(Near)-Optimal Algorithms for Sparse Separable Convex Integer Programs

We study the general integer programming (IP) problem of optimizing a separable convex function over the integer points of a polytope: min{f(x)∣Ax=b,l≤x≤u,x∈Zⁿ}. The number of variables n is a variable part of the input, and we consider the regime where the constraint matrix A has small coefficients ‖A‖_∞ and small primal or dual treedepth td_P(A) or td_D(A), respectively. Equivalently, we consider block-structured matrices, in particular n-fold, tree-fold, 2-stage and multi-stage matrices. We ask about the possibility of near-linear algorithms in the general case of (non-linear) separable convex functions. The techniques of previous works for the linear case are inherently limited to it; in fact, no strongly-polynomial algorithm may exist due to a simple unconditional information-theoretic lower bound of nlog‖u-l‖_∞, where l,u are the vectors of lower and upper bounds. Our first result is that with parameters td_P(A) and ‖A‖_∞, this lower bound can be matched (up to dependency on the parameters). Second, with parameters td_D(A) and ‖A‖_∞, the situation is more involved, and we design an algorithm with complexity g(td_D(A),‖A‖_∞)nlognlog‖u-l‖_∞ where g is some computable function. We conjecture that a stronger lower bound is possible in this regime, and our algorithm is in fact optimal. Our algorithms combine ideas from scaling, proximity, and sensitivity of integer programs, together with a new dynamic data structure allowing fast sparse updates.