Maximum Flow Without the Outer IPM

Abstract: We show that the balancing weights technique of Li (2026) actually produces an approximate pseudo-circulation of a directed, capacitated graph in $m<sup>{1+o(1)}$ time. Together with standard flow techniques, we obtain an $m<sup>{1+o(1)}$ time maximum flow algorithm that avoids the interior-point method framework of recent almost-linear time algorithms (Chen et al. FOCS 2022, van den Brand et al. FOCS 2024).

• The paper shows that balancing weights combined with a dynamic min-ratio cut data structure can deterministically compute a 2-pseudo-circulation with value at least F/6 in m^{1+o(1)} time.
• The method constructs flow from vertex potentials, uses anchor arcs and strongly connected preprocessing to prove the value guarantee, and controls approximation through stability bounds and an energy-decrease argument.
• Standard reductions convert the pseudo-circulation into an exact maximum flow in m^{1+o(1)} time, although current versions incur constant-factor losses, require polynomially bounded integral capacities, and use randomized balanced-graph routing.

Context and contribution

The almost-linear time maximum flow algorithms of Chen et al. and van den Brand et al. share a common architecture: an outer interior-point method (IPM) reduces max-flow to $m\,\mathrm{polylog}\,n$ specialized subproblems, each solved by a dynamic data structure with $n^{o(1)}$ amortized update time (2608.17384). This paper demonstrates that this outer IPM loop is not necessary. The authors show that the balancing weights technique of Li (2026), when combined with the dynamic min-ratio cut data structure from van den Brand et al., directly produces an approximate pseudo-circulation in $m^{1+o(1)}$ time, which can then be converted to an exact maximum flow by standard techniques.

The central object is the $\beta$-pseudo-circulation: a flow $f$ satisfying $f(\partial^+S)\le\beta\cdot f(\partial^-S)$ for all vertex subsets $S$ (the condition is two-sided by complementing $S$). The main theorem is:

Given a capacitated digraph with integral, polynomially bounded capacities (except an infinite-capacity arc $(t,s)$) whose max-flow value is $F$, there is a deterministic $n^{o(1)}$0 time algorithm computing a 2-pseudo-circulation with $n^{o(1)}$1–$n^{o(1)}$2 value at least $n^{o(1)}$3.

The result is deterministic, matching the determinism of the underlying min-ratio cut data structure — a notable property given that several components of the surrounding literature are randomized.

From pseudo-circulation to exact flow

The reduction from the main theorem to max-flow is elementary but effective. A 2-pseudo-circulation induces a capacitated subgraph $n^{o(1)}$4 that is 2-balanced, meaning every cut satisfies $n^{o(1)}$5. By the max-flow/min-cut theorem, since $n^{o(1)}$6 lies in $n^{o(1)}$7 for any $n^{o(1)}$8–$n^{o(1)}$9 cut and carries at least $m^{1+o(1)}$0 units, every cut in $m^{1+o(1)}$1 has capacity at least $m^{1+o(1)}$2. Hence $m^{1+o(1)}$3 supports an $m^{1+o(1)}$4–$m^{1+o(1)}$5 flow of value at least $m^{1+o(1)}$6.

The algorithm then invokes the known result that $m^{1+o(1)}$7-balanced directed graphs admit approximate max-flow in randomized $m^{1+o(1)}$8 time (2608.17384). With $m^{1+o(1)}$9, this yields a flow of value at least $\beta$0 on the original graph. Rounding to integrality, updating the residual graph, and iterating $\beta$1 times gives an exact maximum flow; integral capacities in $\beta$2 add a $\beta$3 factor. Thus the entire pipeline is $\beta$4, with constant-factor approximation losses absorbed into logarithmically many iterations.

The potential-based flow and why it approximates

The pseudo-circulation is induced by a potential function $\beta$5 via

$\beta$6

Arcs with $\beta$7 send full capacity; otherwise flow decays inversely with the potential gap plus a capacity-dependent offset. The authors frame this as a continuous analogue of push-relabel, with the advantage that the flow is fully determined by the potential function — though it is neither a proper flow nor even a preflow, hence the target is a pseudo-circulation rather than a feasible flow.

A preprocessing step is essential for the value guarantee. The graph is reduced to the strongly connected component containing both $\beta$8 and $\beta$9 (if none exists, the answer is trivially zero), and the infinite-capacity arc $f$0 is replaced by $f$1 parallel "anchor" arcs of capacity $f$2 each, where $f$3 bounds all finite capacities. The key lemma shows any potential inducing a 2-pseudo-circulation on this preprocessed graph has anchor flow at least $f$4. The proof integrates $f$5 over level sets $f$6 between $f$7 and $f$8. Each such level set is an $f$9–$f(\partial^+S)\le\beta\cdot f(\partial^-S)$0 cut whose outgoing arcs are all "downhill" and thus saturated, giving $f(\partial^+S)\le\beta\cdot f(\partial^-S)$1 and hence $f(\partial^+S)\le\beta\cdot f(\partial^-S)$2. Anchors contribute at least $f(\partial^+S)\le\beta\cdot f(\partial^-S)$3 to the integral while all $f(\partial^+S)\le\beta\cdot f(\partial^-S)$4 non-anchor arcs contribute at most $f(\partial^+S)\le\beta\cdot f(\partial^-S)$5 each, so anchors carry at least one third of the integral, yielding $f(\partial^+S)\le\beta\cdot f(\partial^-S)$6–$f(\partial^+S)\le\beta\cdot f(\partial^-S)$7 value $f(\partial^+S)\le\beta\cdot f(\partial^-S)$8. The argument relies on the preprocessing assumption that anchors constitute exactly half the arc set; without it, the accounting fails.

Computing the pseudo-circulation via min-ratio cuts

The construction generalizes Section 3 of Li's balancing weights framework to capacitated graphs. Arc directions are encoded as a gradient $f(\partial^+S)\le\beta\cdot f(\partial^-S)$9 on the underlying undirected graph, so that $S$0. A minimum ratio cut with ratio $S$1 implies $S$2 for all $S$3 — i.e., balance of the weight function.

The algorithm maintains van den Brand et al.'s $S$4-approximate dynamic min-ratio cut data structure ($S$5, amortized $S$6 per operation). While the maintained implicit cut has ratio $S$7, the algorithm calls ToggleCut with step size $S$8; when the ratio exceeds $S$9, the weights are balanced up to slack. Edge weights are refreshed only when returned by the data structure's change-detection mechanism with threshold $S$0.

Two lemmas control the resulting error. A stability lemma shows that if two potential gaps differ by at most $S$1, the corresponding weights differ by a factor within $S$2; combined with the detection threshold, this gives $S$3 for every arc. Consequently, once ToggleCut calls cease, $S$4 is a $S$5-pseudo-circulation; running with rescaled $S$6 recovers the stated $S$7 guarantee.

Running time analysis

The analysis tracks an energy functional per arc,

$S$8

where $S$9 is the maximum potential gap over the run. Each ToggleCut call decreases total energy by $(t,s)$0: the implicit cut satisfies $(t,s)$1, and after translating through the weight-approximation factors, the net energy change is bounded by $(t,s)$2 using the fact that $(t,s)$3 (otherwise the data structure's approximation guarantee would be violated).

Since initial total energy is at most $(t,s)$4, the number of ToggleCut calls is $(t,s)$5, and the data structure's output-sensitivity bound ($(t,s)$6 edges returned) yields total time $(t,s)$7. Bounding $(t,s)$8: if some potential gap reached super-polynomial magnitude, strong connectivity would force an intermediate arc with large gap and hence large energy, contradicting the monotone energy budget. With $(t,s)$9, both $F$0 and $F$1 contribute only $F$2 factors.

Limitations and open questions

Several caveats qualify the result. The constants are weak: the pipeline loses factors of 6, 12, and 24 across the pseudo-circulation, balanced-subgraph, and half-approximation stages, so roughly $F$3 iterations are needed even for moderate accuracy. The conversion step uses a randomized balanced-graph routing algorithm, so the end-to-end max-flow algorithm inherits randomization despite the deterministic pseudo-circulation subroutine. The analysis requires integral, polynomially bounded capacities, and the anchor construction depends on the specific preprocessing that fixes the number of anchors at $F$4. Finally, the running-time exponents hide $F$5 terms inherited from the underlying data structure, so the practical gap to $F$6 remains unquantified. An open question is whether the balanced-graph flow step itself can be made deterministic within the same time bound, which would yield a fully deterministic $F$7 max-flow algorithm without the outer IPM.

Conclusion

This paper establishes that the balancing weights method, executed against the dynamic min-ratio cut data structure of recent IPM-based algorithms, suffices to compute a constant-approximate pseudo-circulation in $F$8 deterministic time, and that standard reductions lift this to an exact maximum flow. The contribution is architectural: it removes the outer IPM loop from the almost-linear time max-flow pipeline while retaining its inner data structure, replacing the IPM's role with a direct energy-potential argument reminiscent of continuous push-relabel.