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Wikipedia / Maximum flow problem

In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate.

The maximum flow problem can be seen as a special case of more complex network flow problems, such as the circulation problem. The maximum value of an s-t flow (i.e., flow from source s to sink t) is equal to the minimum capacity of an s-t cut (i.e., cut severing s from t) in the network, as stated in the max-flow min-cut theorem.

History

The maximum flow problem was first formulated in 1954 by T. E. Harris and F. S. Ross as a simplified model of Soviet railway traffic flow. In 1955, Lester R. Ford, Jr. and Delbert R. Fulkerson created the first known algorithm, the Ford–Fulkerson algorithm. In their 1955 paper, Ford and Fulkerson wrote that the problem of Harris and Ross is formulated as follows (see p.5):Consider a rail network connecting two cities by way of a number of intermediate cities, where each link of the network has a number assigned to it representing its capacity. Assuming a steady state condition, find a maximal flow from one given city to the other.In their book Flows in Networks, in 1962, Ford and Fulkerson wrote:It was posed to the authors in the spring of 1955 by T. E. Harris, who, in conjunction with General F. S. Ross (Ret.), had formulated a simplified model of railway traffic flow, and pinpointed this particular problem as the central one suggested by the model [11].where [11] refers to the 1955 secret report Fundamentals of a Method for Evaluating Rail net Capacities by Harris and Ross (see p.5).

Over the years, various improved solutions to the maximum flow problem were discovered, notably the shortest augmenting path algorithm of Edmonds and Karp and independently Dinitz; the blocking flow algorithm of Dinitz; the push-relabel algorithm of Goldberg and Tarjan; and the binary blocking flow algorithm of Goldberg and Rao. The algorithms of Sherman and Kelner, Lee, Orecchia and Sidford, respectively, find an approximately optimal maximum flow but only work in undirected graphs.

In 2013 James B. Orlin algorithm.

In 2022 Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva published an almost-linear time algorithm for the minimum-cost flow problem of which the maximum flow problem is a particular case. For the single-source shortest path (SSSP) problem with negative weights – another particular case of a minimum-cost flow problem – an algorithm running in almost-linear time has also been reported. Both algorithms were deemed best papers at the 2022 Symposium on Foundations of Computer Science. A derandomized version of the 2022 algorithm by Chen et al was presented at the 2023 Symposium on Foundations of Computer Science, establishing that minimum-cost flow could be solved deterministically in almost-linear time.

Definition

First we establish some notation:

respectively.

Definition. The capacity

Definition. A flow that satisfies the following:

Capacity constraint
Conservation of flows. The sum of the flows entering a node must equal the sum of the flows exiting that node, except for the source and the sink. Or:

Remark

Definition. The value of flow it is given by:

Definition. The maximum flow problem with maximum value.

Algorithms

The following tables show the historical development of algorithms for solving the maximum flow problem. Many of the listed publications include similar tables comparing their results to earlier works.

Strongly polynomial

A strongly-polynomial time). The complexity of each algorithm is stated using big O notation.

Pseudo-polynomial and weakly polynomial

In parallel to the development of strongly-polynomial flow algorithms, there has been a long line of pseudo-polynomial refers to the largest edge capacity after rescaling all capacities to integer values. (If the network contains irrational.

Integral flow theorem

The integral flow theorem states that

If each edge in a flow network has integral capacity, then there exists an integral maximal flow.

The claim is not only that the value of the flow is an integer, which follows directly from the max-flow min-cut theorem, but that the flow on every edge is integral. This is crucial for many combinatorial applications (see below), where the flow across an edge may encode whether the item corresponding to that edge is to be included in the set sought or not.

Application

Multi-source multi-sink maximum flow problem

. We can transform the multi-source multi-sink problem into a maximum flow problem by adding a consolidated source and a consolidated sink (also known as supersource and supersink) with infinite capacity on each edge (See Fig. 4.1.1.).

Maximum cardinality bipartite matching

Given a bipartite graph, we are to find a maximum cardinality matching, where

.
.
(See Fig. 4.3.1).

in an integral max-flow.

Minimum path cover in directed acyclic graph

Given a directed acyclic graph, we are to find the minimum number of vertex-disjoint paths, where

instead.

is vertex-disjoint, consider the following:

can either be non-matched; or it can be matched.
.

are vertex-disjoint.

Maximum flow with vertex capacities

has to satisfy not only the capacity constraint and the conservation of flows, but also the vertex capacity constraint

(see Fig. 4.4.1). In this expanded network, the vertex capacity constraint is removed and therefore the problem can be treated as the original maximum flow problem.

Maximum number of paths from s to t

. This problem has several variants:

edge-disjoint paths.

. Most variants of this problem are NP-complete.

Closure problem

A closure of a directed graph is a set of vertices C, such that no edges leave C. The closure problem is the task of finding the maximum-weight or minimum-weight closure in a vertex-weighted directed graph. It may be solved in polynomial time using a reduction to the maximum flow problem.

Real world applications

Baseball elimination

In the baseball elimination problem there are n teams competing in a league. At a specific stage of the league season, wi is the number of wins and ri is the number of games left to play for team i and rij is the number of games left against team j. A team is eliminated if it has no chance to finish the season in the first place. The task of the baseball elimination problem is to determine which teams are eliminated at each point during the season. Schwartz proposed a method which reduces this problem to maximum network flow. In this method a network is created to determine whether team k is eliminated.

Let G = (V, E) be a network with being the source and the sink respectively. One adds a game nodeij – which represents the number of plays between these two teams. We also add a team node for each team and connect each game node with i < j to V, and connects each of them from s by an edge with capacity rij – which represents the number of plays between these two teams. We also add a team node for each team and connect each game node with two team nodes i and j to ensure one of them wins. One does not need to restrict the flow value on these edges. Finally, edges are made from team node i to the sink node t and the capacity of is set to prevent team i from winning more than . Let S be the set of all teams participating in the league and let

In this method it is claimed team k is not eliminated if and only if a flow value of size r(S − {k}) exists in network G. In the mentioned article it is proved that this flow value is the maximum flow value from s to t.

Airline scheduling

In the airline industry a major problem is the scheduling of the flight crews. The airline scheduling problem can be considered as an application of extended maximum network flow. The input of this problem is a set of flights F which contains the information about where and when each flight departs and arrives. In one version of airline scheduling the goal is to produce a feasible schedule with at most k crews.

To solve this problem one uses a variation of the circulation problem called bounded circulation which is the generalization of network flow problems, with the added constraint of a lower bound on edge flows.

Let G = (V, E) be a network with as the source and the sink nodes. For the source and destination of every flight i, one adds two nodes to V, node si as the source and node di as the destination node of flight i. One also adds the following edges to E:

An edge with capacity [0, 1] between s and each si.
An edge with capacity [0, 1] between each di and t.
An edge with capacity [1, 1] between each pair of si and di.
An edge with capacity [0, 1] between each di and sj, if source sj is reachable with a reasonable amount of time and cost from the destination of flight i.
An edge with capacity [0, ∞] between s and t.

In the mentioned method, it is claimed and proved that finding a flow value of k in G between s and t is equal to finding a feasible schedule for flight set F with at most k crews.

Another version of airline scheduling is finding the minimum needed crews to perform all the flights. To find an answer to this problem, a bipartite graph is created where each flight has a copy in set A and set B. If the same plane can perform flight j after flight i, is connected to . A matching in induces a schedule for F and obviously maximum bipartite matching in this graph produces an airline schedule with minimum number of crews. As it is mentioned in the Application part of this article, the maximum cardinality bipartite matching is an application of maximum flow problem.

Circulation–demand problem

There are some factories that produce goods and some villages where the goods have to be delivered. They are connected by a networks of roads with each road having a capacity for maximum goods that can flow through it. The problem is to find if there is a circulation that satisfies the demand. This problem can be transformed into a maximum-flow problem.

Add a source node and add edges from it to every factory node with capacity where is the production rate of factory .
Add a sink node and add edges from all villages to with capacity where is the demand rate of village .

Let G = (V, E) be this new network. There exists a circulation that satisfies the demand if and only if :

If there exists a circulation, looking at the max-flow solution would give the answer as to how much goods have to be sent on a particular road for satisfying the demands.

The problem can be extended by adding a lower bound on the flow on some edges.

Image segmentation

In their book, Kleinberg and Tardos present an algorithm for segmenting an image. They present an algorithm to find the background and the foreground in an image. More precisely, the algorithm takes a bitmap as an input modelled as follows: ai ≥ 0 is the likelihood that pixel i belongs to the foreground, bi ≥ 0 in the likelihood that pixel i belongs to the background, and pij is the penalty if two adjacent pixels i and j are placed one in the foreground and the other in the background. The goal is to find a partition (A, B) of the set of pixels that maximize the following quantity

Indeed, for pixels in A (considered as the foreground), we gain ai; for all pixels in B (considered as the background), we gain bi. On the border, between two adjacent pixels i and j, we loose pij. It is equivalent to minimize the quantity

because

We now construct the network whose nodes are the pixel, plus a source and a sink, see Figure on the right. We connect the source to pixel i by an edge of weight ai. We connect the pixel i to the sink by an edge of weight bi. We connect pixel i to pixel j with weight pij. Now, it remains to compute a minimum cut in that network (or equivalently a maximum flow). The last figure shows a minimum cut.

Extensions

1. In the minimum-cost flow problem, each edge (u,v) also has a cost-coefficient auv in addition to its capacity. If the flow through the edge is fuv, then the total cost is auvfuv. It is required to find a flow of a given size d, with the smallest cost. In most variants, the cost-coefficients may be either positive or negative. There are various polynomial-time algorithms for this problem.

2. The maximum-flow problem can be augmented by disjunctive constraints: a negative disjunctive constraint says that a certain pair of edges cannot simultaneously have a nonzero flow; a positive disjunctive constraints says that, in a certain pair of edges, at least one must have a nonzero flow. With negative constraints, the problem becomes strongly NP-hard even for simple networks. With positive constraints, the problem is polynomial if fractional flows are allowed, but may be strongly NP-hard when the flows must be integral.