Solving Integer Linear Programs with a Small Number of Global Variables and Constraints

Abstract

Integer Linear Programming (ILP) has a broad range of applications in various areas of artificial intelligence. Yet in spite of recent advances, we still lack a thorough understanding of which structural restrictions make ILP tractable. Here we study ILP instances consisting of a small number of "global" variables and/or constraints such that the remaining part of the instance consists of small and otherwise independent components; this is captured in terms of a structural measure we call fracture backdoors which generalizes, for instance, the well-studied class of N -fold ILP instances. Our main contributions can be divided into three parts. First, we formally develop fracture backdoors and obtain exact and approximation algorithms for computing these. Second, we exploit these backdoors to develop several new parameterized algorithms for ILP; the performance of these algorithms will naturally scale based on the number of global variables or constraints in the instance. Finally, we complement the developed algorithms with matching lower bounds. Altogether, our results paint a near-complete complexity landscape of ILP with respect to fracture backdoors.

Mixed integer programming with convex/concave constraints: Fixed-parameter tractability and applications to multicovering and voting

“…In spite of this restrictive structure, n-fold integer programming found a number of applications, e.g., in scheduling [34] and voting [38]. See also the works of Dvorák et al [15] and Knop et al [35] for very recent generalizations and extensions of the this technique. It is quite non-obvious how the technique of n-fold integer programming compares to ours.…”

Solving Integer Linear Programs by Exploiting Variable-Constraint Interactions: A Survey

“…Of course, ILP has also been studied through the lens of structural parameters that are different than treewidth. The first example of such a parameter is the fracture number of Dvořák et al [43], which captures the "distance" of an ILP instance from being fractured into small independent components.…”

“…Intuitively, this means that the fracture number can be viewed as a stronger restriction than treedepth. Dvořák et al [43] showed that the fracture number can be used to obtain XP-algorithms for ILP in settings which would remain NP-hard if treedepth were used instead (see Theorem 12). See Figure 3 for an illustration of the relationships between the different variants of fracture number as well as their relation to treewidth and treedepth.…”

Solving Integer Quadratic Programming via Explicit and Structural Restrictions

“…So far, the tractability of IQP has been explored primarily through the lens of the explicit properties listed above, but we are not aware of many results which would exploit the latter, structural properties. This contrasts with the recent developments for ILP, which saw the introduction of several new algorithms and lower bounds that take into account the specific properties of variable-variable and variable-constraint interactions in instances (Ganian, Ordyniak, and Ramanujan 2017;Dvořák et al 2017;Ganian and Ordyniak 2018;Eiben et al 2018;Jansen and Kratsch 2015;Koutecký, Levin, and Onn 2018). In all of these works, the authors captured the way variables interacted with each other via constraints through suitably defined graph representations, and the primary research question was to identify natural properties of these graphs (formalized via a structural parameter-an integer k) which allow an instance of size n to be solved in time f (k) · n O(1) for some computable function k. Note that this goal represents a stronger and more fine-grained notion of tractability than merely polynomialtime solvability for each fixed value of k; in particular, algorithms with running time of this form are called fixedparameter algorithms and are central to the parameterized complexity paradigm (Cygan et al 2013;Niedermeier 2006;Flum and Grohe 2006;Downey and Fellows 2013).…”

“…The above lower bounds (and especially the latter one) have critical implications for our endeavor. Indeed, they clearly show that any structural parameters which we hope to use to efficiently solve IQP must restrict both the objective function and the constraints; in contrast, for ILP it is often sufficient to use structural parameters merely for restricting the constraints (Ganian, Ordyniak, and Ramanujan 2017;Dvořák et al 2017;Eiben et al 2018;Jansen and Kratsch 2015;Koutecký, Levin, and Onn 2018). With this insight, we can now proceed to a high-level description of our results, divided into four settings based on explicit restrictions of instances (a quick glance of our results is also presented in Table 1).…”