Kantor — Optimal-Transport (Wasserstein) Oracle 🪨

Kantor sells the proof, not just the plan. Given a source distribution a, a sink distribution b and a ground cost C (supplied directly, or computed from point coordinates for the p-Wasserstein W_p), it solves the discrete Kantorovich optimal-transport problem exactly and returns the optimal transport plan P, the cost, and the Kantorovich dual potentials (u, v). The plan and potentials together form a two-sided certificate any client can check in O(m·n) to confirm the cost is globally optimal, without re-solving and without trusting the oracle — the plan bounds the optimum from above, the potentials from below, and they meet at the cost.

This is the optimal-transport analogue of Fermat's least-time dual certificate: there the eikonal potential T(v) witnesses shortest-path optimality; here the Kantorovich potentials (u, v) witness transport optimality.

minimise  Σ_ij P_ij·C_ij     subject to  row sums(P) = a,  col sums(P) = b,  P ≥ 0
dual:     maximise  Σ_i a_i·u_i + Σ_j b_j·v_j     s.t.  u_i + v_j ≤ C_ij  for all i,j

Exact, by min-cost flow (no Sinkhorn, no scipy)

The transport LP is a minimum-cost-flow problem on a bipartite network, solved exactly in pure Python by successive shortest paths (one Bellman-Ford to seed node potentials, then Dijkstra on reduced costs). The integer node potentials produced by the flow are the LP duals — so the Kantorovich certificate falls out of the solver for free. Mass is quantised to integer supplies/demands at a common scale Q with largest-remainder rounding so the marginals match exactly.

An explicitly approximate entropic Sinkhorn path is also offered (method:"sinkhorn", regulariser eps); its objective is an upper bound on the true optimum and it is returned as method:"sinkhorn-approx" — never passed off as exact.

# Exact transport: the 2x2 "swap" case, optimal cost 0.2
curl -s -X POST http://localhost:9314/ai-market/v2/invoke -H "Content-Type: application/json" \
  -d '{"capability_id":"kantor.transport@v1","input":{"a":[0.6,0.4],"b":[0.4,0.6],"cost":[[0,1],[1,0]]}}'

# p-Wasserstein from points (default squared-Euclidean via euclidean, p=2): W_2 = 0.4
curl -s -X POST http://localhost:9314/ai-market/v2/invoke -H "Content-Type: application/json" \
  -d '{"capability_id":"kantor.transport@v1","input":{"a":[0.5,0.5],"b":[0.5,0.5],"source_points":[[0],[1]],"sink_points":[[0.4],[0.6]]}}'

Verifiable by duality, not asserted — but both sides are required

A dual certificate alone is not enough. By LP weak duality any dual-feasible (u,v) gives Σ a_i·u_i + Σ b_j·v_j ≤ optimal_cost — a lower bound only. So a dual-only check (feasibility + claimed_cost = dual_objective) would accept any claimed_cost equal to a feasible-but-suboptimal dual value, and could not catch an under-reported cost: with u = v = 0 the dual is feasible whenever C ≥ 0 and its objective is 0, so a dishonest oracle could otherwise "prove" a cost of 0 (or negative).

kantor.verify therefore requires both the plan P and the potentials (u,v), and checks, in one O(m·n) sweep:

• primal feasibility P ≥ 0, row-sums(P) = a, col-sums(P) = b, and ⟨P,C⟩ = claimed_cost — a feasible plan, so its cost is an upper bound; and
• dual feasibility u_i + v_j ≤ C_ij on every (i,j) pair (no shortcut exists), with strong duality claimed_cost = Σ_i a_i·u_i + Σ_j b_j·v_j — a lower bound.

Upper bound = lower bound = claimed_cost pins the exact optimum (and a transport cost is ≥ 0, so a negative claim is rejected outright). Returns {valid, feasible, strong_duality, primal_feasible, primal_cost_matches, dual_objective, primal_cost, claimed_cost, max_violation, …} — no re-solve, no trust in the oracle.

Built on oracle-core (AIMarket Protocol v2). Part of the AICOM oracle family. MIT.