HOME

TheInfoList



OR:

Approximate
max-flow min-cut theorem In computer science and optimization theory, the max-flow min-cut theorem states that in a flow network, the maximum amount of flow passing from the ''source'' to the ''sink'' is equal to the total weight of the edges in a minimum cut, i.e., the ...
s are mathematical propositions in network flow theory. They deal with the relationship between maximum flow rate ("max-flow") and
minimum cut In graph theory, a minimum cut or min-cut of a graph is a cut (a partition of the vertices of a graph into two disjoint subsets) that is minimal in some metric. Variations of the minimum cut problem consider weighted graphs, directed graphs, term ...
("min-cut") in a
multi-commodity flow problem The multi-commodity flow problem is a network flow problem with multiple commodities (flow demands) between different source and sink nodes. Definition Given a flow network \,G(V,E), where edge (u,v) \in E has capacity \,c(u,v). There are \,k comm ...
. The theorems have enabled the development of
approximation algorithm In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solu ...
s for use in
graph partition In mathematics, a graph partition is the reduction of a graph to a smaller graph by partitioning its set of nodes into mutually exclusive groups. Edges of the original graph that cross between the groups will produce edges in the partitioned graph ...
and related problems.


Multicommodity flow problem

A "commodity" in a network flow problem is a pair of source and sink
node In general, a node is a localized swelling (a "knot") or a point of intersection (a vertex). Node may refer to: In mathematics *Vertex (graph theory), a vertex in a mathematical graph *Vertex (geometry), a point where two or more curves, lines, ...
s. In a multi-commodity flow problem, there are commodities, each with its own source s_, sink t_, and demand D_. The objective is to simultaneously route D_ units of commodity from s_ to t_ for each , such that the total amount of all commodities passing through any edge is no greater than its capacity. (In the case of undirected edges, the sum of the flows in both directions cannot exceed the capacity of the edge). Specially, a 1-commodity (or single commodity) flow problem is also known as a
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 ...
. According to the Ford–Fulkerson algorithm, the max-flow and min-cut are always equal in a 1-commodity flow problem.


Max-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 fD_ 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 \varphi 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 v \in V in graph G=(V,E). 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 u \in V.


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 1/k 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 \varphi for which f\le O\left(\frac\right).'' Theorem 2. ''For any uniform multicommodity flow problem, \Omega\left(\frac\right)\le f\le\varphi, where is the max-flow and \varphi 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, \Omega\left(\frac\right)\le f\le \varphi, where is the max-flow and \varphi 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, \Omega\left(\frac\right)\le f\le \varphi, where is the max-flow and \varphi 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, \Omega\left(\frac\right)\le f\le\varphi, where is the max-flow and \varphi is the directed min-cut of the product multicommodity flow problem.''


Applications to approximation algorithms

The above theorems are very useful to design
approximation algorithm In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solu ...
s for
NP-hard In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard pr ...
problems, such as the
graph partition In mathematics, a graph partition is the reduction of a graph to a smaller graph by partitioning its set of nodes into mutually exclusive groups. Edges of the original graph that cross between the groups will produce edges in the partitioned graph ...
problem and its variations. Here below we briefly introduce a few examples, and the in-depth elaborations can be found in Leighton and Rao (1999).


Sparsest cuts

A sparsest cut of a graph G=(V,E) 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 O(\log n) factor using Theorem 2. Also, a sparsest cut problem with weighted edges, weighted nodes or directed edges can be approximated within an O(\log p) 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 G=(V,E) that partitions the graph into nearly equal-size pieces. We usually call a cut ''b-balanced'' or a -''separator'' (for ) if b\pi(V)\le\pi(U)\le(1-b)\pi(V) where \pi(U) is the sum of the node weights in . This is also an
NP-hard In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard pr ...
problem. An approximation algorithm has been designed for this problem, and the core idea is that has a -balanced cut of size , then we find a -balanced cut of size O\left(S\log \frac n b -b'\right) for any where and . Then we repeat the process then finally obtain the result that total weight of the edges in the cut is at most O\left(\frac\right).


VLSI 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 O(\log^6 n) times optimal.


Forwarding index problem

Given an -node graph and an embedding of K_n in , Chung et al. defined the ''forwarding index'' of the embedding to be the maximum number of paths (each corresponding to an edge of K_n) 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 O(\log n)-factor for every graph .


Planar edge deletion

Tragoudas uses the approximation algorithm for balanced separators to find a set of O\left((R\log n + \sqrt)\log\frac\right) 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 a
polylog In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natura ...
times optimal approximation algorithm for .


References

{{reflist Network flow problem Mathematical theorems