Search

A primal quadratic simplex algorithm tailored to the optimization over the vertices of a polytope is presented. Starting from a feasible vertex, it performs either strictly improving or admissible non-deteriorating steps in order to determine a locally optimum basic feasible solution in terms of the quadratic objective function. The algorithm so generalizes over local improvement methods for according applications, including in particular quadratic optimization problems whose feasible solutions correspond to vertices of a 0-1 polytope. Computational experiments for unconstrained binary quadratic programs, maximum cut, and the quadratic assignment problem serve as a proof of concept and underline the importance of a pivoting rule that is able to accept at least a restricted class of degenerate steps. [ABSTRACT FROM AUTHOR]

The exact solution of the NP-hard (nondeterministic polynomial-time hard) maximum cut problem is important in many applications across, for example, physics, chemistry, neuroscience, and circuit layout—which is also due to its equivalence to the unconstrained binary quadratic optimization problem. Leading solution methods are based on linear or semidefinite programming and require the separation of the so-called odd-cycle inequalities. In their groundbreaking research, F. Barahona and A. R. Mahjoub have given an informal description of a polynomial-time algorithm for this problem. As pointed out recently, however, additional effort is necessary to guarantee that the inequalities obtained correspond to facets of the cut polytope. In this paper, we shed more light on a so enhanced separation procedure and investigate experimentally how it performs in comparison with an ideal setting where one could even employ the sparsest, most violated, or geometrically most promising facet-defining odd-cycle inequalities. Summary of Contribution: This paper aims at a better capability to solve binary quadratic optimization or maximum cut problems and their various applications using integer programming techniques. To this end, the paper describes enhancements to a well-known algorithm for the central separation problem arising in this context; it is demonstrated experimentally that these enhancements are worthwhile from a computational point of view. The linear relaxations of the aforementioned problems are typically solved using fewer iterations and cutting planes than with a nonenhanced approach. It is also shown that the enhanced procedure is only slightly inferior to an ideal, enumerative, and, in practice, intractable global cutting-plane selection. [ABSTRACT FROM AUTHOR]

• Addressing the worst-case performance of labeling methods like the Coffman-Graham algorithm. • Focus on delayed precedence constraints in the single- and multi-processor case. • New lower bounds on worst-case approximation guarantees are derived. • The results extend to more general level-based or critical-path scheduling techniques. • An open problem by Bernstein, Rodeh, and Gertner from 1989 is resolved. There is some evidence that labeling schemes as employed for instance in the famous Coffman-Graham algorithm may provide superior worst-case approximation guarantees than purely path- or level-based list schedules in the presence of (delayed) precedence constraints. In 1989, Bernstein, Rodeh, and Gertner proved that this also holds true for their labeling scheme targeting unit execution time tasks on a single processor provided that all delays imposed by a single task on all of its successors are uniformly either zero or some fixed positive integer. They further conjectured that this superiority is preserved when allowing the delays imposed by a task to differ successor-wise. It is shown in this note however that their labeling scheme as well as more general ones may perform as bad as any list schedule in this case. Moreover, a new lower bound on the worst-case performance of labeling methods in the multiprocessor setting is derived. [ABSTRACT FROM AUTHOR]