Parameterized complexity of the MINCCA problem on graphs of bounded decomposability

In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The "Minimum Changeover Cost Arborescence" (MINCCA) problem consists in finding an arborescence with a given root vertex such that the total changeover cost of the internal vertices is minimized. It has been recently proved by G\"oz\"upek et al. [TCS 2016] that the problem is FPT when parameterized by the treewidth and the maximum degree of the input graph. In this article we present the following results for the MINCCA problem: - the problem is W[1]-hard parameterized by the treedepth of the input graph, even on graphs of average degree at most 8. In particular, it is W[1]-hard parameterized by the treewidth of the input graph, which answers the main open problem of G\"oz\"upek et al. [TCS 2016]; - it is W[1]-hard on multigraphs parameterized by the tree-cutwidth of the input multigraph; - it is FPT parameterized by the star tree-cutwidth of the input graph, which is a slightly restricted version of tree-cutwidth. This result strictly generalizes the FPT result given in G\"oz\"upek et al. [TCS 2016]; - it remains NP-hard on planar graphs even when restricted to instances with at most 6 colors and 0/1 symmetric costs, or when restricted to instances with at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs.Comment: 25 pages, 11 figure

SDP-based bounds for the Quadratic Cycle Cover Problem via cutting plane augmented Lagrangian methods and reinforcement learning

We study the Quadratic Cycle Cover Problem (QCCP), which aims to find a node-disjoint cycle cover in a directed graph with minimum interaction cost between successive arcs. We derive several semidefinite programming (SDP) relaxations and use facial reduction to make these strictly feasible. We investigate a nontrivial relationship between the transformation matrix used in the reduction and the structure of the graph, which is exploited in an efficient algorithm that constructs this matrix for any instance of the problem. To solve our relaxations, we propose an algorithm that incorporates an augmented Lagrangian method into a cutting plane framework by utilizing Dykstra's projection algorithm. Our algorithm is suitable for solving SDP relaxations with a large number of cutting planes. Computational results show that our SDP bounds and our efficient cutting plane algorithm outperform other QCCP bounding approaches from the literature. Finally, we provide several SDP-based upper bounding techniques, among which a sequential Q-learning method that exploits a solution of our SDP relaxation within a reinforcement learning environment

Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes

Semi-Preemptive Routing on Trees

We study a variant of the pickup-and-delivery problem (PDP) in which the objects that have to be transported can be reloaded at most d times, for a given integer d. This problem is known to be polynomially solvable on paths or cycles and NP-complete on trees. We present a (4/3+epsilon)-approximation algorithm if the underlying graph is a tree. By using a result of Charikar et al. (1998), this can be extended to a O(log n log log n)-approximation for general graphs