Partitioning Vectors into Quadruples: Worst-Case Analysis of a Matching-Based Algorithm

Consider a problem where 4k given vectors need to be partitioned into k clusters of four vectors each. A cluster of four vectors is called a quad, and the cost of a quad is the sum of the component-wise maxima of the four vectors in the quad. The problem is to partition the given 4k vectors into k quads with minimum total cost. We analyze a straightforward matching-based algorithm, and prove that this algorithm is a (3/2)-approximation algorithm for this problem. We further analyze the performance of this algorithm on a hierarchy of special cases of the problem, and prove that, in one particular case, the algorithm is a (5/4)-approximation algorithm. Our analysis is tight in all cases except one.

[1]  Dániel Gerbner,et al.  Edge-decomposition of Graphs into Copies of a Tree with Four Edges , 2012, Electron. J. Comb..

[2]  Dorit S. Hochbaum,et al.  Covering the Edges of Bipartite Graphs Using K 2, 2 Graphs , 2007, WAOA.

[3]  Frits C. R. Spieksma,et al.  Approximation Algorithms for the Wafer to Wafer Integration Problem , 2012, WAOA.

[4]  Carsten Thomassen,et al.  Edge-decompositions of highly connected graphs into paths , 2008 .

[5]  W-L. Hsu,et al.  Partitioning vectors into quadruples : Worst-case analysis of a matching-based algorithm , 2018 .

[6]  Dorit S. Hochbaum,et al.  Covering the edges of bipartite graphs using K2, 2 graphs , 2010, Theor. Comput. Sci..

[7]  Tao Jiang,et al.  Approximate Clustering of Fingerprint Vectors with Missing Values , 2005, CATS.

[8]  Fuji Zhang,et al.  On the number of perfect matchings of line graphs , 2013, Discret. Appl. Math..

[9]  David P. Williamson,et al.  The Design of Approximation Algorithms , 2011 .

[10]  Ian Holyer,et al.  The NP-Completeness of Some Edge-Partition Problems , 1981, SIAM J. Comput..

[11]  Sherief Reda,et al.  Maximizing the Functional Yield of Wafer-to-Wafer 3-D Integration , 2009, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[12]  Leonard J. Schulman,et al.  The Vector Partition Problem for Convex Objective Functions , 2001, Math. Oper. Res..

[13]  Gerhard Reinelt,et al.  On partitioning the edges of graphs into connected subgraphs , 1985, J. Graph Theory.

[14]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[15]  Frits C. R. Spieksma,et al.  Multi-dimensional vector assignment problems , 2014, Discret. Optim..