PQ-type adjacency polytopes of join graphs

PQ-type adjacency polytopes $\nabla^{\rm PQ}_G$ are lattice polytopes arising from finite graphs $G$. There is a connection between $\nabla^{\rm PQ}_G$ and the engineering problem known as power-flow study, which models the balance of electric power on a network of power generation. In particular, the normalized volume of $\nabla^{\rm PQ}_G$ plays a central role. In the present paper, we focus the case where $G$ is a join graph. In fact, formulas of the $h^*$-polynomial and the normalized volume of $\nabla^{\rm PQ}_G$ of a join graph $G$ are presented. Moreover, we give explicit formulas of the $h^*$-polynomial and the normalized volume of $\nabla^{\rm PQ}_G$ when $G$ is a complete multipartite graph or a wheel graph.Comment: 18 pages, 1 figur

Relative-Interior Solution for (Incomplete) Linear Assignment Problem with Applications to Quadratic Assignment Problem

We study the set of optimal solutions of the dual linear programming formulation of the linear assignment problem (LAP) to propose a method for computing a solution from the relative interior of this set. Assuming that an arbitrary dual-optimal solution and an optimal assignment are available (for which many efficient algorithms already exist), our method computes a relative-interior solution in linear time. Since LAP occurs as a subproblem in the linear programming relaxation of quadratic assignment problem (QAP), we employ our method as a new component in the family of dual-ascent algorithms that provide bounds on the optimal value of QAP. To make our results applicable to incomplete QAP, which is of interest in practical use-cases, we also provide a linear-time reduction from incomplete LAP to complete LAP along with a mapping that preserves optimality and membership in the relative interior. Our experiments on publicly available benchmarks indicate that our approach with relative-interior solution is frequently capable of providing superior bounds and otherwise is at least comparable

Robust Assignments via Ear Decompositions and Randomized Rounding

Many real-life planning problems require making a priori decisions before all parameters of the problem have been revealed. An important special case of such problem arises in scheduling problems, where a set of tasks needs to be assigned to the available set of machines or personnel (resources), in a way that all tasks have assigned resources, and no two tasks share the same resource. In its nominal form, the resulting computational problem becomes the \emph{assignment problem} on general bipartite graphs. This paper deals with a robust variant of the assignment problem modeling situations where certain edges in the corresponding graph are \emph{vulnerable} and may become unavailable after a solution has been chosen. The goal is to choose a minimum-cost collection of edges such that if any vulnerable edge becomes unavailable, the remaining part of the solution contains an assignment of all tasks. We present approximation results and hardness proofs for this type of problems, and establish several connections to well-known concepts from matching theory, robust optimization and LP-based techniques.Comment: Full version of ICALP 2016 pape

A network flow approach to a common generalization of Clar and Fries numbers

Clar number and Fries number are two thoroughly investigated parameters of plane graphs emerging from mathematical chemistry to measure stability of organic molecules. We consider first a common generalization of these two concepts for bipartite plane graphs, and then extend it to a framework on general (not necessarily planar) directed graphs. The corresponding optimization problem can be transformed into a maximum weight feasible tension problem which is the linear programming dual of a minimum cost network flow (or circulation) problem. Therefore the approach gives rise to a min-max theorem and to a strongly polynomial algorithm that relies exclusively on standard network flow subroutines. In particular, we give the first network flow based algorithm for an optimal Fries structure and its variants

Bulk-robust assignment problems: hardness, approximability and algorithms

This thesis studies robust assignment problems with focus on computational complexity. Assignment problems are well-studied combinatorial optimization problems with numerous practical applications, for instance in production planning. Classical approaches to optimization expect the input data for a problem to be given precisely. In contrast, real-life optimization problems are modeled using forecasts resulting in uncertain problem parameters. This fact can be taken into account using the framework of robust optimization. An instance of the classical assignment problem is represented using a bipartite graph accompanied by a cost function. The goal is to find a minimum-cost assignment, i.e., a set of resources (edges or nodes in the graph) defining a maximum matching. Most models for robust assignment problems suggested in the literature capture only uncertainty in the costs, i.e., the task is to find an assignment minimizing the cost in a worst-case scenario. The contribution of this thesis is the introduction and investigation of the Robust Assignment Problem (RAP) which models edge and node failures while the costs are deterministic. A scenario is defined by a set of resources that may fail simultaneously. If a scenario emerges, the corresponding resources are deleted from the graph. RAP seeks to find a set of resources of minimal cost which is robust against all possible incidents, i.e., a set of resources containing an assignment for all scenarios. In production planning for example, lack of materials needed to complete an order can be encoded as an edge failure and production line maintenance corresponds to a node failure. The main findings of this thesis are hardness of approximation and NP-hardness results for both versions of RAP, even in case of single edge (or node) failures. These results are complemented by approximation algorithms matching the theoretical lower bounds asymptotically. Additionally, we study a new related problem concerning k-robust matchings. A perfect matching in a graph is $k$-robust if the graph remains perfectly matchable after the deletion of any k matching edges from the graph. We address the following question: How many edges have to be added to a graph to make a fixed perfect matching k-robust? We show that, in general, this problem is as hard as both aforementioned variants of RAP. From an application point of view, this result implies that robustification of an existent infrastructure is not easier than designing a new one from scratch.Diese Dissertation behandelt robuste Zuordnungsprobleme mit dem Schwerpunkt auf deren komlexitätstheoretischen Eigenschaften. Zuordnungsprobleme sind gut untersuchte kombinatorische Optimierungsprobleme mit vielen praktischen Anwendungen, z. B. in der Produktionsplanung. Klassische Ansätze der Optimierung gehen davon aus, dass die Inputdaten eines Problems exakt gegeben sind, wohingegen Optimierungsprobleme aus der Praxis mit Hilfe von Voraussagen modelliert werden. Daraus folgen unsichere Problemparameter, woran die Robuste Optimierung ansetzt. Die Unsicherheit wird mit Hilfe einer Szenarienmenge modelliert, die alle möglichen Ausprägungen der Problemparameter beschreibt. Eine Instanz des klassischen Zordnungsproblems wird mit Hilfe eines Graphen und einer Kostenfunktion beschrieben. Die Aufgabe besteht darin, eine Zuordnung mit minimalen Kosten zu finden. Eine Zuordnung ist eine Teilmenge an Ressourcen (Kanten oder Knoten des Graphen), die ein kardinalitätsmaximales Matching induziert. In der Literatur sind überwiegend robuste Zuordnungsprobleme untersucht, die Unsicherheit in den Kosten behandeln, in diesem Fall besteht die Aufgabe darin, eine Zuordnung mit minimalen Kosten im Worst-Case-Szenario zu finden. Diese Dissertation dient der Einführung und Untersuchung des Robust Assignment Problem (RAP) welches Kanten- und Knotenausfälle modelliert; wobei die Kosten determinisitsch sind. Ein Szenario ist durch jene Teilmenge an Ressourcen definiert, welche gleichzeitig ausfallen können. Wenn ein Szenario eintritt, werden die jeweils ausfallenden Ressourcen aus dem Graphen entfernt. In RAP besteht das Ziel darin, eine Menge an Ressourcen mit minimalen Kosten zu finden, die robust gegenüber allen möglichen Ereignissen ist, d. h. eine Ressourcenmenge die für alle Szenarien eine gültige Zuordnung enthält. So kann beispielsweise in der Produktionsplanung der Mangel an Materialien, die für einen Auftrag benötigt werden, als Kantenausfall und die wartungsbedingte Abschaltung einer Produktionslinie als Knotenausfall modelliert werden. Die Hauptergebnisse dieser Arbeit sind Nichtapproximierbarkeits- und NP-Schwierigkeitsresultate beider RAP-Versionen, die bereits für die Einschränkung zutreffen, dass nur einzelne Kanten oder Knoten ausfallen können. Diese Ergebnisse werden durch Approximationsalgorithmen ergänzt, die die theoretischen Approximationsschranken asymptotisch erreichen. Zusätzlich wird ein neues, verwandtes Optimierungsproblem untersucht, welches sich mit k-robusten Matchings beschäftigt. Ein perfektes Matching in einem Graphen ist k-robust, wenn der Graph nach dem Löschen von k Matchingkanten weiterhin ein perfektes Matching besitzt. Es wird der Frage nachgegangen, wie viele Kanten zum Graphen hinzugefügt werden müssen, um ein gegebenes Matching k-robust zu machen. Dabei wird gezeigt, dass dieses Problem im Allgemeinen aus komplexitätstheoretischer Sicht genauso schwierig ist, wie die zuvor erwähnten RAP-Varianten. Aus der Anwendungsperspektive bedeutet dieses Resultat, dass die Robustifikation einer bestehender Infrastruktur nicht einfacher ist, als sie von Grund auf neu zu entwerfen

The Convex Hull of Two Core Capacitated Network Design Problems

The network loading problem (NLP) is a specialized capacitated network design problem in which prescribed point-to-point demand between various pairs of nodes of a network must be met by installing (loading) a capacitated facility. We can load any number of units of the facility on each of the arcs at a specified arc dependent cost. The problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost. This paper studies two core subproblems of the NLP. The first problem, motivated by a Lagrangian relaxation approach for solving the problem, considers a multiple commodity, single arc capacitated network design problem. The second problem is a three node network; this specialized network arises in larger networks if we aggregate nodes. In both cases, we develop families of facets and completely characterize the convex hull of feasible solutions to the integer programming formulation of the problems. These results in turn strengthen the formulation of the NLP

Polyhedral techniques in combinatorial optimization II: computations

Combinatorial optimization problems appear in many disciplines ranging from management and logistics to mathematics, physics, and chemistry. These problems are usually relatively easy to formulate mathematically, but most of them are computationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable progress in techniques based on the polyhedral description of combinatorial problems. leading to a large increase in the size of several problem types that can be solved. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. Ideally we can then solve the problem as a linear programming problem, which can be done efficiently. The purpose of this manuscript is to give an overview of the developments in polyhedral theory, starting with the pioneering work by Dantzig, Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also present some modern applications, and computational experience