Tight complexity lower bounds for integer linear programming with few constraints

Abstract

We consider the standard ILP Feasibility problem: given an integer linear program of the form {Ax = b, x 0}, where A is an integer matrix with k rows and columns, x is a vector of variables, and b is a vector of k integers, we ask whether there exists x ∈ N that satisfies Ax = b. Each row of A specifies one linear constraint on x; our goal is to study the complexity of ILP Feasibility when both k, the number of constraints, and A ∞, the largest absolute value of an entry in A, are small.Papadimitriou [32] was the first to give a fixed-parameter algorithm for ILP Feasibility under parameterization by the number of constraints that runs in time ((. This was very recently improved by Eisenbrand and Weismantel [11], who used the Steinitz lemma to design an algorithm with running time (k∞ , which was subsequently improved by Jansen and Rohwedder [19] 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 k) would contradict the Exponential Time Hypothesis (ETH).This improves previous non-tight lower bounds of Fomin et al. [12].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 tdD(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. [26] that ILP Feasibility can be solved in time 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 + ) O(1) would contradict the ETH.

An Algorithmic Theory of Integer Programming

“…Finally, we show double-exponential lower bounds for (IP) with parameters td P (A) or td D (A) based on the Exponential Time Hypothesis (ETH) (Theorems 114 and 110). No such bound was known for td P (A), and our bound for td D (A) improves and refines the recent lower bound of Knop et al [44].…”

“…Jansen, Lassota, and Rohwedder [36] showed a near-linear time algorithm for n-fold IP, which has a slightly better parameter dependence but slightly worse dependence on n when compared with our algorithms, and only applies to the case of (ILP) while our algorithm also solves (IP) and generalizes to tree-fold IP. Knop, Pilipczuk, and Wrochna [44] gave lower bounds for (ILP) with few rows and also (ILP) parameterized by td D (A); our lower bound of Theorem 114 generalizes their latter bound. Klein [40] proved a lemma (Proposition 27) which allowed him to give a double-exponential (in terms of the parameters) algorithm for 2-stage stochastic IP, when prior work had no concrete bounds on the parameter dependence.…”

“…Our definition of K allows us to recover the currently best known upper bounds on д 1 (A) from Lemma 28. Specifically, Knop et al [44,Lemma 10] show that д 1 (A) ≤ (2∥A∥ ∞ +1) 2 td D (A) −1 . This pertains to the worst case when th(F ) = height(F ) = td D (A).…”

“…, a n , b) of Subset Sum is balanced if the encoding length of b is roughly n, i.e., if n ∈ Θ(log 2 b). We will use the following ETH-based lower bound for Subset Sum: Proposition 106 ( [44]). Unless ETH fails, there is no algorithm for Subset Sum which would solve every balanced instance in time 2 o(n+log b) .…”

Proximity results and faster algorithms for Integer Programming using the Steinitz Lemma

“…We also want to mention a recent tight lower bound for integer programming. Knop et al [23] prove that even for {0, 1}-matrices, the running time of our algorithm is probably optimal. In a nutshell, an algorithm with better asymptotic running time in the exponent for unbounded integer programs would contradict the exponential time hypothesis.…”