Tight Complexity Lower Bounds for Integer Linear Programming with Few Constraints

Abstract

We consider the standard ILP F easibility problem: given an integer linear program of the form {A x = b, x ⩾ 0}, where A is an integer matrix with k rows and ℓ columns, x is a vector of ℓ variables, and b is a vector of k integers, we ask whether there exists x ∈ N ℓ that satisfies Ax = b. Each row of A specifies one linear constraint on x; our goal is to study the complexity of ILP F easibility when both k , the number of constraints, and ‖A‖ ∞ , the largest absolute value of an entry in A , are small. Papadimitriou was the first to give a fixed-parameter algorithm for ILP F easibility under parameterization by the number of constraints that runs in time ((‖A‖ ∞ + ‖b‖ ∞ ) ⋅ k ) O ( k 2 ) . This was very recently improved by Eisenbrand and Weismantel, who used the Steinitz lemma to design an algorithm with running time ( k ‖A‖ ∞ ) O ( k ) ⋅ log ‖b‖ ∞ , which was subsequently refined by Jansen and Rohwedder to O (√ k ‖A‖ ∞ ) k ⋅ log (‖ A‖ ∞ + ‖b‖ ∞ ) ⋅ log ‖A‖ ∞ . We prove that for {0, 1}-matrices A , the running time of the algorithm of Eisenbrand and Weismantel is probably optimal: an algorithm with running time 2 o ( k log k ) ⋅ (ℓ + ‖b‖ ∞ ) o ( k ) would contradict the exponential time hypothesis. This improves previous non-tight lower bounds of Fomin et al. We then consider integer linear programs that may have many constraints, but they need to be structured in a “shallow” way. Precisely, we consider the parameter dual treedepth of the matrix A , denoted td D ( A ), which is the treedepth of the graph over the rows of A , where two rows are adjacent if in some column they simultaneously contain a non-zero entry. It was recently shown by Koutecký et al. that ILP F easibility can be solved in time ‖A‖ ∞ 2 O (td D ( A )) ⋅ ( k + ℓ + log ‖b‖ ∞ ) O (1) . We present a streamlined proof of this fact and prove that, again, this running time is probably optimal: even assuming that all entries of A and b are in {−1, 0, 1}, the existence of an algorithm with running time 2 2 o (td D ( A )) ⋅ ( k + ℓ) O (1) would contradict the exponential time hypothesis.