Typically, graph partition problems fall under the category of NP-hard problems. 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. Furthermore, the relaxation is exact if I= V. A second result shows that indeed our proposed optimization problem (3) is a relaxation of the balanced k-cut problem (1). optimal balanced 2-cut partition. Balanced Cut. Back to problem description. Note that rounding trivially yields a solution in the setting of the previous theorem.

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. This problem differs from the balanced cut problems that are usually studied (see e.g. The most famous approach is spectral clustering [7], which corresponds to the spectral relaxation of the �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�$�=>��ѡn�P7r�+rC�4�t߯4! By continuing you agree to the Copyright © 2020 Elsevier B.V. or its licensors or contributors. Problem statistics. These results also hold for directed graphs. This objective function favors solutions that are both sparse (few edges crossing the cut) and balanced … We give a PTAS for the edge cut variant and for (i). ScienceDirect ® is a registered trademark of Elsevier B.V.Copyright © 2009 Elsevier B.V. All rights reserved.ScienceDirect ® is a registered trademark of Elsevier B.V. Ǜlx����[9:i�"+�V"Ҏh��굦Ȓx4�l��uv7D'�ي(��f�H*ɕ�����?�D��OGr=sM�net0m�Z9����'�K}� �}����[��Sj�T�ɭ�I�+jHc̭=����ęVq��t�R6$��7ִ�K&~�F{��v3u���p�$�h��-I�=E��BF�T�&����pFz�{��OnKޭ\�Wg�;vE����F_cЫ��;ې��8�4,��HrM�u��)@O�lUM1_�0�ieg�H�Hş�}g=Ι�{Ŵ0��@�H��n:��:~�Ur��6�z�j��I�n��x᪲Sg�MHy��+[���z�xa�,,K�GR4 �������z�VF��F������D~�]:a�����v��F��v�S�A��s�nD�~��nj����Ñ~�����QO>��4�$����:���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��!

We consider the problem of finding most balanced cuts among minimum All of these problems are NP-hard. 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. Today we will describe an application of Sparsest Cut to the Balanced Cut problem. However, uniform graph partitioning or a balanced graph partition problem can be shown to be NP-complete to approximate within any finite factor. Problem complexity. ;fhAR���]�����d��wfa���c��2B����)����PVDn�0��dZ��+�lLЪ�\��WnD��F�b'������+'&a�I}�Fk��{Ej]r���\�g��{�>*E Solutions to these problems are generally derived using heuristics and approximation algorithms. %PDF-1.2 <> stream 5 0 obj 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�~� %�쏢 Gc�����]���]���i�������./~��E��˯.�f���I�v.�E_���b���h'c�6��b '\�����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���!
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.

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