Exact Solution Methods for the kk-item Quadratic Knapsack Problem

The purpose of this paper is to solve the 0-1 $k$-item quadratic knapsack problem $(kQKP)$, a problem of maximizing a quadratic function subject to two linear constraints. We propose an exact method based on semidefinite optimization. The semidefinite relaxation used in our approach includes simple rank one constraints, which can be handled efficiently by interior point methods. Furthermore, we strengthen the relaxation by polyhedral constraints and obtain approximate solutions to this semidefinite problem by applying a bundle method. We review other exact solution methods and compare all these approaches by experimenting with instances of various sizes and densities.Comment: 12 page

Semidefinite programming and integer programming

We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems

Mutations of the ret protooncogene in German multiple endocrine neoplasia families: Relation between genotype and phenotype.

It has been suggested that not only the position but also the nature of the mutations of the ret protooncogene strongly correlate with the clinical manifestation of the multiple endocrine neoplasm type 2 (MEN 2) syndrome. In particular, individuals with a Cys634-Arg substitution should have a greater risk of developing parathyroid disease. We, therefore, analyzed 94 unrelated families from Germany with inherited medullary thyroid carcinoma (MTC) for mutation of the ret protooncogene. In all but 1 of 59 families with MEN 2A, germline mutations in the extracellular domain of the ret protein were found. Some 81% of the MEN 2A mutations affected codon 634. Phenotype-genotype correlations suggested that the prevalence of pheochromocytoma and hyperparathyroidism is significantly higher in families with codon 634 mutations, but there was no correlation with the nature of the mutation. In all but 1 of 27 familial MTC (FMTC) families, mutations were detected in 1 of 4 cysteines in the extracellular domain of the ret protooncogene. Half of the FMTC mutations affected codon 634. Mutations outside of codon 634 occurred more often in FMTC families than in MEN 2A families. In all but 1 of 8 MEN 2B patients, de novo mutations in codon 918 were found. These data confirm the preferential localization of MEN 2-associated mutations and the correlation between disease phenotype and the position of the ret mutation, but there was no correlation between the occurrence of hyperparathyroidism or pheochromocytoma and the nature of the mutation

Connections between semidefinite relaxations of the max-cut and stable set problems

We describe the links existing between a recently introduced semidefinite relaxation for the max-cut problem and the well known semidefinite relaxation for the stable set problem underlying the Lovász's theta function. It turns out that the connection between the convex bodies defining the semidefinite relaxations mimicks the connection existing between the corresponding polyhedra. We also show how the semidefinite relaxations can be combined with the classical linear relaxations in order to obtain tighter relaxations

Engineering Branch-and-Cut Algorithms for the Equicut Problem

A minimum equicut of an edge-weighted graph is a partition of the nodes of the graph into two sets of equal size such hat the sum of the weights of edges joining nodes in different partitions is minimum. We compare basic linear and semidefnite relaxations for the equicut problem, and and that linear bounds are competitive with the corresponding semidefnite ones but can be computed much faster. Motivated by an application of equicut in theoretical physics, we revisit an approach by Brunetta et al. and present an enhanced branch-and-cut algorithm. Our computational results suggest that the proposed branch-andcut algorithm has a better performance than the algorithm of Brunetta et al.. Further, it is able to solve to optimality in reasonable time several instances with more than 200 nodes from the physics application