FPT algorithms for a special block-structured integer program with applications in scheduling

Abstract

In this paper, a special case of the generalized 4-block n-fold IPs is investigated, where \(B_i=B\) and B has a rank at most 1. Such IPs, called almost combinatorial 4-block n-fold IPs, include the generalized n-fold IPs as a subcase. We are interested in fixed parameter tractable (FPT) algorithms by taking as parameters the dimensions of the blocks and the largest coefficient. For almost combinatorial 4-block n-fold IPs, we first show that there exists some \(\lambda \le g(\gamma )\) such that for any nonzero kernel element \({\textbf{g}}\) , \(\lambda {\textbf{g}}\) can always be decomposed into kernel elements in the same orthant whose \(\ell _{\infty }\) -norm is bounded by \(g(\gamma )\) (while \({\textbf{g}}\) itself might not admit such a decomposition), where g is a computable function and \(\gamma \) is an upper bound on the dimensions of the blocks and the largest coefficient. Based on this, we are able to bound the \(\ell _{\infty }\) -norm of Graver basis elements by \({\mathcal {O}}(g(\gamma )n)\) and develop an \({\mathcal {O}}(g(\gamma )n^{3+o(1)}\hat{L}^2)\) -time algorithm (here \(\hat{L}\) denotes the logarithm of the largest absolute value occurring in the input). Additionally, we show that the \(\ell _{\infty }\) -norm of Graver basis elements is \(\varOmega (n)\) . As applications, almost combinatorial 4-block n-fold IPs can be used to model generalizations of classical problems, including scheduling with rejection, bi-criteria scheduling, and a generalized delivery problem. Therefore, our FPT algorithm establishes a general framework to settle these problems.