Joachim Schauer

The Knapsack Problem with Conflict Graphs

We extend the classical 0-1 knapsack problem by introducing disjunctive constraints for pairs of items which are not allowed to be packed together into the knapsack. These constraints are represented by edges of a conict graph whose vertices correspond to the items of the knapsack problem. Similar conditions were treated in the literature for bin packing and scheduling problems. For the knapsack problem with conict graphs, exact and heuristic algorithms were proposed in the past. While the problem is strongly NP-hard in general, we present pseudopolynomial algorithms for two special graph classes, namely graphs of bounded treewidth (including trees and series-parallel graphs) and chordal graphs. From these algorithms we can easily derive fully polynomial time approximation schemes (FPTAS).

Paths, trees and matchings under disjunctive constraints

Abstract We study the minimum spanning tree problem, the maximum matching problem and the shortest path problem subject to binary disjunctive constraints: A negative disjunctive constraint states that a certain pair of edges cannot be contained simultaneously in a feasible solution. It is convenient to represent these negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the edges of the underlying graph, and whose edges encode the constraints. We prove that the minimum spanning tree problem is strongly NP -hard, even if every connected component of the conflict graph is a path of length two. On the positive side, this problem is polynomially solvable if every connected component is a single edge (that is, a path of length one). The maximum matching problem is NP -hard for conflict graphs where every connected component is a single edge. Furthermore we will also investigate these graph problems under positive disjunctive constraints: In this setting for certain pairs of edges, a feasible solution must contain at least one edge from every pair. We establish a number of complexity results for these variants including APX-hardness for the shortest path problem.

The maximum flow problem with disjunctive constraints

We study the maximum flow problem subject to binary disjunctive constraints in a directed graph: A negative disjunctive constraint states that a certain pair of arcs in a digraph cannot be simultaneously used for sending flow in a feasible solution. In contrast to this, positive disjunctive constraints force that for certain pairs of arcs at least one arc has to carry flow in a feasible solution. It is convenient to represent the negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the arcs of the underlying graph, and whose edges encode the constraints. Analogously we represent the positive disjunctive constraints by a so-called forcing graph.For conflict graphs we prove that the maximum flow problem is strongly

The Subset Sum game

\mathcal{NP}

The Shortest Path Game: Complexity and Algorithms

-hard, even if the conflict graph consists only of unconnected edges. This result still holds if the network consists only of disjoint paths of length three. For forcing graphs we also provide a sharp line between polynomially solvable and strongly

Determining a Minimum Spanning Tree with Disjunctive Constraints

\mathcal{NP}

Approximating the Quadratic Knapsack Problem on Special Graph Classes

-hard instances for the case where the flow values are required to be integral. Moreover, our hardness results imply that no polynomial time approximation algorithm can exist for both problems. In contrast to this we show that the maximum flow problem with a forcing graph can be solved efficiently if fractional flow values are allowed.

ILP Models for a Nurse Scheduling Problem

Highlights • A game theoretic version of the Subset Sum problem is considered.• Two agents take turns to fill a shared knapsack with their items.• Natural heuristic strategies are proposed and analyzed from a worst-case perspective.

Approximation of knapsack problems with conflict and forcing graphs

In this work we address a game theoretic variant of the shortest path problem, in which two decision makers (agents/players) move together along the edges of a graph from a given starting vertex to a given destination. The two players take turns in deciding in each vertex which edge to traverse next. The decider in each vertex also has to pay the cost of the chosen edge. We want to determine the path where each player minimizes its costs taking into account that also the other player acts in a selfish and rational way. Such a solution is a subgame perfect equilibrium and can be determined by backward induction in the game tree of the associated finite game in extensive form.

On the Shortest Path Game

For the classical minimum spanning tree problem we introduce disjunctive constraints for pairs of edges which can not be both included in the spanning tree at the same time. These constraints are represented by a conflict graph whose vertices correspond to the edges of the original graph. Edges in the conflict graph connect conflicting edges of the original graph. It is shown that the problem becomes strongly