Strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives

A well-studied nonlinear extension of the minimum-cost flow problem is to minimize the objective $\sum_{ij\in E} C_{ij}(f_{ij})$ over feasible flows $f$, where on every arc $ij$ of the network, $C_{ij}$ is a convex function. We give a strongly polynomial algorithm for the case when all $C_{ij}$’s are convex quadratic functions, settling an open problem raised e.g. by Hochbaum [1994]. We also give strongly polynomial algorithms for computing market equilibria in Fisher markets with linear utilities and with spending constraint utilities, that can be formulated in this framework (see Shmyrev [2009], Devanur et al. [2011]). For the latter class this resolves an open question raised by Vazirani [2010]. The running time is $O(m^4\log m)$ for quadratic costs, $O(n^4+n^2(m+n\log n)\log n)$ for Fisher’s markets with linear utilities and $O(mn^3 +m^2(m+n\log n)\log m)$ for spending constraint utilities. All these algorithms are presented in a common framework that addresses the general problem setting. Whereas it is impossible to give a strongly polynomial algorithm for the general problem even in an approximate sense (see Hochbaum [1994]), we show that assuming the existence of certain black-box oracles, one can give an algorithm using a strongly polynomial number of arithmetic operations and oracle calls only. The particular algorithms can be derived by implementing these oracles in the respective settings.

💡 Research Summary

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The paper tackles a long‑standing open problem in network flow optimization: designing a strongly polynomial algorithm for minimum‑cost flow problems with separable convex objective functions. While Hochbaum showed that for general convex costs (even for cubic polynomials) no strongly polynomial ε‑approximation algorithm exists, the authors focus on the special case where each arc cost is a convex quadratic function, i.e., C₍ij₎(α)=c₍ij₎α²+d₍ij₎α with c₍ij₎≥0. In this setting the optimal solution is unique and rational, which makes exact computation feasible.

The authors build on two classic ideas. First, the Edmonds‑Karp scaling framework, which repeatedly solves a series of Δ‑phases where a fixed amount Δ of flow is sent along shortest‑cost augmenting paths in a residual network. Second, Orlin’s strongly polynomial algorithm for linear minimum‑cost flow, which repeatedly identifies “large‑flow” arcs (those carrying more than 4nΔ units) and contracts them, thereby reducing the problem size.

To extend these ideas to the quadratic case, the paper introduces several new technical notions:

1. Oracle model – Each cost function is accessed through a black‑box that returns its value and derivative in constant time. Two additional oracles are assumed: (i) a “Trial” oracle that, given a set F of arcs and a modified demand vector (\hat b), solves the KKT system for the subproblem where arcs in F are allowed to violate non‑negativity; (ii) an “Adjust” routine that modifies the current flow to maintain feasibility when Δ is halved.

2. (Δ,F)-residual graph – For a scaling parameter Δ and a set F of “revealed” arcs, this graph contains all arcs that can be increased by Δ while preserving the property that arcs outside F remain non‑negative. The algorithm maintains the invariant that the (Δ,F)-residual graph has no negative cycles with respect to the linearized cost (C’{ij}(f{ij}+Δ)).

3. F‑pseudo‑flow – A flow that respects non‑negativity on all arcs except those in F, which are temporarily unrestricted. This relaxation allows the algorithm to treat revealed arcs as if they were already part of the optimal support.

The algorithm proceeds as follows. It starts with Δ large enough that the zero flow is (Δ,∅)-feasible. In each Δ‑phase it repeatedly sends Δ units of flow along shortest paths in the (Δ,F)-residual graph, updating the flow while preserving the feasibility invariant. When no more Δ‑augmenting paths exist, Δ is halved and the Adjust routine is invoked, which perturbs the flow by at most mΔ/2 to regain feasibility for the new Δ.

A crucial analytical insight is that after each Δ‑phase, any arc whose flow exceeds ((2n+m+1)Δ) must belong to the optimal support F*. This is a quadratic analogue of Orlin’s 4nΔ bound. Consequently, every O(log n) phases a new arc can be added to F, and after O(m log n) phases we have F=F*. At that point the Trial oracle solves the KKT linear system in O(n²) time, yielding the exact optimal flow f* and the corresponding dual potentials π. A final max‑flow computation converts the dual solution into a primal optimal solution.

Beyond the pure quadratic flow problem, the authors show how two important market‑equilibrium models fit into the same framework. The Fisher market with linear utilities can be reformulated (via Shmyrev’s transformation) as a separable convex flow problem where each arc cost is of the form α log α − α, which is handled by the same oracles. For Fisher markets with spending‑constraint utilities, each buyer‑good pair contributes a piecewise linear‑quadratic cost; again the oracle implementation satisfies the required properties. Thus the algorithm computes market equilibria in strongly polynomial time:

• Quadratic minimum‑cost flow: O(m⁴ log m)
• Linear‑utility Fisher market: O(n⁴ + n²(m + n log n) log n)
• Spending‑constraint Fisher market: O(m n³ + m²(m + n log n) log m)

These bounds improve upon previous polynomial‑time combinatorial algorithms and, for the spending‑constraint case, answer an open question posed by Vazirani. The paper also clarifies why the quadratic case is essentially the only non‑linear class admitting a strongly polynomial algorithm: higher‑degree convex polynomials lead to irrational optima and, by Hochbaum’s hardness result, preclude any strongly polynomial approach.

In summary, the work delivers a unified, oracle‑based framework that yields the first strongly polynomial algorithms for separable convex quadratic flow problems and for two important classes of Fisher market equilibria, bridging a gap between linear flow theory and convex optimization while respecting the stringent requirements of the strong polynomial model.