Inductive Linearization for Binary Quadratic Programs with Linear Constraints: A Computational Study

The computational performance of inductive linearizations for binary quadratic programs in combination with a mixed-integer programming solver is investigated for several combinatorial optimization problems and established benchmark instances. Apparently, a few of these are solved to optimality for the first time

Exact Facetial Odd-Cycle Separation for Maximum Cut and Binary Quadratic Optimization

The exact solution of the NP-hard Maximum Cut Problem is important in many applications across, e.g., Physics, Chemistry, Neuroscience, and Circuit Layout – which is also due to its equivalence to the unconstrained Binary Quadratic Optimization Problem. Leading solution methods are based on linear or semidefinite programming, and require the separation of the so-called odd-cycle inequalities. In their groundbreaking work, F. Barahona and A.R. Mahjoub have given an informal description of a polynomial-time algorithm for this problem. As pointed out recently, however, additional effort is necessary to guarantee that the obtained inequalities correspond to facets of the cut polytope. In this paper, we shed more light on a so enhanced separation procedure and investigate experimentally how it performs in comparison to an ideal setting where one could even employ the sparsest, most violated, or geometrically most promising facet-defining odd-cycle inequalities

Inductive linearization for binary quadratic programs with linear constraints

A linearization technique for binary quadratic programs (BQPs) that comprise linear constraints is presented. The technique, called “inductive linearization”, extends concepts for BQPs with particular equation constraints, that have been referred to as “compact linearization” before, to the general case. Quadratic terms may occur in the objective function, in the set of constraints, or in both. For several relevant applications, the linear programming relaxations obtained from applying the technique are proven to be at least as strong as the one obtained with a well-known classical linearization. It is also shown how to obtain an inductive linearization automatically. This might be used, e.g., by general-purpose mixed-integer programming solvers

A Natural Quadratic Approach to the Generalized Graph Layering Problem

We propose a new exact approach to the generalized graph layering problem that is based on a particular quadratic assignment formulation. It expresses, in a natural way, the associated layout restrictions and several possible objectives, such as a minimum total arc length, minimum number of reversed arcs, and minimum width, or the adaptation to a specific drawing area. Our computational experiments show a competitive performance compared to prior exact models

Performance of a Quantum Annealer for Ising Ground State Computations on Chimera Graphs

Quantum annealing is getting increasing attention in combinatorial optimization. The quantum processing unit by D-Wave is constructed to approximately solve Ising models on so-called Chimera graphs. Ising models are equivalent to quadratic unconstrained binary optimization (QUBO) problems and maximum cut problems on the associated graphs. We have tailored branch-and-cut as well as semidefinite programming algorithms for solving Ising models for Chimera graphs to provable optimality and use the strength of these approaches for comparing our solution values to those obtained on the current quantum annealing machine D-Wave 2000Q. This allows for the assessment of the quality of solutions produced by the D-Wave hardware. It has been a matter of discussion in the literature how well the D-Wave hardware performs at its native task, and our experiments shed some more light on this issue

Generalized Hose uncertainty in single-commodity robust network design

Single-commodity network design considers an edge-weighted, undirected graph with a supply/demand value at each node. It asks for minimum weight capacities such that each node can exactly send (or receive) its supply (or demand). In the robust variant, the supply or demand values may assume any realization in a given uncertainty set. One popular set is the well-known Hose polytope, which specifies an interval for the supply/demand at each node, while ensuring that the total supply and demand are balanced across the whole network. While previous work has established the Hose uncertainty set as a tractable choice, it can yield unnecessarily expensive solutions because it admits many unlikely supply and demand scenarios. In this paper, we propose a generalization of the Hose polytope that more realistically captures existing interdependencies among nodes in real life networks, and we show how to extend the state-of-the-art cutting plane algorithm for solving the single-commodity robust network design problem in view of this new uncertainty set. Our computational studies across multiple robust network design instances illustrate that the new set can provide significant cost savings without sacrificing numerical tractability

Odd-Cycle Separation for Maximum Cut and Binary Quadratic Optimization

Solving the NP-hard Maximum Cut or Binary Quadratic Optimization Problem to optimality is important in many applications including Physics, Chemistry, Neuroscience, and Circuit Layout. The leading approaches based on linear/semidefinite programming require the separation of so-called odd-cycle inequalities for solving relaxations within their associated branch-and-cut frameworks. In their groundbreaking work, F. Barahona and A.R. Mahjoub have given an informal description of a polynomial-time separation procedure for the odd-cycle inequalities. Since then, the odd-cycle separation problem has broadly been considered solved. However, as we reveal, a straightforward implementation is likely to generate inequalities that are not facet-defining and have further undesired properties. Here, we present a more detailed analysis, along with enhancements to overcome the associated issues efficiently. In a corresponding experimental study, it turns out that these are worthwhile, and may speed up the solution process significantly

Compact Linearization for Binary Quadratic Problems Comprising Linear Constraints

In this paper, the compact linearization approach originally proposed for binary quadratic programs with assignment constraints is generalized to such programs with arbitrary linear equations and inequalities that have positive coefficients and right hand sides. Quadratic constraints may exist in addition, and the technique may as well be applied if these impose the only nonlinearities, i.e., the objective function is linear. We present special cases of linear constraints (along with prominent combinatorial optimization problems where these occur) such that the associated compact linearization yields a linear programming relaxation that is provably as least as strong as the one obtained with a classical linearization method. Moreover, we show how to compute a compact linearization automatically which might be used, e.g., by general-purpose mixed-integer programming solvers