Vector Balancing Games with Aging

In this article we study an extension of the vector balancing game investigated by Spencer and Olson (which corresponds to the on-line version of the discrepancy problem for matrices). We assume that decisions in earlier rounds become less and less important as the game continues. For an aging parameter q?1 we define the current move to be q times more important than the previous one. We consider two variants of this problem: First, the objective is a balanced partition at the end of the game, and second, it is to ensure a balanced partition throughout the game. We concentrate on the case q?2. We give an optimal solution for the first problem and a nearly optimal one for the second.

[1]  László Lovász,et al.  Discrepancy of Set-systems and Matrices , 1986, Eur. J. Comb..

[2]  Imre Bárány,et al.  On a Class of Balancing Games , 1979, J. Comb. Theory A.

[3]  John E. Olson,et al.  A Balancing Strategy , 1985, J. Comb. Theory, Ser. A.

[4]  J. Spencer.  Ten lectures on the probabilistic method , 1987 .

[5]  J. Beck,et al.  Discrepancy Theory , 1996 .

[6]  I. Bárány,et al.  On some combinatorial questions in finite-dimensional spaces , 1981 .

[7]  Joel H. Spencer,et al.  Randomization, Derandomization and Antirandomization: Three Games , 1994, Theor. Comput. Sci..

[8]  Benjamin Doerr.  Linear And Hereditary Discrepancy , 2000, Comb. Probab. Comput..

[9]  Catherine H. Yan,et al.  Balancing Game with a Buffer , 1998 .

[10]  Joel H. Spencer,et al.  Integral approximation sequences , 1984, Math. Program..

[11]  R. Graham,et al.  Handbook of Combinatorics , 1995 .