📐algorithm-minimum-cost-flow

The groupant beside algorithms/src/minimum_cost_flow/mod.rs in the Mova repository: what the code computes, what it never does, how to go back from its outputs, its invariants, cost and checks.

Facts

Components

Obernants

The typed relations around the forward model: the behaviour complement (what the code never does), the reverse reconstruction (every input compatible with an output) and specialised views.

Model text

Migration: consolidated into model.md, 2026-11-04, operator decision; fresh hash-bound references to the former files were rebound to the sections below.

descriptant

minimum_cost_flow mod

Original std-only Rust. Prospective model is retained beside implementation. Source citations identify algorithm provenance; no copied implementation. Exact bounded graph semantics are useful independently of any unmeasured optimization. Finite oracle checks and mutant discrimination improve confidence; groupant count alone proves no improvement.

model

minimum_cost_flow mod model

Forward

Successive shortest augmenting paths, Bellman-Ford over residual arcs. Bound n<=64,m<=512,requested<=1024; reject any original positive-capacity negative cycle globally before even zero requested flow. Flow integral and bounded; shortest residual augmentation preserves minimum cost for current value. Maximize value<=requested then minimize signed i128 cost. Input edge order resolves strict relaxation ties. Checked arithmetic. Cost O((limit+1)V(E+V)), storage O(V+E). No speed claim.

Invariant

Each residual pair sums to original capacity. Internal flow conservation holds. No negative residual cycle; each augmentation attains a shortest residual path and increases integer value.

Cost

Successive shortest augmenting paths, Bellman-Ford over residual arcs. Bound n<=64,m<=512,requested<=1024; reject any original positive-capacity negative cycle globally before even zero requested flow. Flow integral and bounded; shortest residual augmentation preserves minimum cost for current value. Maximize value<=requested then minimize signed i128 cost. Input edge order resolves strict relaxation ties. Checked arithmetic. Cost O((limit+1)V(E+V)), storage O(V+E). No speed claim.

nonbehavior

Type: behavior_complement. Let R be the deterministic relation specified by Forward including validation precedence and tie rules, X the declared input type and Y its Result type. Behavior complement is (X times Y) minus R. Specifically outputs violating Each residual pair sums to original capacity. Internal flow conservation holds. No negative residual cycle; each augmentation attains a shortest residual path and increases integer value. are excluded; a valid-input Err or a wrong successful witness is excluded. Resource, cycle and structural rejections are determined before algorithmic output.

compatible_inputs

Type: reverse_reconstruction. Given (value,cost,edge_flows), antecedents satisfy capacity bounds, conservation, sum f_e*c_e=cost and lexicographic optimality under limit. Different unused capacities/costs yield the same output. InvalidEdge identifies first bad index; NegativeCycle recovers existence, not location.

potential_transport

Type: potential_reweighting. Costs c'(u,v)=c(u,v)+h(u)-h(v), all fitting i64, preserve cycle sums and optimal flow value; optimal cost changes by value*(h(source)-h(sink)). Chosen witness can differ only among ties.

zero_capacity

Type: irrelevant_input. Adding a zero-capacity edge cannot change value or cost, though edge_flows gains a zero. Its endpoints must still validate.

capacity_certificate

Type: feasibility_certificate. For every output, 0<=f_e<=capacity_e; incoming minus outgoing equals value at sink, -value at source and zero elsewhere. Objective is exact sum f_e*c_e.

limit_monotonicity

Type: monotone_extension. For limits L<=L' within 1024, achieved value cannot decrease. Cost need not increase when costs are negative. Capped value is min(L, maximum feasible flow).

Limits

Partial syntax omits macro expansion and type resolution. No executable approximate profile: categorical graph semantics and feasibility have no justified percentage relaxation here. Integer operations are bounded by caps; flow uses checked i128/u64 arithmetic, with Overflow defensive under current caps. No callbacks. Allocator failure is outside typed status.

evidence

Evidence — 2026-10-30

Eleven tests across five algorithms, including independent exhaustive small-domain characterization. This file belongs to minimum_cost_flow. Two debug and two optimized runs matched normalized statuses. Strict clippy-driver --test -D warnings passed. A compiled production mutant failed a named assertion; exact bytes restored and positive rerun passed. Raw receipts: $MOVA_DATA/runs/octopus-wave25-graphs-2026-10-30/minimum_cost_flow-*.txt. No timing or whole-system improvement claim. Source SHA-256: a6de6ae849a183b1b508ebf475e7b1517b27bbf5c38a04dc6dbe1394e6885697.

prospective

Prospective model — 2026-10-30

Successive shortest augmenting paths, Bellman-Ford over residual arcs. Bound n<=64,m<=512,requested<=1024; reject any original positive-capacity negative cycle globally before even zero requested flow. Flow integral and bounded; shortest residual augmentation preserves minimum cost for current value. Maximize value<=requested then minimize signed i128 cost. Input edge order resolves strict relaxation ties. Checked arithmetic. Cost O((limit+1)VE), storage O(V+E). No speed claim. Allocation failure is outside typed errors. Resource caps precede vertex/edge validation. No callbacks. Exact categorical output has no justified percentage surrogate; executable approximation profiles unsupported.

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