Approximate max-flow min-(multi)cut theorems and their applications

Consider the multicommodity flow problem in which the object is to maximize the sum of commodities routed. We prove the following approximate max-flow min-multicut theorem: min multicut < max flow < min multicut, O(log k) where k is the number of commodities. Our proof is constructive; it enables us to find a multicut within O(logk) of the max flow (and hence also the optimal multicut). In addition, the proof technique provides a unified framework in which one can also analyse the case of flows with specified demands of Leighton and Rao and Klein et al. and thereby obtain an improved bound for the latter problem.

[1]  Mihalis Yannakakis,et al.  Edge-Deletion Problems , 1981, SIAM J. Comput..

[2]  Peter Elias,et al.  A note on the maximum flow through a network , 1956, IRE Trans. Inf. Theory.

[3]  Vijay V. Vazirani,et al.  A polyhedron with alls—t cuts as vertices, and adjacency of cuts , 1995, Math. Program..

[4]  Spyros Tragoudas.  VLSI partitioning approximation algorithms based on multicommodity flow and other techniques , 1991 .

[5]  Frank Thomson Leighton,et al.  An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[6]  D. R. Fulkerson,et al.  Maximal Flow Through a Network , 1956 .

[7]  B. Rothschild,et al.  MULTICOMMODITY NETWORK FLOWS. , 1969 .

[8]  William H. Cunningham.  The Optimal Multiterminal Cut Problem , 1989, Reliability Of Computer And Communication Networks.

[9]  Mihalis Yannakakis,et al.  Primal-Dual Approximation Algorithms for Integral Flow and Multicut in Trees, with Applications to Matching and Set Cover , 1993, ICALP.

[10]  Baruch Awerbuch,et al.  Complexity of network synchronization , 1985, JACM.

[11]  Mihalis Yannakakis,et al.  The complexity of multiway cuts (extended abstract) , 1992, STOC '92.

[12]  Éva Tardos,et al.  Improved bounds on the max-flow min-cut ratio for multicommodity flows , 1993, Comb..

[13]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[14]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[15]  Philip N. Klein,et al.  Excluded minors, network decomposition, and multicommodity flow , 1993, STOC.

[16]  Philip N. Klein,et al.  Leighton-Rao might be practical: faster approximation algorithms for concurrent flow with uniform capacities , 1990, STOC '90.

[17]  R. Ravi,et al.  Approximation through multicommodity flow , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[18]  Éva Tardos,et al.  Improved Bounds for the Max-Flow Min-Multicut Ratio for Planar and K_r, r-Free Graphs , 1993, Inf. Process. Lett..

[19]  Mihalis Yannakakis,et al.  Cutting and Partitioning a Graph aifter a Fixed Pattern (Extended Abstract) , 1983, ICALP.