From few components to an Eulerian graph by adding arcs

Abstract Eulerian Extension (EE) is the problem to make an arcweighted directed multigraph Eulerian by adding arcs of minimum total cost. EE is NP-hard and has been shown fixed-parameter tractable with respect to the number of arc additions. Complementing this result, on the way to answering an open question, we show that EE is fixedparameter tractable with respect to the combined parameter “number of connected components in the underlying undirected multigraph ” and “sum of indeg(v) − outdeg(v) over all vertices v in the input multigraph where this value is positive. ” Moreover, we show that EE is unlikely to admit a polynomial-size problem kernel for this parameter combination and for the parameter “number of arc additions”.

Efficient Algorithms for Eulerian Extension

Eulerian extension problems aim at making a given (directed) (multi-)graph Eulerian by adding a minimum-cost set of edges (arcs). These problems have natural applications in scheduling and routing and are closely related to the Chinese Postman and Rural Postman problems. Our main result is to show that the NP-hard Weighted Multigraph Eulerian Extension is fixed-parameter tractable with respect to the number k of extension edges (arcs). For an n-vertex multigraph, the corresponding running time amounts to O(4 k · n 3). This implies a fixed-parameter tractability result for the “equivalent” Rural Postman problem. In addition, we present several polynomial-time algorithms for natural Eulerian extension problems

Zoledronic Acid in Reducing Clinical Fracture and Mortality after Hip Fracture.

BACKGROUND: Mortality is increased after a hip fracture, and strategies that improve outcomes are needed. METHODS: In this randomized, double-blind, placebo-controlled trial, 1065 patients were assigned to receive yearly intravenous zoledronic acid (at a dose of 5 mg), and 1062 patients were assigned to receive placebo. The infusions were first administered within 90 days after surgical repair of a hip fracture. All patients received supplemental vitamin D and calcium. The median follow-up was 1.9 years. The primary end point was a new clinical fracture. RESULTS: The rates of any new clinical fracture were 8.6% in the zoledronic acid group and 13.9% in the placebo group, a 35% risk reduction (P = 0.001); the respective rates of a new clinical vertebral fracture were 1.7% and 3.8% (P = 0.02), and the respective rates of new nonvertebral fractures were 7.6% and 10.7% (P = 0.03). In the safety analysis, 101 of 1054 patients in the zoledronic acid group (9.6%) and 141 of 1057 patients in the placebo group (13.3%) died, a reduction of 28% in deaths from any cause in the zoledronic-acid group (P = 0.01). The most frequent adverse events in patients receiving zoledronic acid were pyrexia, myalgia, and bone and musculoskeletal pain. No cases of osteonecrosis of the jaw were reported, and no adverse effects on the healing of fractures were noted. The rates of renal and cardiovascular adverse events, including atrial fibrillation and stroke, were similar in the two groups. CONCLUSIONS: An annual infusion of zoledronic acid within 90 days after repair of a low-trauma hip fracture was associated with a reduction in the rate of new clinical fractures and improved survival. (ClinicalTrials.gov number, NCT00046254.)