Sparse Integer Programming Is Fixed-Parameter Tractable

We study the general integer programming problem where the number of variables n is a variable part of the input. We consider two natural parameters of the constraint matrix A: its numeric measure a and its sparsity measure d. We present an algorithm for solving integer programming in time [Formula: see text], where g is some computable function of the parameters a and d, and L is the binary encoding length of the input. In particular, integer programming is fixed-parameter tractable parameterized by a and d, and is solvable in polynomial time for every fixed a and d. Our results also extend to nonlinear separable convex objective functions. Funding: F. Eisenbrand, C. Hunkenschröder, and K.-M. Klein were supported by the Swiss National Science Foundation (SNSF) within the project “Convexity, geometry of numbers, and the complexity of integer programming” [Grant 163071]. A. Levin and S. Onn are partially supported by the Israel Science Foundation [Grant 308/18]. A. Levin is also partially supported by the Israel Science Foundation [Grant 1467/22]. S. Onn is also partially supported by the Dresner Chair at the Technion. M. Koutecký is partially supported by Charles University project UNCE 24/SCI/008, and by the project 22-22997S of the Grantová Agentura České Republiky (GA ČR).