Multiflow-Multicut Gap Overview

• Multiflow-multicut gap is defined as the ratio between an optimal multicut and its corresponding multiflow relaxation, reflecting the integrality gap in multicommodity flow formulations.
• Graph topology, terminal placement, and capacity models actively influence the gap through various LP relaxations and dual interpretations.
• Advanced techniques such as metric embeddings, region growing, and separator recursion underpin the derivation of approximation bounds in planar, directed, and bounded-width networks.

The multiflow-multicut gap is the ratio between an optimal multicut and the corresponding optimal multiflow relaxation, or, in concurrent-flow language, between sparsest cut and maximum concurrent flow. In LP terms, it is the integrality gap of the natural covering formulation for multicut or sparsest cut. The classical max-flow min-cut theorem gives gap $1$ for a single source-sink pair, and Hu’s theorem preserves gap $1$ for two pairs, but for general multicommodity instances the gap can be much larger; a major line of research asks how graph topology, terminal placement, directionality, and the capacity model control this ratio (Friedrich et al., 2022, Kalantarzadeh et al., 9 Jul 2025, Chekuri et al., 2010).

1. Formulations and dual interpretations

A standard setting uses a supply graph $G=(V,E)$ with capacities and a demand graph $H=(V,F)$ on the same vertex set. A multiflow assigns nonnegative weight to supply paths realizing the demand pairs, while a multicut is a set of supply edges or vertices whose removal disconnects every demanded pair. In the plane multiflow formulation, if $P$ denotes the set of all demand paths, then a feasible multiflow is a function $f:P\to \mathbb{R}_+$ satisfying edge-capacity constraints, and its value is $|f|=\sum_{P\in P} f(P)$; a multicut is a set $M\subseteq E$ intersecting every demand path, with capacity $c(M)=\sum_{e\in M} c(e)$. Every feasible multiflow and every multicut satisfy the basic inequality $|f|\le c(M)$ (Garg et al., 2020).

The natural LP relaxation for multicut is the path-covering LP

$1$0

and its dual is a maximum multicommodity flow LP (Chekuri et al., 2016). In the node-weighted planar-digraph setting, the analogous relaxation is

$1$1

with the dual corresponding to maximum throughput multicommodity flow; in that formulation, the integrality gap of the multicut LP is exactly the flow-cut gap (Chekuri et al., 27 Jan 2026).

For sparsest-cut formulations, one instead compares the maximum concurrent flow to the best cut sparsity. In planar edge-capacitated networks with terminal set $1$2, the worst-case ratio is

$1$3

and the same work identifies this gap with the terminal $1$4-distortion quantity $1$5 (Krauthgamer et al., 2018).

A parallel notion is the integer flow-cut gap. In the formulation of integer and fractional multiflows, the fractional gap asks for the minimum congestion $1$6 such that a cut-feasible instance is fractionally routable in $1$7, while the integer gap imposes integrality on the path variables themselves (Chekuri et al., 2010). This distinction is central in classes where fractional routing is well understood but integral routing is not.

2. Exact regimes and gap-one phenomena

Several graph classes admit exact max-flow/min-cut-type behavior. For one source-sink pair, the classical theorem gives gap $1$8, and Hu’s theorem extends this to two pairs (Friedrich et al., 2022, Kalantarzadeh et al., 9 Jul 2025). Beyond these cases, exactness is typically structural rather than universal.

For undirected series-parallel supply graphs, the decisive obstruction is the odd spindle. A pair $1$9 is cut-sufficient exactly when every capacity-demand assignment satisfying the cut condition is routable, and in this setting cut-sufficiency is equivalent to flow-cut gap $G=(V,E)$0. The complete characterization is that if $G=(V,E)$1 is series-parallel, then $G=(V,E)$2 is cut-sufficient iff it does not contain an odd spindle as a minor; here an odd spindle is a $G=(V,E)$3-spindle with $G=(V,E)$4 odd, where $G=(V,E)$5 and $G=(V,E)$6 consists of a cycle on the $G=(V,E)$7 degree-$G=(V,E)$8 vertices together with an edge joining the two hubs (Chakrabarti et al., 2012). In the Eulerian case, the same work proves an integral routing theorem and a polynomial-time algorithm.

Planar terminal placement can also force exactness. In the single-face edge-capacitated setting, the Okamura-Seymour theorem gives exact gap $G=(V,E)$9 (Krauthgamer et al., 2018). For planar graphs with three holes, Karzanov’s theorem identifies an exact solvability criterion: the instance is feasible iff the cut condition and the $H=(V,F)$0-metric condition both hold, and if the data are Eulerian then an integer solution exists (Babenko et al., 2014). The sharpened version reduces verification to regular cuts and regular $H=(V,F)$1-metrics, giving a precise topological obstruction theory for that regime.

Directed series-parallel graphs behave differently. The cut condition is necessary for feasibility of a fractional multiflow,

$H=(V,F)$2

but is not sufficient in general (Almoghrabi et al., 2024). Nevertheless, the same work establishes strong rounding and integrality statements for total arc loads: if demands are integers, the total arc-flow vector of any multiflow is a convex combination of total arc flows of integer multiflows with

$H=(V,F)$3

and every fractional multiflow can be expressed as a convex combination of unsplittable multiflows satisfying

$H=(V,F)$4

This is not a direct multicut theorem, but it is a strong integrality phenomenon on the flow side (Almoghrabi et al., 2024).

3. Undirected planar, fully planar, and face-local bounds

In planar undirected graphs, the gap depends strongly on how terminals occupy the embedding. If $H=(V,F)$5 is the minimum number of faces whose union contains all terminals, then the edge-capacitated flow-cut gap satisfies

$H=(V,F)$6

improving earlier $H=(V,F)$7 and $H=(V,F)$8 bounds, and this logarithmic dependence is tight up to constants (Krauthgamer et al., 2018). The same work also proves that for planar networks with polymatroid or vertex capacities and terminals on at most $H=(V,F)$9 faces,

$P$0

so the vertex-capacitated gap is $P$1 rather than $P$2 (Krauthgamer et al., 2018).

A different planar regime is the fully planar case, where the union $P$3 is planar. For the maximization version of edge-disjoint paths, there is a polynomial-time $P$4-approximation, and the same duality machinery yields an approximate max-multiflow min-multicut theorem: $P$5 The associated LP gaps are constant: the packing-side gap is at most $P$6, and the covering-side gap is at most $P$7 (Huang et al., 2020).

For plane multiflow maximization in the undirected setting, where again $P$8 is planar, there exists a feasible multiflow $P$9 and a multicut $f:P\to \mathbb{R}_+$0 such that

$f:P\to \mathbb{R}_+$1

with both computable in polynomial time. This gives a fractional flow-cut gap at most $f:P\to \mathbb{R}_+$2. The same work then rounds arbitrary fractional multiflows to half-integer multiflows with loss at most $f:P\to \mathbb{R}_+$3, and half-integer multiflows to integer multiflows with another loss at most $f:P\to \mathbb{R}_+$4, yielding the chain

$f:P\to \mathbb{R}_+$5

Accordingly, the half-integer flow-cut gap is at most $f:P\to \mathbb{R}_+$6, the integer flow-cut gap is at most $f:P\to \mathbb{R}_+$7, and the maximum integer plane multiflow admits a $f:P\to \mathbb{R}_+$8-approximation (Garg et al., 2020).

Half-integrality can itself be optimal in planar union instances. In Seymour instances, meaning $f:P\to \mathbb{R}_+$9 planar, modified WGMV primal-dual analysis yields a half-integral dual and therefore a feasible half-integral flow of value $|f|=\sum_{P\in P} f(P)$0 together with a multicut of value $|f|=\sum_{P\in P} f(P)$1 such that

$|f|=\sum_{P\in P} f(P)$2

This factor $|f|=\sum_{P\in P} f(P)$3 is tight, and converting the half-integral flow to an integral flow gives

$|f|=\sum_{P\in P} f(P)$4

for integral flow (Garg et al., 2020).

4. Directed planar graphs and planar digraphs

Directed planar graphs long resisted any analogue of the undirected planar theory. A major step is the result that the uniform multicommodity flow-cut gap on planar digraphs is $|f|=\sum_{P\in P} f(P)$5, while the non-uniform gap is $|f|=\sum_{P\in P} f(P)$6 (Kawarabayashi et al., 2021). The same work derives a polynomial-time $|f|=\sum_{P\in P} f(P)$7-approximation for uniform directed sparsest cut on planar digraphs, a polynomial-time $|f|=\sum_{P\in P} f(P)$8-approximation for general directed sparsest cut, and a polynomial-time $|f|=\sum_{P\in P} f(P)$9-approximation for directed multicut (Kawarabayashi et al., 2021).

The technical route is geometric. Planar digraph shortest-path quasimetrics admit $M\subseteq E$0-bounded, $M\subseteq E$1-Lipschitz random quasipartitions, samplable in polynomial time, and these imply an embedding into directed $M\subseteq E$2 with distortion $M\subseteq E$3 (Kawarabayashi et al., 2021). The construction uses shortest-path separators and layered decomposition in the spirit of planar separator theory.

Node weights require additional work. For planar digraph multicut, Kawarabayashi and Sidiropoulos obtained an $M\subseteq E$4-approximation in the edge-weighted setting, and the node-weighted extension proves a deterministic efficient $M\subseteq E$5-approximation for node-weighted multicut via the natural LP. Because the LP dual is maximum throughput multicommodity flow, this yields an $M\subseteq E$6 upper bound on the multicommodity flow-cut gap in node-weighted planar digraphs as well (Chekuri et al., 27 Jan 2026). The same LP framework and the standard reduction from multicut to nonuniform sparsest cut give a deterministic $M\subseteq E$7-approximation for node-weighted sparsest cut on planar digraphs (Chekuri et al., 27 Jan 2026).

Planarity is essential in this line of work. The node-weighted extension cannot be obtained by a black-box node-splitting reduction, because the standard transformation does not preserve planarity (Chekuri et al., 27 Jan 2026). The proofs rely on Thorup’s planar separator theorem, applied to shortest-path trees in planar digraphs, and on recursive cutting around three directed shortest paths. The deterministic algorithm replaces randomized ball growing and random shifts with a deterministic region-growing lemma of Garg-Vazirani-Yannakakis; this may lose a logarithmic factor in the layering step, but the final asymptotic guarantee remains $M\subseteq E$8 (Chekuri et al., 27 Jan 2026).

5. Bounded width, directed minor-free classes, and generalized capacities

Outside planarity, bounded-width structure still controls the gap. For a treewidth-$M\subseteq E$9 graph with edge capacities and source-sink pairs, there exists a polynomial-time computable multicommodity flow of value $c(M)=\sum_{e\in M} c(e)$0 and a multicut of value $c(M)=\sum_{e\in M} c(e)$1 such that

$c(M)=\sum_{e\in M} c(e)$2

The multiflow-multicut gap in treewidth-$c(M)=\sum_{e\in M} c(e)$3 graphs is therefore $c(M)=\sum_{e\in M} c(e)$4, and this is asymptotically tight (Friedrich et al., 2022). The constructive proof rounds the optimal fractional multicut solution by modified region growing; the logarithmic dependence comes from bounding how often an edge can be charged, with a crucial shadow multiplicity bound of $c(M)=\sum_{e\in M} c(e)$5 (Friedrich et al., 2022).

Directed minor-free analogues are more limited but still nontrivial. For uniform demands, the multi-commodity flow-cut gap is $c(M)=\sum_{e\in M} c(e)$6 on directed series-parallel graphs and on directed graphs of bounded pathwidth. For non-uniform demands, the gap is $c(M)=\sum_{e\in M} c(e)$7 on directed trees and directed cycles (Salmasi et al., 2017). These were presented as the first constant upper bounds of this type for nontrivial directed families, obtained through Lipschitz quasipartitions and embeddings into convex combinations of directed cut metrics (Salmasi et al., 2017).

Polymatroidal networks generalize edge- and node-capacitated models by imposing submodular capacity constraints on incident edges at each node. In undirected polymatroidal networks with $c(M)=\sum_{e\in M} c(e)$8 source-sink pairs, the maximum concurrent flow versus sparsest-cut gap is $c(M)=\sum_{e\in M} c(e)$9, and the maximum throughput flow versus multicut gap is also $|f|\le c(M)$0 (Chekuri et al., 2011). In directed polymatroidal networks with symmetric demands, the minimum multicut is at most

$|f|\le c(M)$1

when $|f|\le c(M)$2 is the maximum concurrent flow value, while the sparsest cut is at most

$|f|\le c(M)$3

The directed results rely on reducing the polymatroidal dual to a standard edge-capacitated directed instance, whereas the undirected results rely on Lovász extensions and line embeddings with low average distortion (Chekuri et al., 2011).

6. Lower bounds, obstructions, and hardness thresholds

The general gap can be very large. For directed graphs, Chuzhoy and Khanna proved a $|f|\le c(M)$4 lower bound even for directed acyclic graphs, a benchmark repeatedly used to contrast planar-digraph upper bounds (Kawarabayashi et al., 2021, Chekuri et al., 27 Jan 2026).

A classical family exhibiting large directed separation is the Saks et al. construction. Using network coding, the strong-product analysis of $|f|\le c(M)$5 shows that

$|f|\le c(M)$6

and that the optimal network coding rate is exactly the same: $|f|\le c(M)$7 For this family, the coding rate and the multicut are a factor $|f|\le c(M)$8 larger than the multicommodity flow rate, sharpening the Saks $|f|\le c(M)$9 gap (Blasiak, 2013). The key mechanism is the notion of a $1$00-certifiable linear network code, for which every multicut $1$01 satisfies

$1$02

In planar graphs, lower bounds are also nontrivial. The longstanding lower bound of $1$03 has been improved to $1$04, and this already holds for cactus graphs: $1$05 Since cactus graphs are planar, this implies

$1$06

The same work gives an explicit integrality-gap instance with

$1$07

and develops a framework that translates multicut integrality gaps into the nonexistence of suitably light distributions over small-diameter decompositions (Kalantarzadeh et al., 9 Jul 2025).

Demand-graph structure can itself determine approximability. For directed multicut with a fixed directed bipartite demand graph $1$08, if $1$09 denotes the worst-case flow-cut gap, then, assuming UGC, there is no polynomial-time $1$10-approximation for any fixed $1$11 (Chekuri et al., 2016). For non-bipartite fixed $1$12, the hardness degrades by a $1$13 factor in the denominator, and for $1$14 disjoint directed demand edges the problem is hard to approximate within $1$15 (Chekuri et al., 2016). Conversely, if $1$16 excludes an induced $1$17-matching-extension, then the directed flow-cut gap is at most $1$18 and a polynomial-time rounding algorithm achieves this bound (Chekuri et al., 2016). These results make the LP gap itself a hardness threshold for broad directed families.

7. Proof paradigms and conceptual viewpoints

Several proof paradigms recur across the literature. One is LP duality plus metarounding. In the planar-digraph multicut framework, if an $1$19-approximation exists via the natural LP, then Carr-Vempala style arguments yield a distribution over integral solutions cutting each edge or vertex with probability at most $1$20 times its LP value (Chekuri et al., 27 Jan 2026). This viewpoint directly connects approximation factors, integrality gaps, and randomized or deterministic decompositions.

A second paradigm is metric and quasimetric embedding. For planar graphs with terminals on $1$21 faces, stochastic embeddings of terminal metrics into trees with expected distortion $1$22 imply the corresponding flow-cut bound because tree metrics embed isometrically into $1$23 (Krauthgamer et al., 2018). For directed planar graphs, the analogous object is a quasimetric rather than a metric, and $1$24-bounded $1$25-Lipschitz quasipartitions yield embeddings into directed $1$26 with distortion $1$27, which in turn produce multicut and sparsest-cut approximations (Kawarabayashi et al., 2021).

A third paradigm is region growing and separator recursion. The treewidth-$1$28 theorem rounds LP solutions by bounded-treewidth-specific region growing, while the node-weighted planar-digraph multicut algorithm combines removal of large LP variables, a layering decomposition with cost

$1$29

and a separator lemma that ultimately yields

$1$30

(Friedrich et al., 2022, Chekuri et al., 27 Jan 2026). In planar digraphs this is enabled by separators consisting of three directed shortest paths.

A fourth paradigm is primal-dual uncrossing and half-integrality. The WGMV framework for uncrossable cut cover gives a $1$31-approximation in general, and its half-integral refinement for integral edge costs produces half-integral duals that correspond to half-integral multiflows in Seymour instances (Garg et al., 2020). In the plane multiflow setting, uncrossing produces laminar multiflows, total unimodularity gives half-integral solutions, and planar auxiliary graphs plus $1$32-coloring yield integral rounding (Garg et al., 2020).

Finally, planarity often admits topological relaxations stricter than ordinary fractional flow. An uncrossed multiflow is one whose support paths do not cross in the plane. In general planar graphs, an uncrossed fractional multiflow can always be rounded to an integral multiflow with a constant fraction of its value in the maximization model, and a strongly uncrossed feasible fractional multiflow can be rounded in the congestion model to an integral multiflow with edge congestion $1$33, or to an unsplittable multiflow with edge load strictly less than $1$34 (Chekuri et al., 31 Oct 2025). This is not a multicut theorem, but it is a closely related fractional-to-integral gap phenomenon on the routing side.

Taken together, these results show a highly stratified landscape. Exact gap $1$35 survives in a few rigid topologies, planar and bounded-width families often admit constant or polylogarithmic bounds, and general directed instances can exhibit polynomially growing separations. The modern theory therefore treats the multiflow-multicut gap not as a single number, but as a structural invariant shaped by embeddings, separators, minor structure, face covers, demand-graph obstructions, and the choice between edge, node, and submodular capacity models.