Odd-Cycle Separation for Maximum Cut and Binary Quadratic Optimization

Solving the NP-hard Maximum Cut or Binary Quadratic Optimization Problem to optimality is important in many applications including Physics, Chemistry, Neuroscience, and Circuit Layout. The leading approaches based on linear/semidefinite programming require the separation of so-called odd-cycle inequalities for solving relaxations within their associated branch-and-cut frameworks. In their groundbreaking work, F. Barahona and A.R. Mahjoub have given an informal description of a polynomial-time separation procedure for the odd-cycle inequalities. Since then, the odd-cycle separation problem has broadly been considered solved. However, as we reveal, a straightforward implementation is likely to generate inequalities that are not facet-defining and have further undesired properties. Here, we present a more detailed analysis, along with enhancements to overcome the associated issues efficiently. In a corresponding experimental study, it turns out that these are worthwhile, and may speed up the solution process significantly

Improved Mixed-Integer Programming Models for Multiprocessor Scheduling with Communication Delays

We revise existing and introduce new mixed-integer programming models for the Multiprocessor Scheduling Problem with Communication Delays. At first, we show how to provably reduce the number of product variables necessary to explicitly linearize the so-called packing formulation that contains bilinear terms. Then, we reveal that the feasible region of almost all existing formulations contains redundant solutions and formulate new constraints in order to exclude these. At the same time, by exploiting further structural properties, the models are improved concerning their size, strength, and modeling complexity. The discussion of these improvements leads to new much more compact formulations which are then experimentally compared with each other and with other formulations from the literature. We set up a realistic scenario with a preprocessing of the task graphs, delivering the gained information equally to all the tested models and evaluate not only running times but also the obtained lower and upper bounds on the makespan objective for unsolved instances of a large scale benchmark set

On separation pairs and split components of biconnected graphs

The decomposition of a biconnected graph G into its triconnected components is fundamental in graph theory and has a wide range of applications. Based on a palm tree of G, the algorithm by Hopcroft and Tarjan is able to compute them in linear time if some corrections are applied. Today, the algorithm is still considered very hard to understand and proofs of its correctness are technical and challenging. The article at hand provides a more comprehensive description of the algorithm, making it easier to understand and implement. Its correctness is validated by explicitly mapping the algorithmic detection criteria to the graph-theoretic characterization of type-1 and type-2 separation pairs. Further, it reveals further errors and inaccuracies in the common definitions. This includes the description and proofs of further properties and relationships of separation pairs. The presented results also answer the question whether and under which preconditions type-1 and type-2 pairs can be computed separately from each other

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

Compact Hierarchical Graph Drawings via Quadratic Layer Assignment

We propose a new mixed-integer programming formulation that very naturally expresses the layout restrictions of a layered (hierarchical) graph drawing and several associated objectives, such as a minimum total arc length, number of reversed arcs, and width, or the adaptation to a specific drawing area, as a special quadratic assignment problem. Our experiments show that it is competitive to another formulation that we slightly simplify as well

A practical mixed-integer programming model for the vertex separation number problem

We present a novel mixed-integer programming formulation for the vertex separation number problem in general directed graphs. The model is conceptually simple and, to the best of our knowledge, much more compact than existing ones. First experiments give hope that it can solve larger instances than has been possible so far if it is combined with preprocessing techniques to reduce the search space

Solving the Simple Offset Assignment Problem as a Traveling Salesman

In this paper, we present an exact approach to the Simple Offset Assignment problem arising in the domain of address code generation for digital signal processors. It is based on transformations to weighted Hamiltonian cycle problems and integer linear programming. To the best of our knowledge, it is the first approach capable to solve all instances of the established OffsetStone benchmark set to optimality within reasonable time. Therefore, it enables to evaluate the quality of several heuristics relative to the optimum solutions for the first time. Further, using the same transformations, we present a simple and effective improvement heuristic. In addition, we include an existing heuristic into our experiments that has so far not been evaluated with OffsetStone

Improved Scalability By Using Hardware-Aware Thread Affinities

The complexity of an efficient thread management steadily rises with the number of processor cores and heterogeneities in the design of system architectures, e.g., the topologies of execution units and the memory architecture. In this paper, we show that using information about the system topology combined with a hardware-aware thread management is worthwhile. We present such a hardware-aware approach that utilizes thread affinity to automatically steer the mapping of threads to cores and experimentally analyze its performance. Our experiments show that we can achieve significantly better scalability and runtime stability compared to the ordinary dispatching of threads provided by the operating system

An Integer Programming Approach to Optimal Basic Block Instruction Scheduling for Single-Issue Processors

We present a novel integer programming formulation for basic block instruction scheduling on single-issue processors. The problem can be considered as a very general sequential task scheduling problem with delayed precedence-constraints. Our model is based on the linear ordering problem and has, in contrast to the last IP model proposed, numbers of variables and constraints that are strongly polynomial in the instance size. Combined with improved preprocessing techniques and given a time limit of ten minutes of CPU and system time, our branch-and-cut implementation is capable to solve all but eleven of the 369,861 basic blocks of the SPEC 2000 integer and floating point benchmarks to proven optimality. This is competitive to the current state-of-the art constraint programming approach that has also been evaluated on this test suite

Solving the Simple Offset Assignment Problem as a Traveling Salesman

In this paper, we present an exact approach to the Simple Offset Assignment problem arising in the domain of address code generation for digital signal processors. It is based on transformations to weighted Hamiltonian cycle problems and integer linear programming. To the best of our knowledge, it is the first approach capable to solve all instances of the established OffsetStone benchmark set to optimality within reasonable time. Therefore, it enables to evaluate the quality of several heuristics relative to the optimum solutions for the first time. Further, using the same transformations, we present a simple and effective improvement heuristic. In addition, we include an existing heuristic into our experiments that has so far not been evaluated with OffsetStone