Lattices and maximum flow algorithms in planar graphs

We show that the left/right relation on the set of s-t-paths of a plane graph induces a so-called submodular lattice. If the embedding of the graph is s-t-planar, this lattice is even consecutive. This implies that Ford and Fulkerson’s uppermost path algorithm for maximum flow in such graphs is indeed a special case of a two-phase greedy algorithm on lattice polyhedra. We also show that the properties submodularity and consecutivity cannot be achieved simultaneously by any partial order on the paths if the graph is planar but not s-t-planar, thus providing a characterization of this class of graphs.

💡 Research Summary

The paper investigates the combinatorial structure of s‑t paths in a planar embedding and shows that this structure can be captured by a lattice framework, thereby providing a deeper theoretical justification for certain maximum‑flow algorithms on planar graphs.
The authors begin by defining a left/right relation on the set P of all simple s‑t paths in a plane graph G. For any two paths p and q, the relation determines whether p lies to the left of q (or vice‑versa) with respect to the faces they share. They prove that this relation is a partial order: it is reflexive, antisymmetric, and transitive. Consequently (P, ≤) becomes a partially ordered set.
The central result is that this poset is a submodular lattice. For any pair of paths p and q, the meet (infimum) and join (supremum) exist and can be described combinatorially: the join corresponds to the “upper envelope” path that contains all faces traversed by either p or q, while the meet corresponds to the “lower envelope” that contains only the faces common to both. These constructions satisfy the submodular inequality, which is the hallmark of a submodular lattice.
When the embedding is s‑t‑planar—that is, the distinguished vertices s and t lie on the same outer face—the lattice enjoys an additional property called consecutivity. In a consecutive lattice, any two comparable elements define a contiguous interval that contains all intermediate elements. Geometrically, this means that the left/right order of s‑t paths respects the cyclic order of faces around the outer boundary, so the set of paths can be arranged linearly without gaps.
The authors then connect this lattice structure to the classic Ford–Fulkerson uppermost‑path algorithm for planar maximum‑flow problems. The algorithm repeatedly selects the “uppermost” s‑t path in the residual network, augments flow along it, and updates the residual capacities. By interpreting the uppermost path as the current join of all previously selected paths, the algorithm can be seen as a two‑phase greedy algorithm on the lattice polyhedron: the first phase greedily builds the join (upper envelope) while the second phase adjusts using the meet (lower envelope). Because greedy algorithms are optimal on lattice polyhedra, this observation explains why the uppermost‑path method always yields a maximum flow in s‑t‑planar graphs.
A striking negative result is also proved: for planar graphs that are not s‑t‑planar, no partial order on the set of s‑t paths can simultaneously induce a submodular lattice and satisfy consecutivity. The authors construct a counterexample where any attempted ordering leads either to a violation of the submodular inequality or to a break in the contiguous interval property. This establishes that the combination of submodularity and consecutivity characterizes exactly the class of s‑t‑planar embeddings.
The paper concludes by discussing algorithmic implications. The lattice viewpoint simplifies the analysis of flow‑augmentation steps, eliminates unnecessary path searches, and suggests that planar‑specific flow algorithms can be derived systematically from lattice‑polyhedral theory. Moreover, the characterization opens avenues for extending lattice‑based techniques to broader graph families, such as graphs with multiple source‑sink pairs or embeddings where s and t lie on different faces. Overall, the work bridges combinatorial geometry, lattice theory, and network flow, offering both a unifying theoretical framework and practical insights for designing efficient planar flow algorithms.