Sparse Integer Programming is FPT

We report on major progress in integer programming in variable dimension, asserting that the problem, with linear or separable-convex objective, is fixed-parameter tractable parameterized by the numeric measure and sparsity measure of the defining matrix. Integer linear programming, with data w, l, u ∈ Z n , A ∈ Z m×n , and b ∈ Z m , is the problem minwx : Ax = b, l ≤ x ≤ u, x ∈ Z n . (1) It has a very broad expressive power and numerous applications, but is generally NP-hard. A well known result [4] asserts that integer linear programming is fixed-parameter tractable (see [1]) when parameterized by the dimension (number of variables) n, but this does not help in typical situations where the dimension is large and forms a variable part of the input. Here we report on a recent powerful result in integer programming in variable dimension, asserting that the problem is fixed-parameter tractable when parameterized by the numeric measure a := A ∞ := max i,j |A i,j | and the sparsity measure d := mintd(A), td(A T) of A. Here td(A) is the tree-depth of A, defined below, and A T is the transpose. The result holds more generally for integer nonlinear programming where the objective function is separable-convex, that is, of the form f (x) = n i=1 f i (x i) where each f i is a univariate convex function which takes on integer values on integer arguments and which is given by an evaluation oracle. Below we denote by L := log(u − l ∞ + 1) the bit complexity of the lower and upper bounds, and the times are in terms of the number of arithmetic operations and oracle queries. Theorem The linear or separable-convex program (2) is fixed-parameter tractable on a, d; and if d = td(A T) and is fixed, it is polynomial time solvable even if unary encoded a is variable: minf (x) : Ax = b, l ≤ x ≤ u, x ∈ Z n . (2) More specifically, there exist computable functions h 1 and h 2 such that the following hold: 1. [3] When f (x) = wx is linear, the problem is solvable in fixed-parameter tractable time h 1 (a, d)poly(n) if d = td(A) and (a + 1) h 2 (d) poly(n) if d = td(A T) ; 2. [2] When f (x) is separable-convex, it is solvable in fixed-parameter tractable time h 1 (a, d)poly(n)L if d = td(A) and (a + 1) h 2 (d) poly(n)L if d = td(A T). The theorem concerns sparse integer programming in the sense that at least one of A and A T has small tree-depth, a parameter which plays a central role in sparsity, see [5], and which is defined as follows. The height of a rooted tree is the maximum number of vertices on a path from the root to a leaf. Given a graph G = (V, E), a rooted tree on V is valid for G if for each edge j, k ∈ E one of j, k lies on the path from the root to the other. The tree-depth td(G) of G is the smallest height of a rooted tree which is valid for G. The graph of an m × n matrix A is the graph G(A) on [n] where j, k is an edge if and only if there is an i ∈ [m] such that A i,j A i,k = 0. The tree-depth of A is the tree-depth td(A) := td(G(A)) of its graph.