combinatorial optimization Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combina ...

, network flow problems are a class of computational problems in which the input is a

flow network In graph theory, a flow network (also known as a transportation network) is a directed graph where each edge has a capacity and each edge receives a flow. The amount of flow on an edge cannot exceed the capacity of the edge. Often in operations re ...

(a graph with numerical capacities on its edges), and the goal is to construct a flow, numerical values on each edge that respect the capacity constraints and that have incoming flow equal to outgoing flow at all vertices except for certain designated terminals. Specific types of network flow problems include: *The maximum flow problem, in which the goal is to maximize the total amount of flow out of the source terminals and into the sink terminals *The

minimum-cost flow problem The minimum-cost flow problem (MCFP) is an optimization and decision problem to find the cheapest possible way of sending a certain amount of flow through a flow network. A typical application of this problem involves finding the best delivery ro ...

, in which the edges have costs as well as capacities and the goal is to achieve a given amount of flow (or a maximum flow) that has the minimum possible cost *The multi-commodity flow problem, in which one must construct multiple flows for different commodities whose total flow amounts together respect the capacities * Nowhere-zero flow, a type of flow studied in combinatorics in which the flow amounts are restricted to a finite set of nonzero values The max-flow min-cut theorem equates the value of a maximum flow to the value of a minimum cut, a partition of the vertices of the flow network that minimizes the total capacity of edges crossing from one side of the partition to the other. Approximate max-flow min-cut theorems provide an extension of this result to multi-commodity flow problems. The Gomory–Hu tree of an undirected flow network provides a concise representation of all minimum cuts between different pairs of terminal vertices.

Algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...

s for constructing flows include * Dinic's algorithm, a strongly polynomial algorithm for maximum flow *The Edmonds–Karp algorithm, a faster strongly polynomial algorithm for maximum flow *The Ford–Fulkerson algorithm, a greedy algorithm for maximum flow that is not in general strongly polynomial *The network simplex algorithm, a method based on linear programming but specialized for network flow *The out-of-kilter algorithm for minimum-cost flow *The push–relabel maximum flow algorithm, one of the most efficient known techniques for maximum flow Otherwise the problem can be formulated as a more conventional

linear program Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear relationships. Linear ...

or similar and solved using a general purpose optimization solver.

References

Graph algorithms Combinatorial optimization Directed graphs {{sia