# Cologne Excellence Cluster on Cellular Stress Responses in Aging Associated Diseases

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### 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

### 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

### Linear Ordering Based MIP Formulations for the Vertex Separation or Pathwidth Problem

We consider the task to compute the pathwidth of a graph which is known to be equivalent to solving the vertex separation problem. The latter is naturally modeled as a linear ordering problem with respect to the vertices of the graph. Present mixed-integer programs for the vertex separation problem express linear orders using either position or set assignment variables. However, as we show, the lower bound on the pathwidth obtained when solving their linear programming relaxations is zero for any directed graph. This is one reason for their limited utility in solving larger instances to optimality. We then present a new formulation that is based on conventional linear ordering variables and a slightly different perspective on the problem. Its relaxation provably delivers non-zero lower bounds for any graph whose pathwidth is non-zero. Further structural results for and extensions to this formulation are discussed. Finally, an experimental evaluation of three mixed-integer programs, each representing one of the different yet existing modeling approaches, displays their potentials and limitations when used to solve the problem to optimality in practice

### A Local-Search Algorithm for Steiner Forest

In the Steiner Forest problem, we are given a graph and a collection of source-sink pairs, and the goal is to find a subgraph of minimum total length such that all pairs are connected. The problem is APX-Hard and can be 2-approximated by, e.g., the elegant primal-dual algorithm of Agrawal, Klein, and Ravi from 1995.
We give a local-search-based constant-factor approximation for the problem. Local search brings in new techniques to an area that has for long not seen any improvements and might be a step towards a combinatorial algorithm for the more general survivable network design problem. Moreover, local search was an essential tool to tackle the dynamic MST/Steiner Tree problem, whereas dynamic Steiner Forest is still wide open.
It is easy to see that any constant factor local search algorithm requires steps that add/drop many edges together. We propose natural local moves which, at each step, either (a) add a shortest path in the current graph and then drop a bunch of inessential edges, or (b) add a set of edges to the current solution. This second type of moves is motivated by the potential function we use to measure progress, combining the cost of the solution with a penalty for each connected component. Our carefully-chosen local moves and potential function work in tandem to eliminate bad local minima that arise when using more traditional local moves.
Our analysis first considers the case where the local optimum is a single tree, and shows optimality w.r.t. moves that add a single edge (and drop a set of edges) is enough to bound the locality gap. For the general case, we show how to "project" the optimal solution onto the different trees of the local optimum without incurring too much cost (and this argument uses optimality w.r.t. both kinds of moves), followed by a tree-by-tree argument. We hope both the potential function, and our analysis techniques will be useful to develop and analyze local-search algorithms in other contexts