Vector Bin Packing

Abstract

Years and Authors of Summarized Original Work1976; Garey, Graham, Johnson, Yao1977; Kou, Markowski1977; Maruyama, Chang, Tang1981; de la Vega, Lueker1987; Yao1990; Csirik, Frenk, Labbé, Zhang1997; Woeginger2001; Caprara, Toth2004; Chekuri, Khanna2009; Bansal, Caprara, Sviridenko2010; Stillwell, Schanzenback, Vivien, Casanova2011; Panigrahy, Talwar, Uyeda, WiederProblem DefinitionIn the vector bin packing problem, we are given an integral dimension d ≥ 1 and a list L=(x1,x2,…,xn)L=(x1,x2,…,xn)L = (x_{1},x_{2},\ldots ,x_{n}) of items, where each item is a d-dimensional tuple xi = (xi,1,xi,2,…,xi,d)(xi,1,xi,2,…,xi,d)(x_{i,1},x_{i,2},\ldots ,x_{i,d}) with rational entries xi, j ∈ [0, 1]. The goal is to assign the items to a minimum number of multidimensional bins, where if X is the set of items assigned to a bin, we must have, for each j, 1 ≤ j ≤ d, ∑xi∈Xxi,j≤1.∑xi∈Xxi,j≤1.\displaystyle\sum _{x_{i}\in X}x_{i,j} \leq 1. Note that when d = 1, the vector bin packing problem reduces to the classic (one-dimensional) bin packing problem.One potential application of the ...