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Fix #411: [Model] CapacityAssignment (#747)
* Add plan for #411: [Model] CapacityAssignment * Implement #411: [Model] CapacityAssignment * chore: remove plan file after implementation * fix: rustfmt after merge with main * fix: update --capacities help text to cover CapacityAssignment Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com> --------- Co-authored-by: Xiwei Pan <xiwei.pan@connect.hkust-gz.edu.cn> Co-authored-by: Claude Opus 4.6 (1M context) <noreply@anthropic.com> Co-authored-by: Xiwei Pan <90967972+isPANN@users.noreply.github.com>
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‎docs/paper/reductions.typ‎

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"BoundedComponentSpanningForest": [Bounded Component Spanning Forest],
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"BinPacking": [Bin Packing],
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"BoyceCoddNormalFormViolation": [Boyce-Codd Normal Form Violation],
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"CapacityAssignment": [Capacity Assignment],
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"ConsistencyOfDatabaseFrequencyTables": [Consistency of Database Frequency Tables],
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"ClosestVectorProblem": [Closest Vector Problem],
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"ConsecutiveSets": [Consecutive Sets],
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]
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}
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#{
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let x = load-model-example("CapacityAssignment")
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[
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#problem-def("CapacityAssignment")[
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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 $sum_(c in C) d(c, sigma(c)) <= J$.
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][
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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.
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*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.
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#pred-commands(
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"pred create --example " + problem-spec(x) + " -o capacity-assignment.json",
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"pred solve capacity-assignment.json --solver brute-force",
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"pred evaluate capacity-assignment.json --config " + x.optimal_config.map(str).join(","),
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)
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#figure({
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table(
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columns: (auto, auto, auto),
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inset: 4pt,
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align: left,
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table.header([*Link*], [*Cost row*], [*Delay row*]),
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[$c_1$], [$(1, 3, 6)$], [$(8, 4, 1)$],
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[$c_2$], [$(2, 4, 7)$], [$(7, 3, 1)$],
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[$c_3$], [$(1, 2, 5)$], [$(6, 3, 1)$],
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)
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},
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caption: [Canonical Capacity Assignment instance with budgets $K = 10$ and $J = 12$. Each row lists the cost-delay trade-off for one communication link.],
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) <fig:capacity-assignment>
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]
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]
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}
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#{
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let x = load-model-example("PrecedenceConstrainedScheduling")
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let n = x.instance.num_tasks

‎docs/paper/references.bib‎

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

‎problemreductions-cli/src/cli.rs‎

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PathConstrainedNetworkFlow --arcs, --capacities, --source, --sink, --paths, --requirement
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Factoring --target, --m, --n
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BinPacking --sizes, --capacity
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CapacityAssignment --capacities, --cost-matrix, --delay-matrix, --cost-budget, --delay-budget
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SubsetSum --sizes, --target
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SumOfSquaresPartition --sizes, --num-groups, --bound
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PaintShop --sequence
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pred create MIS --graph 0-1,1-2,2-3 --weights 1,1,1
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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 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
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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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/// Edge lengths (e.g., 2,3,1) [default: all 1s]
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#[arg(long)]
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pub edge_lengths: Option<String>,
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/// Edge capacities for multicommodity flow problems (e.g., 1,1,2)
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/// Capacities (edge capacities for flow problems, capacity levels for CapacityAssignment)
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#[arg(long)]
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pub capacities: Option<String>,
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/// Edge lower bounds for lower-bounded flow problems (e.g., 1,1,0,0,1,0,1)
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#[arg(long)]
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pub lower_bounds: Option<String>,
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/// Bundle capacities for IntegralFlowBundles (e.g., 1,1,1)
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#[arg(long)]
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pub bundle_capacities: Option<String>,
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/// Cost matrix for CapacityAssignment (semicolon-separated rows, e.g., "1,3,6;2,4,7")
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#[arg(long)]
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pub cost_matrix: Option<String>,
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/// Delay matrix for CapacityAssignment (semicolon-separated rows, e.g., "8,4,1;7,3,1")
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#[arg(long)]
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pub delay_matrix: Option<String>,
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/// Edge lower bounds for lower-bounded flow problems (e.g., 1,1,0,0,1,0,1)
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#[arg(long)]
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pub lower_bounds: 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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/// Upper bound on total inter-partition arc cost
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#[arg(long)]
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pub cost_bound: Option<i32>,
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/// Budget on total cost for CapacityAssignment
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#[arg(long)]
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pub cost_budget: Option<u64>,
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/// Budget on total delay penalty for CapacityAssignment
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#[arg(long)]
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pub delay_budget: Option<u64>,
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/// Pattern graph edge list for SubgraphIsomorphism (e.g., 0-1,1-2,2-0)
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#[arg(long)]
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pub pattern: Option<String>,

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