Maximum Flow in Networks

• Maximum flow problems concern the transmission of commodities through a capacitated network, with applications in communications networks, logistics, and fluid dynamics, among others.
• Implementation methods include push-relabel, scaling, electrical flows, and interior-point methods, with exact and approximate formulations for diverse graph structures.
• The maximum-flow minimum-cut theorem underpins the validation and optimality of algorithms that estimate or maximize commodity transmission through a network.

Maximum flow is a fundamental problem in combinatorial optimization that asks for the greatest amount of a commodity that can be transmitted from a source $s$ to a sink $t$ through a capacitated network. A feasible flow respects arc or edge capacities and satisfies flow conservation at every intermediate vertex. In its classical scalar form, the maximum-flow/minimum-cut theorem states that the maximum flow value equals the capacity of a minimum $s$–$t$ cut. The problem supports exact, approximate, incremental, dynamic, distributed, GPU, learning-augmented, planar, and multicommodity formulations, with recent research ranging from specialized combinatorial algorithms to almost-linear-time algorithms based on convex optimization and dynamic graph data structures.

1. Mathematical formulation and duality

Let $G=(V,E)$ be a directed network with source $s$, sink $t$, and nonnegative capacity $c(e)$ on every arc. A flow is a function $f:E\to\mathbb{R}$ satisfying

$0\leq f(e)\leq c(e)$

and

$t$0

Its value is the amount entering the sink, equivalently the net amount leaving the source:

$t$1

The maximum-flow problem is therefore

$t$2

subject to capacity and conservation constraints. A more general representation permits forward and backward capacities $t$3 and $t$4, with

$t$5

This notation is convenient for residual networks and for representing undirected edges as oppositely usable directed arcs (Madry, 2016).

For an $t$6–$t$7 cut $t$8, where $t$9 and $s$0, the cut capacity is

$s$1

The max-flow/min-cut theorem gives

$s$2

Thus a flow is optimal precisely when its value equals the capacity of a cut. The theorem also provides a certificate of optimality: after no augmenting path remains in the residual graph, the vertices reachable from $s$3 form the source side of a minimum cut.

Residual networks

Given a feasible flow, the residual capacity of an arc includes both unused forward capacity and the possibility of canceling reverse flow. In a standard directed representation,

$s$4

while the reverse residual capacity increases by the amount of flow that can be canceled. With paired reverse arcs, this is commonly written as

$s$5

An augmenting path is an $s$6–$s$7 path consisting of positive-residual-capacity arcs. Augmenting along its bottleneck preserves feasibility and increases the flow value. If no such path exists, the current flow is maximum.

Integral capacities imply the existence of an integral maximum flow. In a unit-capacity network, the maximum-flow value equals the maximum number of directed edge-disjoint $s$8–$s$9 paths. For undirected unit-capacity networks, the same interpretation applies after choosing orientations for the flow on individual edges.

2. Classical algorithms and algorithmic paradigms

Maximum-flow algorithms are commonly organized around augmenting paths, blocking flows, preflows, scaling, electrical flows, interior-point methods, and residual-network maintenance.

Augmenting paths and blocking flows

Ford–Fulkerson repeatedly augments along residual $t$0–$t$1 paths. With integral capacities, each augmentation increases the value by at least one, but the running time depends on the maximum-flow value. Dinitz’s algorithm, also called Dinic’s algorithm, constructs a layered residual network using BFS distances from $t$2 and repeatedly computes a blocking flow. After every blocking-flow phase, the residual distance from $t$3 to $t$4 strictly increases.

The classical bound quoted for Dinitz is

$t$5

with the tighter unweighted bound

$t$6

due to Even and Karzanov. For unit-capacity graphs, the classical combinatorial bound is

$t$7

due to Karzanov and independently Even–Tarjan (Bernstein et al., 2024).

Push–relabel and preflows

Push–relabel algorithms maintain a preflow rather than a feasible flow. A preflow may have positive excess at intermediate vertices. Each vertex has a height label, and excess is pushed along admissible residual arcs. If no admissible arc exists, the vertex is relabeled.

The standard height invariant is

$t$8

At termination, all internal excesses vanish and the residual height partition identifies a minimum cut. Generic push–relabel has the worst-case bound

$t$9

A strongly polynomial enhanced excess-scaling algorithm, called Enhanced LMES, removes dependence on the numerical capacity magnitude by combining small, medium, and large arc classifications, abundant-arc contractions, special and violating vertices, and a flow-return forest. Its parameterized running time is

$G=(V,E)$0

and the paper states the optimized bound

$G=(V,E)$1

for an appropriate power-of-two choice of $G=(V,E)$2 (Orlin et al., 2019).

Electrical and interior-point methods

Electrical flows minimize a quadratic energy

$G=(V,E)$3

subject to flow-demand constraints. Their computation reduces to solving a weighted Laplacian system. Electrical flows became central to fast approximate and exact maximum-flow algorithms because Laplacian systems admit nearly-linear-time solvers.

The algorithm in “Computing Maximum Flow with Augmenting Electrical Flows” embeds electrical-flow computation directly into a primal-dual augmenting-flow framework. It maintains a feasible flow and a dual vector whose residual-capacity potentials are coupled to edge stretches. Resistance boosting controls high-energy arcs and yields an exact running time

$G=(V,E)$4

where $G=(V,E)$5 is the largest integral capacity (Madry, 2016).

A later interior-point approach replaces quadratic energy maximization by maximization of the full Bregman divergence associated with a weighted logarithmic barrier. The resulting method computes exact maximum flow in

$G=(V,E)$6

time and avoids the $G=(V,E)$7-norm dependence arising in earlier electrical-flow frameworks (Liu et al., 2020).

Other work obtains an almost-linear-time randomized algorithm for exact maximum flow by reducing maximum flow to minimum-cost circulation and maintaining approximate minimum-ratio cycles dynamically. The resulting bound is

$G=(V,E)$8

with high probability (Chen et al., 2022).

3. Planar, separable, and structured networks

Graph structure can substantially improve maximum-flow algorithms. Planarity permits duality-based reductions, while separators, treewidth, tree depth, and minor structure support sparsification and divide-and-conquer.

Planar graphs with vertex capacities

In a planar graph with both arc capacities and internal vertex capacities, the ordinary vertex-splitting reduction preserves the flow value but may destroy planarity. A planar $G=(V,E)$9 with one finite-capacity vertex becomes an underlying graph equivalent to $s$0 after standard splitting, which is nonplanar.

A planarity-preserving reduction replaces a finite-capacity vertex $s$1 of degree

$s$2

by a cycle $s$3 of $s$4 vertices. Each cycle edge is undirected and has capacity $s$5, while incident arcs are attached in their original cyclic order. Any flow crossing from the incoming side to the outgoing side must cross two cycle edges, whose combined capacity is $s$6. The construction takes $s$7 time and preserves the maximum-flow value.

Combining this reduction with directed planar maximum-flow algorithms gives an

$s$8

algorithm in the general planar case. If $s$9 and $t$0 lie on a common face, the graph remains $t$1-planar and the running time becomes

$t$2

(0905.0451).

Separable undirected graphs

Separator-based algorithms obtain $t$3-approximate maximum flow in undirected graphs with recursive separator structures. Their central intermediate problem is grouped $t$4 flow, which aggregates quadratic congestion constraints over groups of edges. Spectral vertex sparsifiers approximately eliminate separator interiors while preserving electrical energies on boundary vertices.

For graphs with $t$5 edges and suitable recursive $t$6-separator structures, the principal bound is

$t$7

For sparse graph families with $t$8, this becomes

$t$9

The framework applies to bounded-genus, minor-free, shallow-minor-free, geometric, and image graphs. If a piece of size $c(e)$0 has boundary size $c(e)$1, the stated savings persist while

$c(e)$2

Thus the improvement weakens for three-dimensional image graphs, whose cube boundaries have order $c(e)$3 (Miller et al., 2012).

Structured strongly polynomial algorithms

A recent framework strengthens Orlin’s approach by reducing arbitrary capacities to polynomially bounded approximate-flow instances while maintaining incremental transitive covers of abundant residual arcs. It replaces full transitive closure by size-bounded transitive covers and postpones rerouting of transitive arcs until the end.

For graphs with few capacitated arcs or nodes, the resulting randomized running time is

$c(e)$4

where $c(e)$5 is the number of nodes plus positive finite-capacity arcs and $c(e)$6 is the matrix-multiplication exponent. When $c(e)$7, this yields

$c(e)$8

algorithms for maximum node-capacitated flow, bipartite $c(e)$9-matching, and maximum closure. For graphs supplied with a tree decomposition of width $f:E\to\mathbb{R}$0, the paper obtains an

$f:E\to\mathbb{R}$1

algorithm up to polylogarithmic factors (Dadush et al., 23 Oct 2025).

4. Approximation, applications, and generalized flow

Maximum flow is used both as an optimization objective and as a primitive inside other algorithms.

Grouped-flow and submodular optimization

Maximum flow can serve as a black-box oracle for decomposable submodular minimization. A submodular function

$f:E\to\mathbb{R}$2

with each component supported on a small set can be optimized through convex optimization over a submodular base polytope. The base-polytope problem is approximated by graph cuts, and a parametric minimum-cut reduction is solved through maximum-flow calls.

When every component acts on $f:E\to\mathbb{R}$3 elements, the total running time is, up to polylogarithmic factors, essentially that of maximum flow in a sparse graph with $f:E\to\mathbb{R}$4 vertices and polynomial integral capacities (Axiotis et al., 2021).

Connectome analysis

Maximum flow has also been proposed as a structural measure for directed connectome graphs. With unit edge capacities, $f:E\to\mathbb{R}$5 equals the maximum number of directed edge-disjoint paths from source $f:E\to\mathbb{R}$6 to target $f:E\to\mathbb{R}$7. The network-wide maximum is

$f:E\to\mathbb{R}$8

The associated minimum-cut interpretation measures directional bottlenecks and redundant parallel connectivity. Experiments on cat, worm, macaque, rat, mouse, fly, and human connectomes found that network-wide maximum flow was highly concentrated in a small number of ordered pairs. For example, the cat graph had maximum flow $f:E\to\mathbb{R}$9 and average flow approximately $0\leq f(e)\leq c(e)$0, while the analyzed human graph had maximum flow approximately $0\leq f(e)\leq c(e)$1 and average flow less than $0\leq f(e)\leq c(e)$2.

The model is deliberately limited: all edges have unit capacity, multiple sources and sinks are not considered, and no biological interpretation of an extremal flow pair is established (Daugulis, 2014).

Multicommodity extensions

In a multicommodity network, each arc has a capacity region

$0\leq f(e)\leq c(e)$3

rather than a scalar capacity. The generalized maximum-flow object is the feasible flow-value region

$0\leq f(e)\leq c(e)$4

Because vectors lack a canonical total ordering, there is generally no single scalar “maximum” flow. Ordinary intersections of cut-capacity regions are insufficient because different cuts may assign incompatible values to shared arcs.

The paper defines mutual capacity as the set of flow values supported by mutually compatible local flows on all cuts. Its generalized max-flow/min-cut theorem states

$0\leq f(e)\leq c(e)$5

where $0\leq f(e)\leq c(e)$6 is the mutual capacity of all $0\leq f(e)\leq c(e)$7–$0\leq f(e)\leq c(e)$8 cuts. Compatibility preserves shared-arc assignments, allowing local cut flows to be glued into global flows (Broussard et al., 2021).

For a prescribed commodity-ratio vector $0\leq f(e)\leq c(e)$9, the ratio maximum-flow problem seeks

$t$00

An augmenting-cycle formulation reduces feasibility to constraints in the cycle space of dimension $t$01, rather than the direct arc-space formulation.

5. Dynamic, incremental, and learning-augmented maximum flow

Dynamic maximum flow studies networks whose topology or capacities change over time. The principal challenge is to repair an existing flow rather than recompute one from an empty state.

Incremental network design

Incremental network design differs from incremental flow maintenance. In each period, at most one potential arc is built, and a maximum flow is computed after previously constructed arcs become available. The objective is cumulative:

$t$02

The problem is strongly NP-hard, with a reduction from Exact Cover by 3-Sets even when existing arcs have capacity $t$03 and potential arcs have capacity $t$04. Two mixed-integer formulations are studied. The period-indexed formulation models every arc-period pair, whereas the flow-level formulation uses the difference

$t$05

between ultimate and initial maximum-flow values. The latter reduces symmetry and is generally more effective computationally (Kalinowski et al., 2013).

Three heuristics are considered: Quickest-increment, Quickest-to-ultimate, and Quickest-to-target. For unit-capacity networks, Quickest-to-ultimate has a $t$06-approximation guarantee and Quickest-increment has a $t$07-approximation guarantee. In incremental maximum matching, Quickest-to-ultimate has a $t$08-approximation guarantee.

Learning-augmented computation

A flow-conserving prediction can accelerate exact maximum-flow computation even when it violates the capacities of the current instance. If $t$09 is a prediction and $t$10 is a maximum flow under the revealed capacities, define

$t$11

The prediction is repaired by canceling flow on cycles or $t$12–$t$13 paths until all capacity constraints hold, after which Ford–Fulkerson augments the repaired flow. The resulting exact maximum flow is computed in

$t$14

time. Running the prediction-based algorithm in parallel with a worst-case algorithm yields the robust bound

$t$15

The same work gives a PAC-learning procedure for finding a flow-conserving prediction with near-optimal expected $t$16 error from sampled capacity vectors (Polak et al., 2022).

Dynamic graph processing

A distributed asynchronous push–relabel algorithm supports vertex and edge additions, deletions, and capacity changes. It maintains residual capacities, excesses, and height labels in a vertex-centric shared-nothing model. Capacity decreases can create negative excess; a second height system and deficient-vertex repair route flow toward vertices requiring incoming flow.

The algorithm is designed for high-rate updates but does not provide a strong worst-case convergence bound under adversarial update streams. Its empirical objective includes throughput, result latency, and solution stability. The method processes evolving graphs with hundreds of millions of edges and reports rates extending to millions of changes per second in favorable cases (Luo et al., 2023).

A separate incremental algorithm maintains a $t$17-approximate flow in undirected unit-capacity graphs under edge insertions. Its total update time is

$t$18

where $t$19 is the final maximum-flow value. It uses residual graph sparsification, Nagamochi–Ibaraki indices, and incremental reachability. The algorithm achieves polylogarithmic amortized update time in dense graphs and more generally when

$t$20

The theorem is restricted to incremental, undirected, uncapacitated, approximate flow (Goranci et al., 13 Feb 2025).

6. Recent directions: dense, distributed, and outer-IPM-free algorithms

Recent work has pursued both specialized combinatorial algorithms and reductions that remove major layers of continuous optimization.

Dense combinatorial maximum flow

A randomized combinatorial algorithm computes exact maximum flow in

$t$21

time with high probability. It uses weighted push–relabel guided by a directed expander hierarchy. Edge weights induce a weighted residual distance, and the algorithm lists useful augmenting paths without continuous optimization or heavy dynamic graph data structures.

For unit-capacity graphs, the method improves over the classical

$t$22

bound when

$t$23

The approach is particularly relevant to dense graphs, where its running time is essentially quadratic in $t$24 and nearly independent of $t$25 (Bernstein et al., 2024).

Distributed planar maximum flow

In the distributed $t$26 model, undirected weighted planar graphs admit a deterministic

$t$27

round algorithm for a $t$28-approximate maximum $t$29–$t$30 flow value, equivalently a $t$31-approximate minimum cut. Here $t$32 is the hop-diameter.

The algorithm implements Reif’s planar minimum-cut method through recursive incisions in the planar dual. A shortest separating dual cycle becomes a shortest path after cutting the dual open along an initial path. Approximate shortest-path computations are performed simultaneously over disjoint annuli using virtual graphs simulated on the primal network.

The algorithm outputs the approximate cut value, cut bipartition, and crossing edges, but does not construct an edge-by-edge approximate flow within the same round bound (Abd-Elhaleem et al., 10 Aug 2026).

Removing the outer interior-point method

A further development constructs an approximate pseudo-circulation directly rather than following an outer interior-point trajectory. A $t$33-pseudo-circulation satisfies

$t$34

The algorithm constructs a $t$35-pseudo-circulation of value at least $t$36 in deterministic

$t$37

time for polynomially bounded integral capacities. It uses potential-induced arc flows,

$t$38

and balances directed cuts through a dynamic approximate minimum-ratio-cut data structure.

The resulting pseudo-circulation yields a constant-factor approximate maximum flow. Repeated residual-network iteration then produces an exact maximum flow in $t$39 time up to polylogarithmic factors and capacity-scaling dependence. The contribution removes the outer IPM loop, although it continues to use dynamic min-ratio-cut technology originally developed for IPM-based algorithms (Li et al., 18 Aug 2026).

Maximum-flow research therefore spans a spectrum of methods. Classical augmenting paths and push–relabel remain important for implementation and dynamic repair; planar duality and separators exploit graph structure; electrical flows, Bregman divergences, and minimum-ratio cycles support fast algebraic algorithms; learning and sparsification reduce repeated-instance costs; and distributed and GPU implementations address scale and hardware parallelism. Across these approaches, the residual network and the max-flow/min-cut theorem remain the central structural principles: feasible flow is increased through residual augmentations, while optimality is certified by the absence of augmenting paths or by a saturated minimum cut.