Exponentially faster fixed-parameter algorithms for high-multiplicity scheduling
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We consider so-called N-fold integer programs (IPs) of the form max{c^T x : Ax = b, ℓ≤ x ≤ u, x ∈ℤ^nt}, whereA ∈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 forN-fold IPs by taking as parameters the numbers of rows and columns of theA- andB-matrices, together with the largest absolute valueΔover 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 submatricesA^(i)andB^(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 anyN-fold IP in timent log(nt)L^2(S_A)^O(r+s)(p_Ap_BΔ)^O(rp_Ap_B+sp_Ap_B), wherep_Aandp_Bare the size of the largest set in such a partition ofA^(i)andB^(j), respectively,S_Ais the number of parts in the partition ofA = (A^(1),..., A^(n)), and L = (log(||u - ℓ||_∞)· (log(max_x:ℓ≤ x ≤ u |c^Tx|)) is a measure of the input. We 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.
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