Exponentially faster fixed-parameter algorithms for high-multiplicity scheduling

Abstract

We consider so-called N -fold integer programs (IPs) of the form max{c T x : Ax = b, ℓ ≤ x ≤ u, x ∈ Z nt }, where A ∈ Z (r+sn)×nt consists of n arbitrary matrices A (i) ∈ Z r×t on a horizontal, and n arbitrary matrices B (j) ∈ Z s×t on a diagonal line. Several recent works design fixed-parameter algorithms for N -fold IPs by taking as parameters the numbers of rows and columns of the A-and B-matrices, together with the largest absolute value ∆ over their entries. These advances provide fast algorithms for several well-studied combinatorial optimization problems on strings, on graphs, and in machine scheduling.In this work, we extend this research by proposing algorithms that additionally harness a partition structure of submatrices A (i) and B (j) , where row indices of non-zero entries do not overlap between any two sets in the partition. Our main result is an algorithm for solving any N -fold IP in time nt log(nt)L(S A ) O(r+s) (p A p B ∆) O(rpApB +spApB ) , where p A and p B are the size of the largest set in such a partition of A (i) and B (j) , respectively, S A is the number of parts in the partition of A = (A (1) , . . . , A (n) ), andWe show that these new structural parameters are naturally small in high-multiplicity scheduling problems, such as makespan minimization on related and unrelated machines, with and without release times, the Santa Claus objective, and the weighted sum of completion times. In essence, we obtain algorithms that are exponentially faster than previous works by Knop et al. (ESA 2017) and Eisenbrand et al./Koutecký et al. (ICALP 2018) in terms of the number of job types. We also improve the complexity of minimum sum coloring on graphs of bounded neighborhood diversity.

Impact of Liquid Crystal Based Interference Mitigation and Precoding on the Multiuser Performance of VLC Massive MIMO Arrays

“…In our case, the K receivers are the children, the N light sources are the presents and the channel gain h k,n is the value of present n for child k. As the Santa Claus problem is NPhard, an optimal solution cannot be calculated in reasonable time. In [34], [35], the authors present exact or approximate algorithms, however, they take long processing times even in small instances, so they are not suited for our application. Therefore, we suggest to use a heuristic algorithm, which computes a solution quickly but does not claim to find an optimal solution.…”