MincutLPbased

MincutLPbased refers to a class of algorithms for computing minimum s-t cuts by formulating the problem as a linear program (LP). The approach exploits linear programming duality with the classic max-flow min-cut theorem: solving an LP that encodes a cut yields the minimum cut value, and, in many graph families, the LP solution is integral and corresponds to a valid cut.

A typical formulation assigns a variable x_e to each directed edge e, representing whether e is included in the cut, with edge capacity c_e and objective to minimize sum c_e x_e. Constraints enforce that every s-t path must contain at least one cut edge, and bounds 0 ≤ x_e ≤ 1. The resulting LP can be solved in polynomial time by interior-point methods. In practice, the LP relaxation is used within a broader optimization framework, and, for standard graphs, its optimum matches the true minimum cut.

Variants of mincutLPbased extend the basic model to directed or undirected graphs, multiple sources or sinks, vertex capacities, or additional side constraints. The method can also be paired with cutting-plane techniques or column generation to handle large-scale instances or to embed the min-cut computation inside a larger mixed-integer program.

Relation to other approaches: while highly tuned max-flow algorithms (for example, push-relabel) remain the fastest for plain min-cut, LP-based formulations offer a flexible, constraint-friendly alternative when the cut problem interacts with other linear constraints or optimization objectives.

See also: maximum flow, min-cut theorem, Gomory-Hu tree, linear programming, cutting planes.