10.2 Tropical network flows

Tropical network flows are a crucial concept in tropical geometry, modeling the movement of commodities or information through networks. They have wide-ranging applications in transportation, communication, and logistics, helping optimize the flow of goods, data, or resources while considering tropical algebra's unique properties.

These flows are governed by capacity constraints and conservation principles. The max-flow min-cut theorem and algorithms like Ford-Fulkerson and Edmonds-Karp are fundamental to solving tropical network flow problems, providing efficient methods for finding maximum flows in complex networks.

Tropical network flows

• Tropical network flows are a fundamental concept in tropical geometry that deals with the movement of commodities or information through a network
• The study of tropical network flows has important applications in various fields such as transportation, communication, and logistics
• Tropical network flows can be used to model and optimize the flow of goods, data, or resources in a network, taking into account the unique properties of tropical algebra

Definition of flows

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• A flow in a tropical network is a function that assigns a non-negative value to each edge, representing the amount of commodity or information flowing through that edge
• The flow value on an edge cannot exceed the capacity of the edge, which is a pre-defined upper limit
• The total flow entering a vertex must be equal to the total flow leaving the vertex, except for the source and sink vertices

Capacity constraints

• Each edge in a tropical network has a capacity, which is the maximum amount of flow that can pass through the edge
• The flow on an edge cannot exceed its capacity, ensuring that the network does not become overloaded
• Capacity constraints play a crucial role in determining the maximum flow that can be achieved in a network

Conservation of flows

• The conservation of flows principle states that the total flow entering a vertex must be equal to the total flow leaving the vertex, except for the source and sink vertices
• This principle ensures that flow is neither created nor destroyed at intermediate vertices, maintaining the balance of flow throughout the network
• The conservation of flows is a fundamental property that must be satisfied by any valid flow in a tropical network

Residual networks

• A residual network is a modified version of the original network that represents the remaining capacity available on each edge
• The residual capacity of an edge is the difference between its original capacity and the current flow on that edge
• Residual networks are used to identify augmenting paths and improve the flow in the original network

Augmenting paths

• An augmenting path is a path from the source to the sink in the residual network along which the flow can be increased
• By finding an augmenting path and increasing the flow along that path, the overall flow in the network can be improved
• Augmenting paths are a key concept in many tropical network flow algorithms, such as the Ford-Fulkerson algorithm

Max-flow min-cut theorem

• The max-flow min-cut theorem states that the maximum flow in a network is equal to the minimum capacity of all cuts separating the source and the sink
• A cut is a partition of the vertices into two disjoint sets, with the source in one set and the sink in the other
• The capacity of a cut is the sum of the capacities of the edges crossing from the source set to the sink set

Ford-Fulkerson algorithm

• The Ford-Fulkerson algorithm is a classic method for finding the maximum flow in a tropical network
• It iteratively finds augmenting paths in the residual network and increases the flow along those paths until no more augmenting paths can be found
• The algorithm terminates when the maximum flow is reached, which is equal to the minimum capacity of all cuts in the network

Edmonds-Karp algorithm

• The Edmonds-Karp algorithm is an improvement over the Ford-Fulkerson algorithm that uses a specific strategy for finding augmenting paths
• It always chooses the shortest augmenting path in terms of the number of edges, which guarantees a polynomial time complexity
• The Edmonds-Karp algorithm is more efficient than the general Ford-Fulkerson algorithm and is widely used in practice

Dinic's algorithm

• Dinic's algorithm is another efficient algorithm for finding the maximum flow in a tropical network
• It uses the concept of layered networks and blocking flows to improve the efficiency of finding augmenting paths
• Dinic's algorithm has a better worst-case time complexity than the Edmonds-Karp algorithm, making it suitable for large-scale networks

Goldberg-Tarjan algorithm

• The Goldberg-Tarjan algorithm, also known as the push-relabel algorithm, is a different approach to finding the maximum flow in a tropical network
• It maintains a labeling of vertices that represents their distance from the sink and uses push and relabel operations to update the flow and labels
• The Goldberg-Tarjan algorithm has a good theoretical time complexity and performs well in practice

Preflow-push algorithm

• The preflow-push algorithm is a variant of the Goldberg-Tarjan algorithm that allows for excess flow at vertices
• It maintains a preflow, which is a flow that satisfies the capacity constraints but may violate the conservation of flows at some vertices
• The algorithm repeatedly pushes excess flow from overflowing vertices to neighboring vertices until a maximum flow is reached

Capacity scaling

• Capacity scaling is a technique used to improve the efficiency of tropical network flow algorithms
• It involves iteratively solving the maximum flow problem on a sequence of networks with scaled capacities
• By starting with a network with reduced capacities and gradually increasing them, the algorithm can find the maximum flow more quickly

Applications of network flows

• Tropical network flows have numerous applications in various domains, such as transportation, logistics, and resource allocation
• They can be used to model and optimize the movement of goods, vehicles, or data through a network
• Network flows can also be applied to problems like bipartite matching, vertex cover, and disjoint paths

Bipartite matching

• Bipartite matching is a problem that can be solved using tropical network flows
• It involves finding a maximum matching in a bipartite graph, where vertices are divided into two disjoint sets and edges only exist between vertices in different sets
• By constructing a flow network and finding the maximum flow, the maximum bipartite matching can be obtained

Vertex cover

• The vertex cover problem asks for a minimum set of vertices in a graph such that every edge is incident to at least one vertex in the set
• This problem can be reduced to a tropical network flow problem by constructing a suitable flow network
• By finding the maximum flow and using the max-flow min-cut theorem, a minimum vertex cover can be determined

Disjoint paths

• The disjoint paths problem involves finding a set of paths in a graph that do not share any vertices or edges
• Tropical network flows can be used to solve this problem by creating a flow network with appropriate capacities
• By finding the maximum flow in the network, the maximum number of disjoint paths can be obtained

Circulations with demands

• Circulations with demands are a variant of tropical network flows where each vertex has a specified demand that must be satisfied
• The flow entering a vertex must be equal to the flow leaving the vertex plus the demand at that vertex
• Circulations with demands can be solved using modified versions of tropical network flow algorithms

Circulations with lower bounds

• Circulations with lower bounds are another extension of tropical network flows where edges have both lower and upper capacity bounds
• The flow on an edge must be greater than or equal to the lower bound and less than or equal to the upper bound
• Algorithms for circulations with lower bounds typically involve transforming the problem into a standard tropical network flow problem

Minimum cost flows

• Minimum cost flows are a variant of tropical network flows where each edge has an associated cost per unit of flow
• The objective is to find a flow that satisfies the capacity constraints and conservation of flows while minimizing the total cost
• Minimum cost flow problems can be solved using algorithms like successive shortest paths, cycle canceling, and cost scaling

Successive shortest paths

• The successive shortest paths algorithm is a method for solving minimum cost flow problems
• It iteratively finds the shortest path from the source to the sink in the residual network, considering the costs of edges
• The flow is augmented along the shortest path until the optimal minimum cost flow is reached

Cycle canceling

• Cycle canceling is another algorithm for solving minimum cost flow problems
• It starts with a feasible flow and iteratively finds negative cost cycles in the residual network
• By canceling negative cost cycles, the algorithm improves the flow until the optimal minimum cost flow is obtained

Cost scaling

• Cost scaling is a technique used to improve the efficiency of minimum cost flow algorithms
• It involves solving a sequence of minimum cost flow problems with scaled costs, starting with a coarse approximation and gradually refining it
• Cost scaling can significantly reduce the number of iterations required to find the optimal minimum cost flow

Capacity scaling

• Capacity scaling can also be applied to minimum cost flow problems
• It involves solving a sequence of minimum cost flow problems with scaled capacities, starting with a network with reduced capacities and gradually increasing them
• Capacity scaling can improve the efficiency of minimum cost flow algorithms, especially when combined with cost scaling

Network simplex algorithm

• The network simplex algorithm is a specialized version of the simplex algorithm for solving minimum cost flow problems
• It exploits the network structure of the problem to efficiently update the basic feasible solution
• The network simplex algorithm maintains a spanning tree of the network and iteratively improves the solution by pivoting along edges

Generalized flows

• Generalized flows are an extension of tropical network flows where edges have gain factors associated with them
• The flow on an edge is multiplied by the gain factor when it traverses the edge, allowing for flow to be gained or lost
• Generalized flow problems can be solved using modified versions of tropical network flow algorithms, such as the generalized Ford-Fulkerson algorithm

Flows over time

• Flows over time, also known as dynamic flows, consider the temporal aspect of flow in a network
• In this setting, flow takes time to travel through edges, and the goal is to optimize the flow over a given time horizon
• Flows over time can be used to model problems like evacuation planning, traffic management, and resource allocation over time

Earliest arrival flows

• Earliest arrival flows are a specific type of flows over time where the objective is to maximize the amount of flow that reaches the sink by any given time
• The earliest arrival flow problem aims to find a flow that sends as much flow as possible to the sink as early as possible
• Algorithms for earliest arrival flows typically involve solving a sequence of static flow problems at different time steps

Universally maximum flows

• Universally maximum flows are a variant of flows over time where the goal is to find a flow that maximizes the amount of flow reaching the sink at all time steps simultaneously
• A universally maximum flow is a flow that is maximum for every time step, providing a robust solution for dynamic flow problems
• Universally maximum flows can be computed using algorithms based on time-expanded networks or by solving a sequence of earliest arrival flow problems

Lexicographically maximum flows

• Lexicographically maximum flows are another variant of flows over time where the objective is to find a flow that maximizes the amount of flow reaching the sink at each time step in a lexicographic order
• A lexicographically maximum flow is a flow that is maximum at the earliest time step, then maximum at the second earliest time step, and so on
• Lexicographically maximum flows can be computed using algorithms based on time-expanded networks or by solving a sequence of earliest arrival flow problems with modified objectives

Dynamic flows

• Dynamic flows, also known as flows over time, consider the temporal aspect of flow in a network
• In dynamic flow problems, flow takes time to travel through edges, and the goal is to optimize the flow over a given time horizon
• Dynamic flows can be used to model problems like evacuation planning, traffic management, and resource allocation over time

Flows with vertex capacities

• Flows with vertex capacities are an extension of tropical network flows where vertices have capacity constraints in addition to edge capacities
• The flow passing through a vertex cannot exceed its capacity, adding an additional constraint to the flow problem
• Flows with vertex capacities can be solved by transforming the problem into a standard tropical network flow problem with additional edges and vertices

Multi-source multi-sink flows

• Multi-source multi-sink flows are a generalization of tropical network flows where there are multiple sources and multiple sinks in the network
• The objective is to find a flow that maximizes the total flow from the sources to the sinks while satisfying the capacity constraints and conservation of flows
• Multi-source multi-sink flow problems can be solved by transforming the network into a single-source single-sink network using a super-source and a super-sink

Undirected network flows

• Undirected network flows are a variant of tropical network flows where the edges in the network are undirected, meaning flow can pass through them in either direction
• In undirected network flows, the capacity constraints apply to the total flow in both directions on an edge
• Undirected network flow problems can be solved by transforming the undirected network into a directed network by replacing each undirected edge with two directed edges in opposite directions, each with the original capacity