Markus Sinnl

Thinning out Steiner trees: a node-based model for uniform edge costs

The Steiner tree problem is a challenging NP-hard problem. Many hard instances of this problem are publicly available, that are still unsolved by state-of-the-art branch-and-cut codes. A typical strategy to attack these instances is to enrich the polyhedral description of the problem, and/or to implement more and more sophisticated separation procedures and branching strategies. In this paper we investigate the opposite viewpoint, and try to make the solution method as simple as possible while working on the modeling side. Our working hypothesis is that the extreme hardness of some classes of instances mainly comes from over-modeling, and that some instances can become quite easy to solve when a simpler model is considered. In other words, we aim at “thinning out” the usual models for the sake of getting a more agile framework. In particular, we focus on a model that only involves node variables, which is rather appealing for the “uniform” cases where all edges have the same cost. In our computational study, we first show that this new model allows one to quickly produce very good (sometimes proven optimal) solutions for notoriously hard instances from the literature. In some cases, our approach takes just few seconds to prove optimality for instances never solved (even after days of computation) by the standard methods. Moreover, we report improved solutions for several SteinLib instances, including the (in)famous hypercube ones. We also demonstrate how to build a unified solver on top of the new node-based model and the previous state-of-the-art model (defined in the space of arc and node variables). The solver relies on local branching, initialization heuristics, preprocessing and local search procedures. A filtering mechanism is applied to automatically select the best algorithmic ingredients for each instance individually. The presented solver is the winner of the DIMACS Challenge on Steiner trees in most of the considered categories.

Redesigning Benders Decomposition for Large-Scale Facility Location

The uncapacitated facility location (UFL) problem is one of the most famous and most studied problems in the operations research literature. Given a set of potential facility locations and a set of customers, the goal is to find a subset of facility locations to open and to allocate each customer to open facilities so that the facility opening plus customer allocation costs are minimized. In our setting, for each customer the allocation cost is assumed to be a linear or separable convex quadratic function. Motivated by recent UFL applications in business analytics, we revise approaches that work on a projected decision space and hence are intrinsically more scalable for large-scale input data. Our working hypothesis is that many of the exact (decomposition) approaches that were proposed decades ago and discarded soon after need to be redesigned to take advantage of the new hardware and software technologies. To this end, we “thin out” the classical models from the literature and use (generalized) Benders ...

Benders decomposition without separability: a computational study for capacitated facility location problems

Benders is one of the most famous decomposition tools for Mathematical Programming, and it is the method of choice e.g., in mixed-integer stochastic programming. Its hallmark is the capability of decomposing certain types of models into smaller subproblems, each of which can be solved individually to produce local information (notably, cutting planes) to be exploited by a centralized “master” problem. As its name suggests, the power of the technique comes essentially from the decomposition effect, i.e., the separability of the problem into a master problem and several smaller subproblems. In this paper we address the question of whether the Benders approach can be useful even without separability of the subproblem, i.e., when its application yields a single subproblem of the same size as the original problem. In particular, we focus on the capacitated facility location problem, in two variants: the classical linear case, and a “congested” case where the objective function contains convex but non-separable quadratic terms. We show how to embed the Benders approach within a modern branch-and-cut mixed-integer programming solver, addressing explicitly all the ingredients that are instrumental for its success. In particular, we discuss some computational aspects that are related to the negative effects derived from the lack of separability. Extensive computational results on various classes of instances from the literature are reported, with a comparison with the state-of-the-art exact and heuristic algorithms. The outcome is that a clever but simple implementation of the Benders approach can be very effective even without separability, as its performance is comparable and sometimes even better than that of the most effective and sophisticated algorithms proposed in the previous literature.

Intersection Cuts for Bilevel Optimization

The exact solution of bilevel optimization problems is a very challenging task that received more and more attention in recent years, as witnessed by the flourishing recent literature on this topic. In this paper we present ideas and algorithms to solve to proven optimality generic Mixed-Integer Bilevel Linear Programs MIBLPs where all constraints are linear, and some/all variables are required to take integer values. In doing so, we look for a general-purpose approach applicable to any MIBLP under mild conditions, rather than ad-hoc methods for specific cases. Our approach concentrates on minimal additions required to convert an effective branch-and-cut MILP exact code into a valid MIBLP solver, thus inheriting the wide arsenal of MILP tools cuts, branching rules, heuristics available in modern solvers.

On the Two-Architecture Connected Facility Location Problem

We introduce a new variant of the connected facility location problem that allows for modeling mixed deployment strategies (FTTC/FTTB/FTTH) in the design of local access telecommunication networks. Several mixed integer programming models and valid inequalities are presented. Computational studies on realistic instances from three towns in Germany are provided.

A Computational Study of Exact Approaches for the Bi-Objective Prize-Collecting Steiner Tree Problem

We introduce the bi-objective prize-collecting Steiner tree problem, whose goal is to find a subtree considering the conflicting objectives of minimizing the edge costs for building that tree, and maximizing the collected node revenues. We consider five iterative mixed-integer programming (MIP) frameworks that identify the complete Pareto front, i.e., one efficient solution for every point on the Pareto front. More precisely, the following methods are studied: an e-constraint method, a two-phase method, a binary search in the objective space, a weighted Chebyshev norm method, and a method of Sylva and Crema. We also investigate how to exploit and recycle information gained during these iterative MIP procedures to accelerate the solution process. We consider (i) additional strengthening valid inequalities, (ii) procedures for initializing feasible solutions (using a solution pool), (iii) procedures for recycling violated cuts (using a cut pool), and (iv) guiding the branching process by previously detected...

Alteration of Golgi Structure by Stress: A Link to Neurodegeneration?

The Golgi apparatus is well-known for its role as a sorting station in the secretory pathway as well as for its role in regulating post-translational protein modification. Another role for the Golgi is the regulation of cellular signaling by spatially regulating kinases, phosphatases, and GTPases. All these roles make it clear that the Golgi is a central regulator of cellular homeostasis. The response to stress and the initiation of adaptive responses to cope with it are fundamental abilities of all living cells. It was shown previously that the Golgi undergoes structural rearrangements under various stress conditions such as oxidative or osmotic stress. Neurodegenerative diseases are also frequently associated with alterations of Golgi morphology and many stress factors have been described to play an etiopathological role in neurodegeneration. It is however unclear whether the stress-Golgi connection plays a role in neurodegenerative diseases. Using a combination of bioinformatics modeling and literature mining, we will investigate evidence for such a tripartite link and we ask whether stress-induced Golgi arrangements are cause or consequence in neurodegeneration.

A Dual Ascent-Based Branch-and-Bound Framework for the Prize-Collecting Steiner Tree and Related Problems

We present a branch-and-bound (B&B) framework for the asymmetric prize-collecting Steiner tree problem (APCSTP). Several well-known network design problems can be transformed to the APCSTP, including the Steiner tree problem (STP), prize-collecting Steiner tree problem (PCSTP), maximum-weight connected subgraph problem (MWCS), and node-weighted Steiner tree problem (NWSTP). The main component of our framework is a new dual ascent algorithm for the rooted APCSTP, which generalizes Wong’s dual ascent algorithm for the Steiner arborescence problem. The lower bounds and dual information obtained from the algorithm are exploited within powerful bound-based reduction tests and for guiding primal heuristics. The framework is complemented by additional alternative-based reduction tests. Extensive computational results on benchmark instances for the PCSTP, MWCS, and NWSTP indicate the framework’s effectiveness, as most instances from literature are solved to optimality within seconds, including most of the (previo...

A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs

Bilevel optimization problems are very challenging optimization models arising in many important practical contexts, including pricing mechanisms in the energy sector, airline and telecommunication industry, transportation networks, critical infrastructure defense, and machine learning. In this paper, we consider bilevel programs with continuous and discrete variables at both levels, with linear objectives and constraints (continuous upper level variables, if any, must not appear in the lower level problem). We propose a general-purpose branch-and-cut exact solution method based on several new classes of valid inequalities, which also exploits a very effective bilevel-specific preprocessing procedure. An extensive computational study is presented to evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature and 60 randomly generated instances. Our new algorithm consistently outperforms (often by a large margin) alternative state-of-the-art metho...

An algorithmic framework for the exact solution of tree-star problems

Many problems arising in the area of telecommunication ask for solutions with a tree-star topology. This paper proposes a general procedure for finding optimal solutions to a family of these problems. The family includes problems in the literature named as connected facility location, rent-or-buy and generalized Steiner tree-star. We propose a solution framework based on a branch-and-cut algorithm which also relies on sophisticated reduction and heuristic techniques. An important ingredient of this framework is a dual ascent procedure for asymmetric connected facility location. This paper shows how this procedure can be exploited in combination with various mixed integer programming formulations. Using the new framework, many benchmark instances in the literature for which only heuristic results were available so far, can be solved to provable optimality within seconds. To better assess the computational performance of the new approach, we additionally consider larger instances and provide optimal solutions for most of them too.