Min sum clustering with penalties

Given a complete graph G=(V,E), a weight function on its edges, and a penalty function on its vertices, the penalized k-min-sum problem is the problem of finding a partition of V to k+1 sets, S1,...,Sk+1, minimizing , where for , and p(S)=[summation operator]i[set membership, variant]Spi. Our main result is a randomized approximation scheme for the metric version of the penalized 1-min-sum problem, when the ratio between the minimal and maximal penalty is bounded. For the metric penalized k-min-sum problem where k is a constant, we offer a 2-approximation.

[1]  Refael Hassin,et al.  The minimum generalized vertex cover problem , 2003, TALG.

[2]  Claire Mathieu,et al.  A Randomized Approximation Scheme for Metric MAX-CUT , 2001, J. Comput. Syst. Sci..

[3]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[4]  Oded Regev.  Improved Inapproximability of Lattice and Coding Problems With Preprocessing , 2004, IEEE Trans. Inf. Theory.

[5]  Dana Ron,et al.  Property testing and its connection to learning and approximation , 1998, JACM.

[6]  Mihalis Yannakakis,et al.  Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications , 1996, SIAM J. Comput..

[7]  Teofilo F. Gonzalez,et al.  P-Complete Approximation Problems , 1976, J. ACM.

[8]  Amit Kumar,et al.  Scheduling with Outliers , 2009, APPROX-RANDOM.

[9]  Refael Hassin,et al.  Approximation Algorithms for Min-sum p-clustering , 1998, Discret. Appl. Math..

[10]  Refael Hassin,et al.  Approximation algorithms for maximum dispersion , 1997, Oper. Res. Lett..

[11]  Marek Karpinski,et al.  Approximation schemes for clustering problems , 2003, STOC '03.

[12]  Marek Karpinski,et al.  Approximation schemes for Metric Bisection and partitioning , 2004, SODA '04.

[13]  S. Safra,et al.  On the hardness of approximating minimum vertex cover , 2005 .

[14]  Subhash Khot,et al.  Vertex cover might be hard to approximate to within 2-/spl epsiv/ , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[15]  Piotr Indyk.  A sublinear time approximation scheme for clustering in metric spaces , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[16]  Dorit S. Hochbaum,et al.  Solving integer programs over monotone inequalities in three variables: A framework for half integrality and good approximations , 2002, Eur. J. Oper. Res..

[17]  Samir Khuller,et al.  Algorithms for facility location problems with outliers , 2001, SODA '01.

[18]  Yen-Liang Chen,et al.  An overlapping cluster algorithm to provide non-exhaustive clustering , 2006, Eur. J. Oper. Res..

[19]  Ke Chen,et al.  A constant factor approximation algorithm for k-median clustering with outliers , 2008, SODA '08.

[20]  Uriel Feige,et al.  The Dense k -Subgraph Problem , 2001, Algorithmica.

[21]  Jinhui Xu,et al.  An LP rounding algorithm for approximating uncapacitated facility location problem with penalties , 2005, Inf. Process. Lett..