Multicommodity flow problem
A "commodity" in a network flow problem is a pair of source and sinkMax-flow and min-cut
In a multicommodity flow problem, ''max-flow'' is the maximum value of , where is the common fraction of each commodity that is routed, such that units of commodity can be simultaneously routed for each without violating any capacity constraints. ''min-cut'' is the minimum of all cuts of the ratio of the capacity of the cut to the demand of the cut. Max-flow is always upper bounded by the min-cut for a multicommodity flow problem.Uniform multicommodity flow problem
In a uniform multicommodity flow problem, there is a commodity for every pair of nodes and the demand for every commodity is the same. (Without loss of generality, the demand for every commodity is set to one.) The underlying network and capacities are arbitrary.Product multicommodity flow problem
In a product multicommodity flow problem, there is a nonnegative weight for each node in graph . The demand for the commodity between nodes and is the product of the weights of node and node . The uniform multicommodity flow problem is a special case of the product multicommodity flow problem for which the weight is set to 1 for all nodes .Duality of linear programming
In general, the dual of a multicommodity flow problem for a graph is the problem of apportioning a fixed amount of weight (where weights can be considered as distances) to the edges of such that to maximize the cumulative distance between the source and sink pairs.History
The research on the relationship between the max-flow and min-cut of multicommodity flow problem has obtained great interest since Ford and Fulkerson's result for 1-commodity flow problems. Hu showed that the max-flow and min-cut are always equal for two commodities. Okamura and Seymour illustrated a 4-commodity flow problem with max-flow equals to 3/4 and min-cut equals 1. Shahrokhi and Matula also proved that the max-flow and min-cut are equal provided the dual of the flow problem satisfies a certain cut condition in a uniform multicommodity flow problem. Many other researchers also showed concrete research results in similar problems For a general network flow problem, the max-flow is within a factor of of the min-cut since each commodity can be optimized separately using of the capacity of each edge. This is not a good result especially in case of large numbers of commodities.Approximate max-flow min-cut theorems
Theorems of uniform multicommodity flow problems
There are two theorems first introduced by Tom Leighton and Satish Rao in 1988 and then extended in 1999. Theorem 2 gives a tighter bound compared to Theorem 1. Theorem 1. ''For any , there is an -node uniform multicommodity flow problem with max-flow and min-cut for which .'' Theorem 2. ''For any uniform multicommodity flow problem, , where is the max-flow and is the min-cut of the uniform multicommodity flow problem.'' To prove Theorem 2, both the max-flow and the min-cut should be discussed. For the max-flow, the techniques from duality theory of linear programming have to be employed. According to the duality theory of linear programming, an optimal distance function results in a total weight that is equal to the max-flow of the uniform multicommodity flow problem. For the min-cut, a 3-stage process has to be followed: Stage 1: Consider the dual of uniform commodity flow problem and use the optimal solution to define a graph with distance labels on the edges. Stage 2: Starting from a source or a sink, grow a region in the graph until find a cut of small enough capacity separating the root from its mate. Stage 3: Remove the region and repeat the process of stage 2 until all nodes get processed.Generalized to product multicommodity flow problem
Theorem 3. ''For any product multicommodity flow problem with commodities, , where is the max-flow and is the min-cut of the product multicommodity flow problem.'' The proof methodology is similar to that for Theorem 2; the major difference is to take node weights into consideration.Extended to directed multicommodity flow problem
In a directed multicommodity flow problem, each edge has a direction, and the flow is restricted to move in the specified direction. In a directed uniform multicommodity flow problem, the demand is set to 1 for every directed edge. Theorem 4. ''For any directed uniform multicommodity flow problem with nodes, , where is the max-flow and is the min-cut of the uniform multicommodity flow problem.'' The major difference in the proof methodology compared to Theorem 2 is that, now the edge directions need to be considered when defining distance labels in stage 1 and for growing the regions in stage 2, more details can be found in. Similarly, for product multicommodity flow problem, we have the following extended theorem: Theorem 5. ''For any directed product multicommodity flow problem with commodities, , where is the max-flow and is the directed min-cut of the product multicommodity flow problem.''Applications to approximation algorithms
The above theorems are very useful to designSparsest cuts
A sparsest cut of a graph is a partition for which the ratio of the number of edges connecting the two partitioned components over the product of the numbers of nodes of both components is minimized. This is a NP-hard problem, and it can be approximated to within factor using Theorem 2. Also, a sparsest cut problem with weighted edges, weighted nodes or directed edges can be approximated within an factor where is the number of nodes with nonzero weight according to Theorem 3, 4 and 5.Balanced cuts and separators
In some applications, we want to find a small cut in a graph that partitions the graph into nearly equal-size pieces. We usually call a cut ''b-balanced'' or a -''separator'' (for ) if where is the sum of the node weights in . This is also anVLSI layout problems
It is helpful to find a layout of minimum size when designing a VLSI circuit. Such a problem can often be modeled as a graph embedding problem. The objective is to find an embedding for which the layout area is minimized. Finding the minimum layout area is also NP-hard. An approximation algorithm has been introduced and the result is times optimal.Forwarding index problem
Given an -node graph and an embedding of in , Chung et al. defined the ''forwarding index'' of the embedding to be the maximum number of paths (each corresponding to an edge of ) that pass through any node of . The objective is to find an embedding that minimizes the forwarding index. Using embedding approaches it is possible to bound the node and edge-forwarding indices to within an -factor for every graph .Planar edge deletion
Tragoudas uses the approximation algorithm for balanced separators to find a set of edges whose removal from a bounded-degree graph results in a planar graph, where is the minimum number of edges that need to be removed from before it becomes planar. It remains an open question if there is aReferences
{{reflist Network flow problem Mathematical theorems