We consider integer programs (IPs) whose constraint matrix has a special block structure. More precisely, we consider IP: min{f (x) : Hcomx = b, l ≤ x ≤ u, x ∈ Z t B +nt A }, in which the objective function f is separable convex and the constraint matrix Hcom is composed of small submatrices Ai, B, C, Di such that the first row of Hcom is (C, D1, D2, . . . , Dn), the first column of Hcom is (C, B, B, . . . , B) ⊤ , the main diagonal of Hcom is (C, A1, A2, . . . , An), and the rest entries are 0. Furthermore, the rank of submatrix B is 1.We study fixed parameter tractable (FPT) algorithms by taking as parameters the number of rows and columns of small submatrices, together with the largest absolute value over their entries.We call the IP studied (almost) combinatorial 4-block n-fold IP. It generalizes n-fold IP and is meanwhile a special case of 4-block n-fold IP. In the literature, existing FPT algorithms for block-structured IP rely on bounding the ℓ1-or ℓ∞-norm of elements of the Graver basis. The existence of FPT algorithms for 4-block n-fold IP is a major open problem and Chen et al. [ESA 2020] showed that the ℓ∞-norm of the Graver basis elements of 4-block n-fold IP is Ω(n). This motivates us to study special cases of 4-block n-fold IP to find structural insights.We show that, the ℓ∞-norm of the Graver basis elements of combinatorial 4-block n-fold IP is also Ω(n). However, there exists some FPT-value λ such that for any nonzero element g ∈ {x : Hcomx = 0}, λg can always be decomposed into Graver basis elements in the same orthant whose ℓ∞-norm is FPT-bounded (while g itself might not admit such a decomposition). This seems to exhibit an "intermediate" phenomenon. Based on this, we are able to bound the ℓ∞-norm of Graver basis elements for combinatorial 4-block n-fold IP by OF P T (n) and develop OF P T (n 3+o(1) )-time algorithms (here the OF P T hides a multiplicative FPT-term).As applications, we show that combinatorial 4-block n-fold IP can be used to model important generalizations of the classical scheduling problems, including scheduling with rejection and bicriteria scheduling, which implies that our FPT algorithm establishes a general framework to settle the classical scheduling problems.
ACM Subject ClassificationTheory of computation → Fixed parameter tractability Keywords and phrases 4-block n-fold IP, Fixed parameter tractable, Scheduling, Integer programming Digital Object Identifier 10.4230/LIPIcs.CVIT.2016.27 Acknowledgements I want to thank . . .