A second result shows that indeed our proposed optimization problem (3) is a relaxation of the balanced k-cut problem (1). Back to problem description. The balanced k-cut problem is known to be NP-hard [4] and thus in practice relaxations [4, 5] or greedy approaches [6] are used for finding the optimal multi-cut.
This objective function favors solutions that are both sparse (few edges crossing the cut) and balanced … x��\K��qV踁�þ�4��Ө�C�a>B�C�%rC:�:� I�4 ��x�W�U��]=�,H)@4z�둕�/3��NLr'����b���˝�&L&�;�)켌S�q煒�x�G�a���O/�e'/�;���?��}���ŧ�v�X�~� Typically, graph partition problems fall under the category of NP-hard problems. ScienceDirect ® is a registered trademark of Elsevier B.V.Copyright © 2009 Elsevier B.V. All rights reserved.ScienceDirect ® is a registered trademark of Elsevier B.V. Problem complexity. �pT.N:��5i�Y�-�N4V6N��S'E=۪@�9��H��&c��K�\H����FԤ�%)���J��9#��Ҽ�� �s�䙋u4��b��ȅn��DU�8� �`ly�����V�-_�3���6�����2�O�G3��ԅ����/I���?�W�$�=>��ѡn�P7r�+rC�4�t߯4!
These results also hold for directed graphs. This problem differs from the balanced cut problems that are usually studied (see e.g. '\�����t,s����u�ܴK)M=&GC��D�[^��-F��_-�����1�����z���(?Y1[�yu�~2�;���|B�I�ݏ��4��%��ފQ������ `$Z�!V��t������H�45��,��|����1F�;68ڻ�f{��`��Һt�yy^��c�� �T`�r�|A*E�B )t!^j��A��*�6��6꣘��K�$����|��� �P$�L��ej=������5�����ǝ�V�������}��y�:�O�.�5�-@�$���%��6�n�rY��n��5��mw�&��)�z��T�7+8E���q�ZlI�6�.kZ�C���d�b��3zy105�8=���f�:+�j7LV��j����j�l���! ;fhAR���]�����d��wfa���c��2B����)����PVDn�0��dZ��+�lLЪ�\��WnD��F�b'������+'&a�I}�Fk��{Ej]r���\�g��{�>*E The most famous approach is spectral clustering [7], which corresponds to the spectral relaxation of the stream We consider the problem of finding most balanced cuts among minimum All of these problems are NP-hard. �������z�VF��F������D~�]:a�����v��F��v�S�A��s�nD�~��nj����Ñ~�����QO>��4�$����:���CF=ig�9#��������r^�Q�����������y^�ᯗ�q���A���_\���VW;I�-ۏ��I�$jN���ÑE)A��>��~&?1�&��S�|B�tyfR�~ڑ��ѓ�f��CT��>!-�G��H$�����7��!
Topic: Balanced Cut, Sparsest Cut, and Metric Embeddings Date: 3/21/2007 In the last lecture, we described an O(logklogD)-approximation algorithm for Sparsest Cut, where k is the number of terminal pairs and D is the total requirement.
Note that rounding trivially yields a solution in the setting of the previous theorem.
%�쏢 The sparsest cut problem is to bipartition the vertices so as to minimize the ratio of the number of edges across the cut divided by the number of vertices in the smaller half of the partition. <> 5 0 obj
Today we will describe an application of Sparsest Cut to the Balanced Cut problem.
We give a 2-approximation for (ii), and show that no non-trivial approximation exists for (iii) unless P=NP.To prove these results we show that we can partition the vertices of We use cookies to help provide and enhance our service and tailor content and ads.
Ǜlx����[9:i�"+�V"Ҏh��굦Ȓx4�l��uv7D'�ي(��f�H*ɕ�����?�D��OGr=sM�net0m�Z9����'�K}� �}����[��Sj�T�ɭ�I�+jHc̭=����ęVq��t�R6$��7ִ�K&~�F{��v3u���p�$�h��-I�=E��BF�T�&����pFz�{��OnKޭ\�Wg�;vE����F_cЫ��;ې��8�4,��HrM�u��)@O�lUM1_�0�ieg�H�Hş�}g=Ι�{Ŵ0��@�H��n:��:~�Ur��6�z�j��I�n��x᪲Sg�MHy��+[���z�xa�,,K�GR4 However, uniform graph partitioning or a balanced graph partition problem can be shown to be NP-complete to approximate within any finite factor. Balanced Cut. Solutions to these problems are generally derived using heuristics and approximation algorithms.
We give a PTAS for the edge cut variant and for (i). Problem statistics.
Gc�����]���]���i�������./~��E��˯.�f���I�v.�E_���b���h'c�6��b Furthermore, the relaxation is exact if I= V.
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optimal balanced 2-cut partition.