+    /// Edge lower bounds for lower-bounded flow problems (e.g., 1,1,0,0,1,0,1)
+    #[arg(long)]
+    pub lower_bounds: Option<String>,
     /// Vertex multipliers in vertex order (e.g., 1,2,3,1)
     #[arg(long)]
     pub multipliers: Option<String>,
@@ -547,6 +555,12 @@ pub struct CreateArgs {
     /// Upper bound on total inter-partition arc cost
     #[arg(long)]
     pub cost_bound: Option<i32>,
+    /// Budget on total cost for CapacityAssignment
+    #[arg(long)]
+    pub cost_budget: Option<u64>,
+    /// Budget on total delay penalty for CapacityAssignment
+    #[arg(long)]
+    pub delay_budget: Option<u64>,
     /// Pattern graph edge list for SubgraphIsomorphism (e.g., 0-1,1-2,2-0)
     #[arg(long)]
     pub pattern: Option<String>,