Packing random items of three colors

Abstract

Consider three colors 1,2,3, and forj≤3, considern items (X i,j)i≤n of colorj. We want to pack these items inn bins of equal capacity (the bin size is not fixed, and is to be determined once all the objects are known), subject to the condition that each bin must contain exactly one item of each color, and that the total item sizes attributed to any given bin does not exceed the bin capacity. Consider the stochastic model where the random variables (X i,jj)i≤n,j≤3 are independent uniformly distributed over [0,1]. We show that there is a polynomial-time algorithm that produces a packing which has a wasted space≤K logn with overwhelming probability.

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Work partially supported by an N.S.F. grant.

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Rhee, W.T., Talagrand, M. Packing random items of three colors. Combinatorica 12, 331–350 (1992). https://doi.org/10.1007/BF01285822

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AMS subject classification code (1991)

  • 60 D 05
  • 90 C 27