Randomly Rounding Rationals with Cardinality Constraints and Derandomizations

We show how to generate randomized roundings of rational vectors that satisfy hard cardinality constraints and allow large deviations bounds. This improves and extends earlier results by Srinivasan (FOCS 2001), Gandhi et al. (FOCS 2002) and the author (STACS 2006). Roughly speaking, we show that also for rounding arbitrary rational vectors randomly or deterministically, it suffices to understand the problem for {0, 1/2} vectors (which typically is much easier). So far, this was only known for vectors with entries in 2-lZ, l ∈ N. To prove the general case, we exhibit a number of results of independent interest, in particular, a quite useful lemma on negatively correlated random variables, an extension of de Werra's (RAIRO 1971) coloring result for unimodular hypergraphs and a sufficient condition for a unimodular hypergraph to have a perfectly balanced non-trivial partial coloring. We also show a new solution for the general derandomization problem for rational matrices.

[1]  Anand Srivastav,et al.  Bounds and constructions for the star-discrepancy via ?-covers , 2005, J. Complex..

[2]  Kunihiko Sadakane,et al.  Combinatorics and Algorithms on Low-Discrepancy Roundings of a Real Sequence , 2001, International Colloquium on Automata, Languages and Programming.

[3]  Anand Srivastav,et al.  Derandomization in Combinatorial Optimization , 2001 .

[4]  Benjamin Doerr.  Non-independent randomized rounding , 2003, SODA '03.

[5]  Rajiv Gandhi,et al.  Dependent rounding in bipartite graphs , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[6]  Alan M. Frieze,et al.  A new rounding procedure for the assignment problem with applications to dense graph arrangement problems , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[7]  Aravind Srinivasan,et al.  Distributions on level-sets with applications to approximation algorithms , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[8]  D. de Werra.  Equitable colorations of graphs , 1971 .

[9]  Benjamin Doerr,et al.  Global roundings of sequences , 2004, Inf. Process. Lett..

[10]  Benjamin Doerr.  Generating Randomized Roundings with Cardinality Constraints and Derandomizations , 2006, STACS.

[11]  Peter Stangier,et al.  Algorithmic Chernoff-Hoeffding inequalities in integer programming , 1996, Random Struct. Algorithms.

[12]  Prabhakar Raghavan,et al.  Multiterminal global routing: A deterministic approximation scheme , 2005, Algorithmica.

[13]  Maxim Sviridenko,et al.  Pipage Rounding: A New Method of Constructing Algorithms with Proven Performance Guarantee , 2004, J. Comb. Optim..

[14]  Prabhakar Raghavan,et al.  Randomized rounding: A technique for provably good algorithms and algorithmic proofs , 1985, Comb..

[15]  Benjamin Doerr,et al.  Unbiased Rounding of Rational Matrices , 2006, FSTTCS.

[16]  Benjamin Doerr.  Nonindependent Randomized Rounding and an Application to Digital Halftoning , 2004, SIAM J. Comput..

[17]  Rajiv Gandhi,et al.  An improved approximation algorithm for vertex cover with hard capacities , 2006, J. Comput. Syst. Sci..

[18]  Aravind Srinivasan,et al.  Randomized Distributed Edge Coloring via an Extension of the Chernoff-Hoeffding Bounds , 1997, SIAM J. Comput..