Divide-and-Conquer Algorithms for Partitioning Hypergraphs and Submodular Systems

The submodular system k-partition problem is a problem of partitioning a given finite set V into k non-empty subsets V1,V2,…,Vk so that $\sum_{i=1}^{k}f(V_{i})$ is minimized where f is a non-negative submodular function on V. In this paper, we design an approximation algorithm for the problem with fixed k. We also analyze the approximation factor of our algorithm for the hypergraph k-cut problem, which is a problem contained by the submodular system k-partition problem.

[1]  Dorit S. Hochbaum,et al.  A Polynomial Algorithm for the k-cut Problem for Fixed k , 1994, Math. Oper. Res..

[2]  Martin D. F. Wong,et al.  A fast hypergraph min-cut algorithm for circuit partitioning , 2000, Integr..

[3]  Andrew V. Goldberg,et al.  A new approach to the maximum flow problem , 1986, STOC '86.

[4]  Andrzej Lingas,et al.  On the Complexity of Constructing Evolutionary Trees , 1999, J. Comb. Optim..

[5]  Mikkel Thorup,et al.  Minimum k-way cuts via deterministic greedy tree packing , 2008, STOC.

[6]  Hiroshi Nagamochi,et al.  Algorithms for the minimum partitioning problems in graphs , 2007 .

[7]  David R. Karger,et al.  A new approach to the minimum cut problem , 1996, JACM.

[8]  Toshihide Ibaraki,et al.  Greedy splitting algorithms for approximating multiway partition problems , 2005, Math. Program..

[9]  Satoru Iwata,et al.  Submodular function minimization , 2007, Math. Program..

[10]  Frank Wagner,et al.  A simple hypergraph min cut algorithm , 1996 .

[11]  Mechthild Stoer,et al.  A simple min-cut algorithm , 1997, JACM.

[12]  Eugene L. Lawler,et al.  Cutsets and partitions of hypergraphs , 1973, Networks.

[13]  Toshihide Ibaraki,et al.  A Unified Framework for Approximating Multiway Partition Problems , 2009, ISAAC.

[14]  YOKO KAMIDOI,et al.  A Deterministic Algorithm for Finding All Minimum k-Way Cuts , 2006, SIAM J. Comput..

[15]  Mihalis Yannakakis,et al.  Suboptimal Cuts: Their Enumeration, Weight and Number (Extended Abstract) , 1992, ICALP.

[16]  Mingyu Xiao.  Finding Minimum 3-Way Cuts in Hypergraphs , 2008, TAMC.

[17]  A. Frank.  Applications of submodular functions , 1993 .

[18]  Mingyu Xiao,et al.  An Improved Divide-and-Conquer Algorithm for Finding All Minimum k-Way Cuts , 2008, ISAAC.

[19]  Toshihide Ibaraki,et al.  Algorithmic Aspects of Graph Connectivity , 2008, Encyclopedia of Mathematics and its Applications.

[20]  Mihalis Yannakakis,et al.  Multiway cuts in node weighted graphs , 2004, J. Algorithms.

[21]  Maurice Queyranne,et al.  A combinatorial algorithm for minimizing symmetric submodular functions , 1995, SODA '95.

[22]  Takuro Fukunaga,et al.  Computing Minimum Multiway Cuts in Hypergraphs from Hypertree Packings , 2010, IPCO.